# Properties

 Label 165.6.a.a Level $165$ Weight $6$ Character orbit 165.a Self dual yes Analytic conductor $26.463$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.4633302691$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + ( - 7 \beta_{2} - 77) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b1 - 2) * q^2 + 9 * q^3 + (b2 + 4*b1 + 7) * q^4 + 25 * q^5 + (-9*b1 - 18) * q^6 + (b2 + 11*b1 - 61) * q^7 + (-7*b2 - 77) * q^8 + 81 * q^9 $$q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + ( - 7 \beta_{2} - 77) q^{8} + 81 q^{9} + ( - 25 \beta_1 - 50) q^{10} - 121 q^{11} + (9 \beta_{2} + 36 \beta_1 + 63) q^{12} + ( - 28 \beta_{2} - 24 \beta_1 - 210) q^{13} + ( - 14 \beta_{2} + 22 \beta_1 - 250) q^{14} + 225 q^{15} + ( - 11 \beta_{2} + 68 \beta_1 - 161) q^{16} + (39 \beta_{2} + 37 \beta_1 - 801) q^{17} + ( - 81 \beta_1 - 162) q^{18} + ( - 29 \beta_{2} - 67 \beta_1 - 935) q^{19} + (25 \beta_{2} + 100 \beta_1 + 175) q^{20} + (9 \beta_{2} + 99 \beta_1 - 549) q^{21} + (121 \beta_1 + 242) q^{22} + ( - 20 \beta_{2} + 220 \beta_1 + 684) q^{23} + ( - 63 \beta_{2} - 693) q^{24} + 625 q^{25} + (108 \beta_{2} + 734 \beta_1 + 896) q^{26} + 729 q^{27} + ( - 12 \beta_{2} + 92 \beta_1 + 1500) q^{28} + (77 \beta_{2} + 107 \beta_1 - 2615) q^{29} + ( - 225 \beta_1 - 450) q^{30} + (38 \beta_{2} + 130 \beta_1 + 146) q^{31} + (189 \beta_{2} + 212 \beta_1 + 263) q^{32} - 1089 q^{33} + ( - 154 \beta_{2} + 64 \beta_1 + 814) q^{34} + (25 \beta_{2} + 275 \beta_1 - 1525) q^{35} + (81 \beta_{2} + 324 \beta_1 + 567) q^{36} + (272 \beta_{2} - 528 \beta_1 - 2866) q^{37} + (154 \beta_{2} + 1562 \beta_1 + 3838) q^{38} + ( - 252 \beta_{2} - 216 \beta_1 - 1890) q^{39} + ( - 175 \beta_{2} - 1925) q^{40} + ( - 473 \beta_{2} + 977 \beta_1 - 3245) q^{41} + ( - 126 \beta_{2} + 198 \beta_1 - 2250) q^{42} + (341 \beta_{2} - 809 \beta_1 - 4621) q^{43} + ( - 121 \beta_{2} - 484 \beta_1 - 847) q^{44} + 2025 q^{45} + ( - 160 \beta_{2} - 784 \beta_1 - 9328) q^{46} + ( - 422 \beta_{2} + 222 \beta_1 - 6538) q^{47} + ( - 99 \beta_{2} + 612 \beta_1 - 1449) q^{48} + (4 \beta_{2} - 964 \beta_1 - 8555) q^{49} + ( - 625 \beta_1 - 1250) q^{50} + (351 \beta_{2} + 333 \beta_1 - 7209) q^{51} + ( - 162 \beta_{2} - 3432 \beta_1 - 19358) q^{52} + (586 \beta_{2} - 962 \beta_1 - 1212) q^{53} + ( - 729 \beta_1 - 1458) q^{54} - 3025 q^{55} + (392 \beta_{2} - 2184 \beta_1 + 1624) q^{56} + ( - 261 \beta_{2} - 603 \beta_1 - 8415) q^{57} + ( - 338 \beta_{2} + 1092 \beta_1 + 2486) q^{58} + (356 \beta_{2} - 2748 \beta_1 - 2200) q^{59} + (225 \beta_{2} + 900 \beta_1 + 1575) q^{60} + (364 \beta_{2} - 3300 \beta_1 - 18926) q^{61} + ( - 244 \beta_{2} - 1052 \beta_1 - 4348) q^{62} + (81 \beta_{2} + 891 \beta_1 - 4941) q^{63} + ( - 427 \beta_{2} - 6076 \beta_1 - 337) q^{64} + ( - 700 \beta_{2} - 600 \beta_1 - 5250) q^{65} + (1089 \beta_1 + 2178) q^{66} + (680 \beta_{2} - 2144 \beta_1 - 12108) q^{67} + ( - 850 \beta_{2} + 492 \beta_1 + 19762) q^{68} + ( - 180 \beta_{2} + 1980 \beta_1 + 6156) q^{69} + ( - 350 \beta_{2} + 550 \beta_1 - 6250) q^{70} + (980 \beta_{2} + 2612 \beta_1 - 25548) q^{71} + ( - 567 \beta_{2} - 6237) q^{72} + (428 \beta_{2} + 2808 \beta_1 - 15750) q^{73} + ( - 288 \beta_{2} - 702 \beta_1 + 27748) q^{74} + 5625 q^{75} + ( - 1096 \beta_{2} - 7436 \beta_1 - 30424) q^{76} + ( - 121 \beta_{2} - 1331 \beta_1 + 7381) q^{77} + (972 \beta_{2} + 6606 \beta_1 + 8064) q^{78} + ( - 775 \beta_{2} - 5417 \beta_1 - 34233) q^{79} + ( - 275 \beta_{2} + 1700 \beta_1 - 4025) q^{80} + 6561 q^{81} + (442 \beta_{2} + 9332 \beta_1 - 33854) q^{82} + (1474 \beta_{2} + 14234 \beta_1 - 32042) q^{83} + ( - 108 \beta_{2} + 828 \beta_1 + 13500) q^{84} + (975 \beta_{2} + 925 \beta_1 - 20025) q^{85} + ( - 214 \beta_{2} + 442 \beta_1 + 41990) q^{86} + (693 \beta_{2} + 963 \beta_1 - 23535) q^{87} + (847 \beta_{2} + 9317) q^{88} + ( - 132 \beta_{2} - 2140 \beta_1 + 56494) q^{89} + ( - 2025 \beta_1 - 4050) q^{90} + (1378 \beta_{2} - 6602 \beta_1 - 8410) q^{91} + (1904 \beta_{2} + 6576 \beta_1 + 22128) q^{92} + (342 \beta_{2} + 1170 \beta_1 + 1314) q^{93} + (1044 \beta_{2} + 13268 \beta_1 - 180) q^{94} + ( - 725 \beta_{2} - 1675 \beta_1 - 23375) q^{95} + (1701 \beta_{2} + 1908 \beta_1 + 2367) q^{96} + ( - 1664 \beta_{2} - 8760 \beta_1 + 56154) q^{97} + (952 \beta_{2} + 10415 \beta_1 + 50902) q^{98} - 9801 q^{99}+O(q^{100})$$ q + (-b1 - 2) * q^2 + 9 * q^3 + (b2 + 4*b1 + 7) * q^4 + 25 * q^5 + (-9*b1 - 18) * q^6 + (b2 + 11*b1 - 61) * q^7 + (-7*b2 - 77) * q^8 + 81 * q^9 + (-25*b1 - 50) * q^10 - 121 * q^11 + (9*b2 + 36*b1 + 63) * q^12 + (-28*b2 - 24*b1 - 210) * q^13 + (-14*b2 + 22*b1 - 250) * q^14 + 225 * q^15 + (-11*b2 + 68*b1 - 161) * q^16 + (39*b2 + 37*b1 - 801) * q^17 + (-81*b1 - 162) * q^18 + (-29*b2 - 67*b1 - 935) * q^19 + (25*b2 + 100*b1 + 175) * q^20 + (9*b2 + 99*b1 - 549) * q^21 + (121*b1 + 242) * q^22 + (-20*b2 + 220*b1 + 684) * q^23 + (-63*b2 - 693) * q^24 + 625 * q^25 + (108*b2 + 734*b1 + 896) * q^26 + 729 * q^27 + (-12*b2 + 92*b1 + 1500) * q^28 + (77*b2 + 107*b1 - 2615) * q^29 + (-225*b1 - 450) * q^30 + (38*b2 + 130*b1 + 146) * q^31 + (189*b2 + 212*b1 + 263) * q^32 - 1089 * q^33 + (-154*b2 + 64*b1 + 814) * q^34 + (25*b2 + 275*b1 - 1525) * q^35 + (81*b2 + 324*b1 + 567) * q^36 + (272*b2 - 528*b1 - 2866) * q^37 + (154*b2 + 1562*b1 + 3838) * q^38 + (-252*b2 - 216*b1 - 1890) * q^39 + (-175*b2 - 1925) * q^40 + (-473*b2 + 977*b1 - 3245) * q^41 + (-126*b2 + 198*b1 - 2250) * q^42 + (341*b2 - 809*b1 - 4621) * q^43 + (-121*b2 - 484*b1 - 847) * q^44 + 2025 * q^45 + (-160*b2 - 784*b1 - 9328) * q^46 + (-422*b2 + 222*b1 - 6538) * q^47 + (-99*b2 + 612*b1 - 1449) * q^48 + (4*b2 - 964*b1 - 8555) * q^49 + (-625*b1 - 1250) * q^50 + (351*b2 + 333*b1 - 7209) * q^51 + (-162*b2 - 3432*b1 - 19358) * q^52 + (586*b2 - 962*b1 - 1212) * q^53 + (-729*b1 - 1458) * q^54 - 3025 * q^55 + (392*b2 - 2184*b1 + 1624) * q^56 + (-261*b2 - 603*b1 - 8415) * q^57 + (-338*b2 + 1092*b1 + 2486) * q^58 + (356*b2 - 2748*b1 - 2200) * q^59 + (225*b2 + 900*b1 + 1575) * q^60 + (364*b2 - 3300*b1 - 18926) * q^61 + (-244*b2 - 1052*b1 - 4348) * q^62 + (81*b2 + 891*b1 - 4941) * q^63 + (-427*b2 - 6076*b1 - 337) * q^64 + (-700*b2 - 600*b1 - 5250) * q^65 + (1089*b1 + 2178) * q^66 + (680*b2 - 2144*b1 - 12108) * q^67 + (-850*b2 + 492*b1 + 19762) * q^68 + (-180*b2 + 1980*b1 + 6156) * q^69 + (-350*b2 + 550*b1 - 6250) * q^70 + (980*b2 + 2612*b1 - 25548) * q^71 + (-567*b2 - 6237) * q^72 + (428*b2 + 2808*b1 - 15750) * q^73 + (-288*b2 - 702*b1 + 27748) * q^74 + 5625 * q^75 + (-1096*b2 - 7436*b1 - 30424) * q^76 + (-121*b2 - 1331*b1 + 7381) * q^77 + (972*b2 + 6606*b1 + 8064) * q^78 + (-775*b2 - 5417*b1 - 34233) * q^79 + (-275*b2 + 1700*b1 - 4025) * q^80 + 6561 * q^81 + (442*b2 + 9332*b1 - 33854) * q^82 + (1474*b2 + 14234*b1 - 32042) * q^83 + (-108*b2 + 828*b1 + 13500) * q^84 + (975*b2 + 925*b1 - 20025) * q^85 + (-214*b2 + 442*b1 + 41990) * q^86 + (693*b2 + 963*b1 - 23535) * q^87 + (847*b2 + 9317) * q^88 + (-132*b2 - 2140*b1 + 56494) * q^89 + (-2025*b1 - 4050) * q^90 + (1378*b2 - 6602*b1 - 8410) * q^91 + (1904*b2 + 6576*b1 + 22128) * q^92 + (342*b2 + 1170*b1 + 1314) * q^93 + (1044*b2 + 13268*b1 - 180) * q^94 + (-725*b2 - 1675*b1 - 23375) * q^95 + (1701*b2 + 1908*b1 + 2367) * q^96 + (-1664*b2 - 8760*b1 + 56154) * q^97 + (952*b2 + 10415*b1 + 50902) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 7 * q^2 + 27 * q^3 + 25 * q^4 + 75 * q^5 - 63 * q^6 - 172 * q^7 - 231 * q^8 + 243 * q^9 $$3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q - 7 * q^2 + 27 * q^3 + 25 * q^4 + 75 * q^5 - 63 * q^6 - 172 * q^7 - 231 * q^8 + 243 * q^9 - 175 * q^10 - 363 * q^11 + 225 * q^12 - 654 * q^13 - 728 * q^14 + 675 * q^15 - 415 * q^16 - 2366 * q^17 - 567 * q^18 - 2872 * q^19 + 625 * q^20 - 1548 * q^21 + 847 * q^22 + 2272 * q^23 - 2079 * q^24 + 1875 * q^25 + 3422 * q^26 + 2187 * q^27 + 4592 * q^28 - 7738 * q^29 - 1575 * q^30 + 568 * q^31 + 1001 * q^32 - 3267 * q^33 + 2506 * q^34 - 4300 * q^35 + 2025 * q^36 - 9126 * q^37 + 13076 * q^38 - 5886 * q^39 - 5775 * q^40 - 8758 * q^41 - 6552 * q^42 - 14672 * q^43 - 3025 * q^44 + 6075 * q^45 - 28768 * q^46 - 19392 * q^47 - 3735 * q^48 - 26629 * q^49 - 4375 * q^50 - 21294 * q^51 - 61506 * q^52 - 4598 * q^53 - 5103 * q^54 - 9075 * q^55 + 2688 * q^56 - 25848 * q^57 + 8550 * q^58 - 9348 * q^59 + 5625 * q^60 - 60078 * q^61 - 14096 * q^62 - 13932 * q^63 - 7087 * q^64 - 16350 * q^65 + 7623 * q^66 - 38468 * q^67 + 59778 * q^68 + 20448 * q^69 - 18200 * q^70 - 74032 * q^71 - 18711 * q^72 - 44442 * q^73 + 82542 * q^74 + 16875 * q^75 - 98708 * q^76 + 20812 * q^77 + 30798 * q^78 - 108116 * q^79 - 10375 * q^80 + 19683 * q^81 - 92230 * q^82 - 81892 * q^83 + 41328 * q^84 - 59150 * q^85 + 126412 * q^86 - 69642 * q^87 + 27951 * q^88 + 167342 * q^89 - 14175 * q^90 - 31832 * q^91 + 72960 * q^92 + 5112 * q^93 + 12728 * q^94 - 71800 * q^95 + 9009 * q^96 + 159702 * q^97 + 163121 * q^98 - 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 35$$ v^2 - 35
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 35$$ b2 + 35

## 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
 7.25531 0.921799 −7.17710
−9.25531 9.00000 53.6607 25.0000 −83.2977 36.4478 −200.476 81.0000 −231.383
1.2 −2.92180 9.00000 −23.4631 25.0000 −26.2962 −85.0105 162.052 81.0000 −73.0450
1.3 5.17710 9.00000 −5.19759 25.0000 46.5939 −123.437 −192.576 81.0000 129.428
 $$n$$: 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 165.6.a.a 3
3.b odd 2 1 495.6.a.e 3
5.b even 2 1 825.6.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.a 3 1.a even 1 1 trivial
495.6.a.e 3 3.b odd 2 1
825.6.a.j 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 7T_{2}^{2} - 36T_{2} - 140$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 7 T^{2} - 36 T - 140$$
$3$ $$(T - 9)^{3}$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} + 172 T^{2} + 2896 T - 382464$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} + 654 T^{2} + \cdots - 317918392$$
$17$ $$T^{3} + 2366 T^{2} + \cdots - 137826264$$
$19$ $$T^{3} + 2872 T^{2} + \cdots + 11543616$$
$23$ $$T^{3} - 2272 T^{2} + \cdots + 3706904576$$
$29$ $$T^{3} + 7738 T^{2} + \cdots + 5220625848$$
$31$ $$T^{3} - 568 T^{2} + \cdots - 289787904$$
$37$ $$T^{3} + 9126 T^{2} + \cdots - 129972509048$$
$41$ $$T^{3} + 8758 T^{2} + \cdots - 1122652557432$$
$43$ $$T^{3} + 14672 T^{2} + \cdots - 518908872384$$
$47$ $$T^{3} + 19392 T^{2} + \cdots - 1508908531200$$
$53$ $$T^{3} + 4598 T^{2} + \cdots + 728896505288$$
$59$ $$T^{3} + 9348 T^{2} + \cdots - 6267836310080$$
$61$ $$T^{3} + 60078 T^{2} + \cdots - 13500896397400$$
$67$ $$T^{3} + 38468 T^{2} + \cdots - 8479952260160$$
$71$ $$T^{3} + 74032 T^{2} + \cdots - 17011990639616$$
$73$ $$T^{3} + 44442 T^{2} + \cdots - 9750515676328$$
$79$ $$T^{3} + 108116 T^{2} + \cdots + 9069370346752$$
$83$ $$T^{3} + \cdots - 739830059345664$$
$89$ $$T^{3} + \cdots - 158914472576552$$
$97$ $$T^{3} + \cdots + 352998320493112$$