# Properties

 Label 165.6.a.b 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,6,Mod(1,165)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$26.4633302691$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.3368.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 15x + 11$$ x^3 - x^2 - 15*x + 11 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (4 \beta_{2} + 8) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 9) q^{6} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (4*b2 + 8) * q^4 - 25 * q^5 + (-9*b1 - 9) * q^6 + (-11*b2 + 14*b1 - 69) * q^7 + (-8*b2 + 4*b1 + 12) * q^8 + 81 * q^9 $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (4 \beta_{2} + 8) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 9) q^{6} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{8} + 81 q^{9} + (25 \beta_1 + 25) q^{10} + 121 q^{11} + (36 \beta_{2} + 72) q^{12} + ( - 31 \beta_{2} - 25 \beta_1 + 152) q^{13} + ( - 34 \beta_{2} + 138 \beta_1 - 444) q^{14} - 225 q^{15} + ( - 128 \beta_{2} + 32 \beta_1 - 400) q^{16} + ( - 57 \beta_{2} + 34 \beta_1 - 81) q^{17} + ( - 81 \beta_1 - 81) q^{18} + (136 \beta_{2} + 119 \beta_1 - 1351) q^{19} + ( - 100 \beta_{2} - 200) q^{20} + ( - 99 \beta_{2} + 126 \beta_1 - 621) q^{21} + ( - 121 \beta_1 - 121) q^{22} + (164 \beta_{2} + 372 \beta_1 - 2284) q^{23} + ( - 72 \beta_{2} + 36 \beta_1 + 108) q^{24} + 625 q^{25} + (162 \beta_{2} - 22 \beta_1 + 916) q^{26} + 729 q^{27} + ( - 132 \beta_{2} + 304 \beta_1 - 2628) q^{28} + (580 \beta_{2} + 93 \beta_1 + 1185) q^{29} + (225 \beta_1 + 225) q^{30} + (764 \beta_{2} - 920 \beta_1 - 764) q^{31} + (384 \beta_{2} + 944 \beta_1 - 848) q^{32} + 1089 q^{33} + ( - 22 \beta_{2} + 400 \beta_1 - 1074) q^{34} + (275 \beta_{2} - 350 \beta_1 + 1725) q^{35} + (324 \beta_{2} + 648) q^{36} + ( - 1312 \beta_{2} + 410 \beta_1 + 1324) q^{37} + ( - 748 \beta_{2} + 790 \beta_1 - 3698) q^{38} + ( - 279 \beta_{2} - 225 \beta_1 + 1368) q^{39} + (200 \beta_{2} - 100 \beta_1 - 300) q^{40} + (140 \beta_{2} - 521 \beta_1 + 3331) q^{41} + ( - 306 \beta_{2} + 1242 \beta_1 - 3996) q^{42} + ( - 37 \beta_{2} - 2 \beta_1 - 11831) q^{43} + (484 \beta_{2} + 968) q^{44} - 2025 q^{45} + ( - 1816 \beta_{2} + 1836 \beta_1 - 12716) q^{46} + (1004 \beta_{2} - 640 \beta_1 - 1248) q^{47} + ( - 1152 \beta_{2} + 288 \beta_1 - 3600) q^{48} + (1510 \beta_{2} - 3622 \beta_1 + 845) q^{49} + ( - 625 \beta_1 - 625) q^{50} + ( - 513 \beta_{2} + 306 \beta_1 - 729) q^{51} + (756 \beta_{2} - 948 \beta_1 - 5408) q^{52} + ( - 2746 \beta_{2} - 2226 \beta_1 - 4102) q^{53} + ( - 729 \beta_1 - 729) q^{54} - 3025 q^{55} + (136 \beta_{2} - 824 \beta_1 + 5376) q^{56} + (1224 \beta_{2} + 1071 \beta_1 - 12159) q^{57} + ( - 1532 \beta_{2} - 3992 \beta_1 - 6552) q^{58} + (844 \beta_{2} + 3292 \beta_1 - 26476) q^{59} + ( - 900 \beta_{2} - 1800) q^{60} + (916 \beta_{2} - 3326 \beta_1 - 22180) q^{61} + (2152 \beta_{2} - 3976 \beta_1 + 34352) q^{62} + ( - 891 \beta_{2} + 1134 \beta_1 - 5589) q^{63} + ( - 448 \beta_{2} - 1152 \beta_1 - 24320) q^{64} + (775 \beta_{2} + 625 \beta_1 - 3800) q^{65} + ( - 1089 \beta_1 - 1089) q^{66} + ( - 4474 \beta_{2} + 496 \beta_1 - 13726) q^{67} + (268 \beta_{2} + 496 \beta_1 - 11868) q^{68} + (1476 \beta_{2} + 3348 \beta_1 - 20556) q^{69} + (850 \beta_{2} - 3450 \beta_1 + 11100) q^{70} + ( - 26 \beta_{2} + 1080 \beta_1 - 21206) q^{71} + ( - 648 \beta_{2} + 324 \beta_1 + 972) q^{72} + (3773 \beta_{2} - 8443 \beta_1 - 3802) q^{73} + (984 \beta_{2} + 5646 \beta_1 - 13378) q^{74} + 5625 q^{75} + ( - 6016 \beta_{2} + 4420 \beta_1 + 18364) q^{76} + ( - 1331 \beta_{2} + 1694 \beta_1 - 8349) q^{77} + (1458 \beta_{2} - 198 \beta_1 + 8244) q^{78} + (10742 \beta_{2} - 5831 \beta_1 + 9101) q^{79} + (3200 \beta_{2} - 800 \beta_1 + 10000) q^{80} + 6561 q^{81} + (1804 \beta_{2} - 4552 \beta_1 + 16568) q^{82} + ( - 2095 \beta_{2} - 4559 \beta_1 - 34496) q^{83} + ( - 1188 \beta_{2} + 2736 \beta_1 - 23652) q^{84} + (1425 \beta_{2} - 850 \beta_1 + 2025) q^{85} + (82 \beta_{2} + 12014 \beta_1 + 12020) q^{86} + (5220 \beta_{2} + 837 \beta_1 + 10665) q^{87} + ( - 968 \beta_{2} + 484 \beta_1 + 1452) q^{88} + ( - 12390 \beta_{2} + 2204 \beta_1 + 24872) q^{89} + (2025 \beta_1 + 2025) q^{90} + ( - 2456 \beta_{2} + 4440 \beta_1 - 7224) q^{91} + ( - 8960 \beta_{2} + 11728 \beta_1 + 19648) q^{92} + (6876 \beta_{2} - 8280 \beta_1 - 6876) q^{93} + (552 \beta_{2} - 4412 \beta_1 + 23196) q^{94} + ( - 3400 \beta_{2} - 2975 \beta_1 + 33775) q^{95} + (3456 \beta_{2} + 8496 \beta_1 - 7632) q^{96} + (3142 \beta_{2} + 10816 \beta_1 - 104024) q^{97} + (11468 \beta_{2} - 12017 \beta_1 + 135883) q^{98} + 9801 q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (4*b2 + 8) * q^4 - 25 * q^5 + (-9*b1 - 9) * q^6 + (-11*b2 + 14*b1 - 69) * q^7 + (-8*b2 + 4*b1 + 12) * q^8 + 81 * q^9 + (25*b1 + 25) * q^10 + 121 * q^11 + (36*b2 + 72) * q^12 + (-31*b2 - 25*b1 + 152) * q^13 + (-34*b2 + 138*b1 - 444) * q^14 - 225 * q^15 + (-128*b2 + 32*b1 - 400) * q^16 + (-57*b2 + 34*b1 - 81) * q^17 + (-81*b1 - 81) * q^18 + (136*b2 + 119*b1 - 1351) * q^19 + (-100*b2 - 200) * q^20 + (-99*b2 + 126*b1 - 621) * q^21 + (-121*b1 - 121) * q^22 + (164*b2 + 372*b1 - 2284) * q^23 + (-72*b2 + 36*b1 + 108) * q^24 + 625 * q^25 + (162*b2 - 22*b1 + 916) * q^26 + 729 * q^27 + (-132*b2 + 304*b1 - 2628) * q^28 + (580*b2 + 93*b1 + 1185) * q^29 + (225*b1 + 225) * q^30 + (764*b2 - 920*b1 - 764) * q^31 + (384*b2 + 944*b1 - 848) * q^32 + 1089 * q^33 + (-22*b2 + 400*b1 - 1074) * q^34 + (275*b2 - 350*b1 + 1725) * q^35 + (324*b2 + 648) * q^36 + (-1312*b2 + 410*b1 + 1324) * q^37 + (-748*b2 + 790*b1 - 3698) * q^38 + (-279*b2 - 225*b1 + 1368) * q^39 + (200*b2 - 100*b1 - 300) * q^40 + (140*b2 - 521*b1 + 3331) * q^41 + (-306*b2 + 1242*b1 - 3996) * q^42 + (-37*b2 - 2*b1 - 11831) * q^43 + (484*b2 + 968) * q^44 - 2025 * q^45 + (-1816*b2 + 1836*b1 - 12716) * q^46 + (1004*b2 - 640*b1 - 1248) * q^47 + (-1152*b2 + 288*b1 - 3600) * q^48 + (1510*b2 - 3622*b1 + 845) * q^49 + (-625*b1 - 625) * q^50 + (-513*b2 + 306*b1 - 729) * q^51 + (756*b2 - 948*b1 - 5408) * q^52 + (-2746*b2 - 2226*b1 - 4102) * q^53 + (-729*b1 - 729) * q^54 - 3025 * q^55 + (136*b2 - 824*b1 + 5376) * q^56 + (1224*b2 + 1071*b1 - 12159) * q^57 + (-1532*b2 - 3992*b1 - 6552) * q^58 + (844*b2 + 3292*b1 - 26476) * q^59 + (-900*b2 - 1800) * q^60 + (916*b2 - 3326*b1 - 22180) * q^61 + (2152*b2 - 3976*b1 + 34352) * q^62 + (-891*b2 + 1134*b1 - 5589) * q^63 + (-448*b2 - 1152*b1 - 24320) * q^64 + (775*b2 + 625*b1 - 3800) * q^65 + (-1089*b1 - 1089) * q^66 + (-4474*b2 + 496*b1 - 13726) * q^67 + (268*b2 + 496*b1 - 11868) * q^68 + (1476*b2 + 3348*b1 - 20556) * q^69 + (850*b2 - 3450*b1 + 11100) * q^70 + (-26*b2 + 1080*b1 - 21206) * q^71 + (-648*b2 + 324*b1 + 972) * q^72 + (3773*b2 - 8443*b1 - 3802) * q^73 + (984*b2 + 5646*b1 - 13378) * q^74 + 5625 * q^75 + (-6016*b2 + 4420*b1 + 18364) * q^76 + (-1331*b2 + 1694*b1 - 8349) * q^77 + (1458*b2 - 198*b1 + 8244) * q^78 + (10742*b2 - 5831*b1 + 9101) * q^79 + (3200*b2 - 800*b1 + 10000) * q^80 + 6561 * q^81 + (1804*b2 - 4552*b1 + 16568) * q^82 + (-2095*b2 - 4559*b1 - 34496) * q^83 + (-1188*b2 + 2736*b1 - 23652) * q^84 + (1425*b2 - 850*b1 + 2025) * q^85 + (82*b2 + 12014*b1 + 12020) * q^86 + (5220*b2 + 837*b1 + 10665) * q^87 + (-968*b2 + 484*b1 + 1452) * q^88 + (-12390*b2 + 2204*b1 + 24872) * q^89 + (2025*b1 + 2025) * q^90 + (-2456*b2 + 4440*b1 - 7224) * q^91 + (-8960*b2 + 11728*b1 + 19648) * q^92 + (6876*b2 - 8280*b1 - 6876) * q^93 + (552*b2 - 4412*b1 + 23196) * q^94 + (-3400*b2 - 2975*b1 + 33775) * q^95 + (3456*b2 + 8496*b1 - 7632) * q^96 + (3142*b2 + 10816*b1 - 104024) * q^97 + (11468*b2 - 12017*b1 + 135883) * q^98 + 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 27 * q^3 + 28 * q^4 - 75 * q^5 - 18 * q^6 - 232 * q^7 + 24 * q^8 + 243 * q^9 $$3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 252 q^{12} + 450 q^{13} - 1504 q^{14} - 675 q^{15} - 1360 q^{16} - 334 q^{17} - 162 q^{18} - 4036 q^{19} - 700 q^{20} - 2088 q^{21} - 242 q^{22} - 7060 q^{23} + 216 q^{24} + 1875 q^{25} + 2932 q^{26} + 2187 q^{27} - 8320 q^{28} + 4042 q^{29} + 450 q^{30} - 608 q^{31} - 3104 q^{32} + 3267 q^{33} - 3644 q^{34} + 5800 q^{35} + 2268 q^{36} + 2250 q^{37} - 12632 q^{38} + 4050 q^{39} - 600 q^{40} + 10654 q^{41} - 13536 q^{42} - 35528 q^{43} + 3388 q^{44} - 6075 q^{45} - 41800 q^{46} - 2100 q^{47} - 12240 q^{48} + 7667 q^{49} - 1250 q^{50} - 3006 q^{51} - 14520 q^{52} - 12826 q^{53} - 1458 q^{54} - 9075 q^{55} + 17088 q^{56} - 36324 q^{57} - 17196 q^{58} - 81876 q^{59} - 6300 q^{60} - 62298 q^{61} + 109184 q^{62} - 18792 q^{63} - 72256 q^{64} - 11250 q^{65} - 2178 q^{66} - 46148 q^{67} - 35832 q^{68} - 63540 q^{69} + 37600 q^{70} - 64724 q^{71} + 1944 q^{72} + 810 q^{73} - 44796 q^{74} + 16875 q^{75} + 44656 q^{76} - 28072 q^{77} + 26388 q^{78} + 43876 q^{79} + 34000 q^{80} + 19683 q^{81} + 56060 q^{82} - 101024 q^{83} - 74880 q^{84} + 8350 q^{85} + 24128 q^{86} + 36378 q^{87} + 2904 q^{88} + 60022 q^{89} + 4050 q^{90} - 28568 q^{91} + 38256 q^{92} - 5472 q^{93} + 74552 q^{94} + 100900 q^{95} - 27936 q^{96} - 319746 q^{97} + 431134 q^{98} + 29403 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 27 * q^3 + 28 * q^4 - 75 * q^5 - 18 * q^6 - 232 * q^7 + 24 * q^8 + 243 * q^9 + 50 * q^10 + 363 * q^11 + 252 * q^12 + 450 * q^13 - 1504 * q^14 - 675 * q^15 - 1360 * q^16 - 334 * q^17 - 162 * q^18 - 4036 * q^19 - 700 * q^20 - 2088 * q^21 - 242 * q^22 - 7060 * q^23 + 216 * q^24 + 1875 * q^25 + 2932 * q^26 + 2187 * q^27 - 8320 * q^28 + 4042 * q^29 + 450 * q^30 - 608 * q^31 - 3104 * q^32 + 3267 * q^33 - 3644 * q^34 + 5800 * q^35 + 2268 * q^36 + 2250 * q^37 - 12632 * q^38 + 4050 * q^39 - 600 * q^40 + 10654 * q^41 - 13536 * q^42 - 35528 * q^43 + 3388 * q^44 - 6075 * q^45 - 41800 * q^46 - 2100 * q^47 - 12240 * q^48 + 7667 * q^49 - 1250 * q^50 - 3006 * q^51 - 14520 * q^52 - 12826 * q^53 - 1458 * q^54 - 9075 * q^55 + 17088 * q^56 - 36324 * q^57 - 17196 * q^58 - 81876 * q^59 - 6300 * q^60 - 62298 * q^61 + 109184 * q^62 - 18792 * q^63 - 72256 * q^64 - 11250 * q^65 - 2178 * q^66 - 46148 * q^67 - 35832 * q^68 - 63540 * q^69 + 37600 * q^70 - 64724 * q^71 + 1944 * q^72 + 810 * q^73 - 44796 * q^74 + 16875 * q^75 + 44656 * q^76 - 28072 * q^77 + 26388 * q^78 + 43876 * q^79 + 34000 * q^80 + 19683 * q^81 + 56060 * q^82 - 101024 * q^83 - 74880 * q^84 + 8350 * q^85 + 24128 * q^86 + 36378 * q^87 + 2904 * q^88 + 60022 * q^89 + 4050 * q^90 - 28568 * q^91 + 38256 * q^92 - 5472 * q^93 + 74552 * q^94 + 100900 * q^95 - 27936 * q^96 - 319746 * q^97 + 431134 * q^98 + 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15x + 11$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 10$$ v^2 - 10
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 10$$ b2 + 10

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.03932 0.723686 −3.76300
−8.07863 9.00000 33.2643 −25.0000 −72.7077 −39.3760 −10.2141 81.0000 201.966
1.2 −1.44737 9.00000 −29.9051 −25.0000 −13.0263 41.5023 89.5997 81.0000 36.1843
1.3 7.52601 9.00000 24.6408 −25.0000 67.7341 −234.126 −55.3856 81.0000 −188.150
 $$n$$: 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.b 3
3.b odd 2 1 495.6.a.d 3
5.b even 2 1 825.6.a.i 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - 60T_{2} - 88$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} + \cdots - 88$$
$3$ $$(T - 9)^{3}$$
$5$ $$(T + 25)^{3}$$
$7$ $$T^{3} + 232 T^{2} + \cdots - 382608$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} - 450 T^{2} + \cdots + 22659488$$
$17$ $$T^{3} + 334 T^{2} + \cdots - 57782448$$
$19$ $$T^{3} + \cdots - 1630951200$$
$23$ $$T^{3} + \cdots - 24275701568$$
$29$ $$T^{3} + \cdots + 65949214584$$
$31$ $$T^{3} + \cdots - 211578448896$$
$37$ $$T^{3} + \cdots - 431879868536$$
$41$ $$T^{3} + \cdots - 7803557208$$
$43$ $$T^{3} + \cdots + 1659712050000$$
$47$ $$T^{3} + \cdots + 52162385088$$
$53$ $$T^{3} + \cdots + 2687939232856$$
$59$ $$T^{3} + \cdots - 3633753791296$$
$61$ $$T^{3} + \cdots - 12904038746056$$
$67$ $$T^{3} + \cdots - 40648408406912$$
$71$ $$T^{3} + \cdots + 8578136735360$$
$73$ $$T^{3} + \cdots - 144432126809632$$
$79$ $$T^{3} + \cdots + 351884592248992$$
$83$ $$T^{3} + \cdots + 5794291383408$$
$89$ $$T^{3} + \cdots - 246103360939432$$
$97$ $$T^{3} + \cdots + 179909862970168$$