# Properties

 Label 165.4.a.e Level $165$ Weight $4$ Character orbit 165.a Self dual yes Analytic conductor $9.735$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.47528.1 Defining polynomial: $$x^{3} - x^{2} - 26 x - 22$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 10 + \beta_{2} ) q^{4} -5 q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 3 + 9 \beta_{1} - 2 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 10 + \beta_{2} ) q^{4} -5 q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 3 + 9 \beta_{1} - 2 \beta_{2} ) q^{8} + 9 q^{9} + ( 5 - 5 \beta_{1} ) q^{10} + 11 q^{11} + ( 30 + 3 \beta_{2} ) q^{12} + ( 38 - 2 \beta_{2} ) q^{13} + ( -28 + 16 \beta_{1} - 6 \beta_{2} ) q^{14} -15 q^{15} + ( 60 - 2 \beta_{1} + 5 \beta_{2} ) q^{16} + ( -34 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -9 + 9 \beta_{1} ) q^{18} + ( -24 + 14 \beta_{1} - 8 \beta_{2} ) q^{19} + ( -50 - 5 \beta_{2} ) q^{20} + ( 12 - 6 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -11 + 11 \beta_{1} ) q^{22} + ( 48 - 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 9 + 27 \beta_{1} - 6 \beta_{2} ) q^{24} + 25 q^{25} + ( -48 + 24 \beta_{1} + 4 \beta_{2} ) q^{26} + 27 q^{27} + ( 238 - 38 \beta_{1} + 12 \beta_{2} ) q^{28} + ( -62 - 34 \beta_{1} ) q^{29} + ( 15 - 15 \beta_{1} ) q^{30} + ( 96 - 40 \beta_{1} - 8 \beta_{2} ) q^{31} + ( -93 + 21 \beta_{1} + 4 \beta_{2} ) q^{32} + 33 q^{33} + ( -10 - 50 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -20 + 10 \beta_{1} - 10 \beta_{2} ) q^{35} + ( 90 + 9 \beta_{2} ) q^{36} + ( 286 - 20 \beta_{1} ) q^{37} + ( 222 - 66 \beta_{1} + 30 \beta_{2} ) q^{38} + ( 114 - 6 \beta_{2} ) q^{39} + ( -15 - 45 \beta_{1} + 10 \beta_{2} ) q^{40} + ( 30 + 66 \beta_{1} ) q^{41} + ( -84 + 48 \beta_{1} - 18 \beta_{2} ) q^{42} + ( 48 - 22 \beta_{1} - 14 \beta_{2} ) q^{43} + ( 110 + 11 \beta_{2} ) q^{44} -45 q^{45} + ( -436 + 52 \beta_{1} - 32 \beta_{2} ) q^{46} + ( 168 - 4 \beta_{2} ) q^{47} + ( 180 - 6 \beta_{1} + 15 \beta_{2} ) q^{48} + ( 117 - 72 \beta_{1} ) q^{49} + ( -25 + 25 \beta_{1} ) q^{50} + ( -102 - 6 \beta_{1} - 6 \beta_{2} ) q^{51} + ( 172 + 4 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 126 - 96 \beta_{1} + 16 \beta_{2} ) q^{53} + ( -27 + 27 \beta_{1} ) q^{54} -55 q^{55} + ( -600 + 156 \beta_{1} - 14 \beta_{2} ) q^{56} + ( -72 + 42 \beta_{1} - 24 \beta_{2} ) q^{57} + ( -516 - 96 \beta_{1} - 34 \beta_{2} ) q^{58} + ( 172 + 32 \beta_{1} + 8 \beta_{2} ) q^{59} + ( -150 - 15 \beta_{2} ) q^{60} + ( 134 + 12 \beta_{1} + 68 \beta_{2} ) q^{61} + ( -816 - 24 \beta_{2} ) q^{62} + ( 36 - 18 \beta_{1} + 18 \beta_{2} ) q^{63} + ( -10 - 28 \beta_{1} - 27 \beta_{2} ) q^{64} + ( -190 + 10 \beta_{2} ) q^{65} + ( -33 + 33 \beta_{1} ) q^{66} + ( -176 + 100 \beta_{1} - 16 \beta_{2} ) q^{67} + ( -558 - 30 \beta_{1} - 38 \beta_{2} ) q^{68} + ( 144 - 72 \beta_{1} + 12 \beta_{2} ) q^{69} + ( 140 - 80 \beta_{1} + 30 \beta_{2} ) q^{70} + ( -324 + 60 \beta_{1} - 28 \beta_{2} ) q^{71} + ( 27 + 81 \beta_{1} - 18 \beta_{2} ) q^{72} + ( 194 + 36 \beta_{1} - 38 \beta_{2} ) q^{73} + ( -626 + 266 \beta_{1} - 20 \beta_{2} ) q^{74} + 75 q^{75} + ( -1002 + 254 \beta_{1} - 62 \beta_{2} ) q^{76} + ( 44 - 22 \beta_{1} + 22 \beta_{2} ) q^{77} + ( -144 + 72 \beta_{1} + 12 \beta_{2} ) q^{78} + ( -212 + 94 \beta_{1} + 8 \beta_{2} ) q^{79} + ( -300 + 10 \beta_{1} - 25 \beta_{2} ) q^{80} + 81 q^{81} + ( 1092 + 96 \beta_{1} + 66 \beta_{2} ) q^{82} + ( 60 - 180 \beta_{1} + 54 \beta_{2} ) q^{83} + ( 714 - 114 \beta_{1} + 36 \beta_{2} ) q^{84} + ( 170 + 10 \beta_{1} + 10 \beta_{2} ) q^{85} + ( -492 - 72 \beta_{1} + 6 \beta_{2} ) q^{86} + ( -186 - 102 \beta_{1} ) q^{87} + ( 33 + 99 \beta_{1} - 22 \beta_{2} ) q^{88} + ( 254 + 28 \beta_{1} + 120 \beta_{2} ) q^{89} + ( 45 - 45 \beta_{1} ) q^{90} + ( -244 - 40 \beta_{1} + 92 \beta_{2} ) q^{91} + ( 776 - 416 \beta_{1} + 84 \beta_{2} ) q^{92} + ( 288 - 120 \beta_{1} - 24 \beta_{2} ) q^{93} + ( -188 + 140 \beta_{1} + 8 \beta_{2} ) q^{94} + ( 120 - 70 \beta_{1} + 40 \beta_{2} ) q^{95} + ( -279 + 63 \beta_{1} + 12 \beta_{2} ) q^{96} + ( 658 + 100 \beta_{1} - 44 \beta_{2} ) q^{97} + ( -1341 + 45 \beta_{1} - 72 \beta_{2} ) q^{98} + 99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{2} + 9q^{3} + 30q^{4} - 15q^{5} - 6q^{6} + 10q^{7} + 18q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 2q^{2} + 9q^{3} + 30q^{4} - 15q^{5} - 6q^{6} + 10q^{7} + 18q^{8} + 27q^{9} + 10q^{10} + 33q^{11} + 90q^{12} + 114q^{13} - 68q^{14} - 45q^{15} + 178q^{16} - 104q^{17} - 18q^{18} - 58q^{19} - 150q^{20} + 30q^{21} - 22q^{22} + 120q^{23} + 54q^{24} + 75q^{25} - 120q^{26} + 81q^{27} + 676q^{28} - 220q^{29} + 30q^{30} + 248q^{31} - 258q^{32} + 99q^{33} - 80q^{34} - 50q^{35} + 270q^{36} + 838q^{37} + 600q^{38} + 342q^{39} - 90q^{40} + 156q^{41} - 204q^{42} + 122q^{43} + 330q^{44} - 135q^{45} - 1256q^{46} + 504q^{47} + 534q^{48} + 279q^{49} - 50q^{50} - 312q^{51} + 520q^{52} + 282q^{53} - 54q^{54} - 165q^{55} - 1644q^{56} - 174q^{57} - 1644q^{58} + 548q^{59} - 450q^{60} + 414q^{61} - 2448q^{62} + 90q^{63} - 58q^{64} - 570q^{65} - 66q^{66} - 428q^{67} - 1704q^{68} + 360q^{69} + 340q^{70} - 912q^{71} + 162q^{72} + 618q^{73} - 1612q^{74} + 225q^{75} - 2752q^{76} + 110q^{77} - 360q^{78} - 542q^{79} - 890q^{80} + 243q^{81} + 3372q^{82} + 2028q^{84} + 520q^{85} - 1548q^{86} - 660q^{87} + 198q^{88} + 790q^{89} + 90q^{90} - 772q^{91} + 1912q^{92} + 744q^{93} - 424q^{94} + 290q^{95} - 774q^{96} + 2074q^{97} - 3978q^{98} + 297q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 26 x - 22$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 17$$

## 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
 −4.06484 −0.906392 5.97123
−5.06484 3.00000 17.6526 −5.00000 −15.1945 27.4348 −48.8887 9.00000 25.3242
1.2 −1.90639 3.00000 −4.36567 −5.00000 −5.71918 −22.9186 23.5738 9.00000 9.53196
1.3 4.97123 3.00000 16.7131 −5.00000 14.9137 5.48376 43.3148 9.00000 −24.8561
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 3
3.b odd 2 1 495.4.a.k 3
5.b even 2 1 825.4.a.r 3
5.c odd 4 2 825.4.c.k 6
11.b odd 2 1 1815.4.a.r 3
15.d odd 2 1 2475.4.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.e 3 1.a even 1 1 trivial
495.4.a.k 3 3.b odd 2 1
825.4.a.r 3 5.b even 2 1
825.4.c.k 6 5.c odd 4 2
1815.4.a.r 3 11.b odd 2 1
2475.4.a.t 3 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2 T_{2}^{2} - 25 T_{2} - 48$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-48 - 25 T + 2 T^{2} + T^{3}$$
$3$ $$( -3 + T )^{3}$$
$5$ $$( 5 + T )^{3}$$
$7$ $$3448 - 604 T - 10 T^{2} + T^{3}$$
$11$ $$( -11 + T )^{3}$$
$13$ $$-37216 + 3712 T - 114 T^{2} + T^{3}$$
$17$ $$8448 + 2792 T + 104 T^{2} + T^{3}$$
$19$ $$65520 - 11496 T + 58 T^{2} + T^{3}$$
$23$ $$148224 - 10736 T - 120 T^{2} + T^{3}$$
$29$ $$-629760 - 14308 T + 220 T^{2} + T^{3}$$
$31$ $$9589248 - 38592 T - 248 T^{2} + T^{3}$$
$37$ $$-18607336 + 223548 T - 838 T^{2} + T^{3}$$
$41$ $$-3013632 - 106596 T - 156 T^{2} + T^{3}$$
$43$ $$1445400 - 44940 T - 122 T^{2} + T^{3}$$
$47$ $$-4372224 + 82192 T - 504 T^{2} + T^{3}$$
$53$ $$-3654264 - 222068 T - 282 T^{2} + T^{3}$$
$59$ $$-1206720 + 57584 T - 548 T^{2} + T^{3}$$
$61$ $$342344792 - 681332 T - 414 T^{2} + T^{3}$$
$67$ $$-8135552 - 206752 T + 428 T^{2} + T^{3}$$
$71$ $$-2867712 + 97888 T + 912 T^{2} + T^{3}$$
$73$ $$26458592 - 100544 T - 618 T^{2} + T^{3}$$
$79$ $$-88503440 - 161224 T + 542 T^{2} + T^{3}$$
$83$ $$-434328048 - 1091340 T + T^{3}$$
$89$ $$1941629400 - 2118532 T - 790 T^{2} + T^{3}$$
$97$ $$98075336 + 967212 T - 2074 T^{2} + T^{3}$$
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