# Properties

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

# Learn more

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: 3.3.18257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 26x + 8$$ x^3 - x^2 - 26*x + 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + 25 q^{5} + (9 \beta_1 - 9) q^{6} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + (2 \beta_{2} + 37 \beta_1 - 21) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 - 9 * q^3 + (b2 - b1 - 14) * q^4 + 25 * q^5 + (9*b1 - 9) * q^6 + (-6*b2 - 20*b1 - 14) * q^7 + (2*b2 + 37*b1 - 21) * q^8 + 81 * q^9 $$q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + 25 q^{5} + (9 \beta_1 - 9) q^{6} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + (2 \beta_{2} + 37 \beta_1 - 21) q^{8} + 81 q^{9} + ( - 25 \beta_1 + 25) q^{10} + 121 q^{11} + ( - 9 \beta_{2} + 9 \beta_1 + 126) q^{12} + (44 \beta_{2} - 24 \beta_1 + 90) q^{13} + (14 \beta_{2} + 68 \beta_1 + 278) q^{14} - 225 q^{15} + ( - 67 \beta_{2} + 35 \beta_1 - 186) q^{16} + ( - 102 \beta_{2} + 236 \beta_1 + 100) q^{17} + ( - 81 \beta_1 + 81) q^{18} + ( - 46 \beta_{2} + 172 \beta_1 - 994) q^{19} + (25 \beta_{2} - 25 \beta_1 - 350) q^{20} + (54 \beta_{2} + 180 \beta_1 + 126) q^{21} + ( - 121 \beta_1 + 121) q^{22} + (64 \beta_{2} + 688 \beta_1 - 464) q^{23} + ( - 18 \beta_{2} - 333 \beta_1 + 189) q^{24} + 625 q^{25} + (68 \beta_{2} - 486 \beta_1 + 850) q^{26} - 729 q^{27} + (138 \beta_{2} + 236 \beta_1 - 318) q^{28} + (330 \beta_{2} + 652 \beta_1 - 1840) q^{29} + (225 \beta_1 - 225) q^{30} + ( - 44 \beta_{2} - 56 \beta_1 - 4956) q^{31} + ( - 166 \beta_{2} - 395 \beta_1 - 645) q^{32} - 1089 q^{33} + ( - 338 \beta_{2} + 818 \beta_1 - 4728) q^{34} + ( - 150 \beta_{2} - 500 \beta_1 - 350) q^{35} + (81 \beta_{2} - 81 \beta_1 - 1134) q^{36} + (648 \beta_{2} - 448 \beta_1 - 2130) q^{37} + ( - 218 \beta_{2} + 1408 \beta_1 - 4286) q^{38} + ( - 396 \beta_{2} + 216 \beta_1 - 810) q^{39} + (50 \beta_{2} + 925 \beta_1 - 525) q^{40} + ( - 330 \beta_{2} - 1804 \beta_1 - 2264) q^{41} + ( - 126 \beta_{2} - 612 \beta_1 - 2502) q^{42} + (618 \beta_{2} + 1340 \beta_1 - 11850) q^{43} + (121 \beta_{2} - 121 \beta_1 - 1694) q^{44} + 2025 q^{45} + ( - 624 \beta_{2} - 112 \beta_1 - 11648) q^{46} + (364 \beta_{2} - 1544 \beta_1 - 7820) q^{47} + (603 \beta_{2} - 315 \beta_1 + 1674) q^{48} + (280 \beta_{2} + 2832 \beta_1 - 5935) q^{49} + ( - 625 \beta_1 + 625) q^{50} + (918 \beta_{2} - 2124 \beta_1 - 900) q^{51} + ( - 854 \beta_{2} - 694 \beta_1 + 6776) q^{52} + ( - 1196 \beta_{2} - 360 \beta_1 - 7126) q^{53} + (729 \beta_1 - 729) q^{54} + 3025 q^{55} + ( - 546 \beta_{2} - 3100 \beta_1 - 12122) q^{56} + (414 \beta_{2} - 1548 \beta_1 + 8946) q^{57} + ( - 322 \beta_{2} - 1130 \beta_1 - 10284) q^{58} + ( - 2864 \beta_{2} - 4928 \beta_1 - 1988) q^{59} + ( - 225 \beta_{2} + 225 \beta_1 + 3150) q^{60} + ( - 616 \beta_{2} + 432 \beta_1 + 8262) q^{61} + (12 \beta_{2} + 5352 \beta_1 - 4356) q^{62} + ( - 486 \beta_{2} - 1620 \beta_1 - 1134) q^{63} + (2373 \beta_{2} + 1019 \beta_1 + 10694) q^{64} + (1100 \beta_{2} - 600 \beta_1 + 2250) q^{65} + (1089 \beta_1 - 1089) q^{66} + ( - 64 \beta_{2} + 3360 \beta_1 + 4524) q^{67} + (2108 \beta_{2} + 218 \beta_1 - 24538) q^{68} + ( - 576 \beta_{2} - 6192 \beta_1 + 4176) q^{69} + (350 \beta_{2} + 1700 \beta_1 + 6950) q^{70} + (3576 \beta_{2} - 3376 \beta_1 + 1552) q^{71} + (162 \beta_{2} + 2997 \beta_1 - 1701) q^{72} + (1028 \beta_{2} - 1656 \beta_1 + 846) q^{73} + (1096 \beta_{2} - 3702 \beta_1 + 10670) q^{74} - 5625 q^{75} + ( - 154 \beta_{2} + 744 \beta_1 + 1842) q^{76} + ( - 726 \beta_{2} - 2420 \beta_1 - 1694) q^{77} + ( - 612 \beta_{2} + 4374 \beta_1 - 7650) q^{78} + ( - 4202 \beta_{2} - 8252 \beta_1 - 8130) q^{79} + ( - 1675 \beta_{2} + 875 \beta_1 - 4650) q^{80} + 6561 q^{81} + (1474 \beta_{2} + 5234 \beta_1 + 25764) q^{82} + (2800 \beta_{2} - 5744 \beta_1 - 15284) q^{83} + ( - 1242 \beta_{2} - 2124 \beta_1 + 2862) q^{84} + ( - 2550 \beta_{2} + 5900 \beta_1 + 2500) q^{85} + ( - 722 \beta_{2} + 6288 \beta_1 - 29686) q^{86} + ( - 2970 \beta_{2} - 5868 \beta_1 + 16560) q^{87} + (242 \beta_{2} + 4477 \beta_1 - 2541) q^{88} + (2640 \beta_{2} + 8896 \beta_1 - 66838) q^{89} + ( - 2025 \beta_1 + 2025) q^{90} + (1436 \beta_{2} - 5496 \beta_1 - 29716) q^{91} + ( - 2560 \beta_{2} - 4752 \beta_1 + 112) q^{92} + (396 \beta_{2} + 504 \beta_1 + 44604) q^{93} + (1908 \beta_{2} + 4544 \beta_1 + 21340) q^{94} + ( - 1150 \beta_{2} + 4300 \beta_1 - 24850) q^{95} + (1494 \beta_{2} + 3555 \beta_1 + 5805) q^{96} + ( - 48 \beta_{2} - 13696 \beta_1 + 87090) q^{97} + ( - 2552 \beta_{2} + 3415 \beta_1 - 51839) q^{98} + 9801 q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 - 9 * q^3 + (b2 - b1 - 14) * q^4 + 25 * q^5 + (9*b1 - 9) * q^6 + (-6*b2 - 20*b1 - 14) * q^7 + (2*b2 + 37*b1 - 21) * q^8 + 81 * q^9 + (-25*b1 + 25) * q^10 + 121 * q^11 + (-9*b2 + 9*b1 + 126) * q^12 + (44*b2 - 24*b1 + 90) * q^13 + (14*b2 + 68*b1 + 278) * q^14 - 225 * q^15 + (-67*b2 + 35*b1 - 186) * q^16 + (-102*b2 + 236*b1 + 100) * q^17 + (-81*b1 + 81) * q^18 + (-46*b2 + 172*b1 - 994) * q^19 + (25*b2 - 25*b1 - 350) * q^20 + (54*b2 + 180*b1 + 126) * q^21 + (-121*b1 + 121) * q^22 + (64*b2 + 688*b1 - 464) * q^23 + (-18*b2 - 333*b1 + 189) * q^24 + 625 * q^25 + (68*b2 - 486*b1 + 850) * q^26 - 729 * q^27 + (138*b2 + 236*b1 - 318) * q^28 + (330*b2 + 652*b1 - 1840) * q^29 + (225*b1 - 225) * q^30 + (-44*b2 - 56*b1 - 4956) * q^31 + (-166*b2 - 395*b1 - 645) * q^32 - 1089 * q^33 + (-338*b2 + 818*b1 - 4728) * q^34 + (-150*b2 - 500*b1 - 350) * q^35 + (81*b2 - 81*b1 - 1134) * q^36 + (648*b2 - 448*b1 - 2130) * q^37 + (-218*b2 + 1408*b1 - 4286) * q^38 + (-396*b2 + 216*b1 - 810) * q^39 + (50*b2 + 925*b1 - 525) * q^40 + (-330*b2 - 1804*b1 - 2264) * q^41 + (-126*b2 - 612*b1 - 2502) * q^42 + (618*b2 + 1340*b1 - 11850) * q^43 + (121*b2 - 121*b1 - 1694) * q^44 + 2025 * q^45 + (-624*b2 - 112*b1 - 11648) * q^46 + (364*b2 - 1544*b1 - 7820) * q^47 + (603*b2 - 315*b1 + 1674) * q^48 + (280*b2 + 2832*b1 - 5935) * q^49 + (-625*b1 + 625) * q^50 + (918*b2 - 2124*b1 - 900) * q^51 + (-854*b2 - 694*b1 + 6776) * q^52 + (-1196*b2 - 360*b1 - 7126) * q^53 + (729*b1 - 729) * q^54 + 3025 * q^55 + (-546*b2 - 3100*b1 - 12122) * q^56 + (414*b2 - 1548*b1 + 8946) * q^57 + (-322*b2 - 1130*b1 - 10284) * q^58 + (-2864*b2 - 4928*b1 - 1988) * q^59 + (-225*b2 + 225*b1 + 3150) * q^60 + (-616*b2 + 432*b1 + 8262) * q^61 + (12*b2 + 5352*b1 - 4356) * q^62 + (-486*b2 - 1620*b1 - 1134) * q^63 + (2373*b2 + 1019*b1 + 10694) * q^64 + (1100*b2 - 600*b1 + 2250) * q^65 + (1089*b1 - 1089) * q^66 + (-64*b2 + 3360*b1 + 4524) * q^67 + (2108*b2 + 218*b1 - 24538) * q^68 + (-576*b2 - 6192*b1 + 4176) * q^69 + (350*b2 + 1700*b1 + 6950) * q^70 + (3576*b2 - 3376*b1 + 1552) * q^71 + (162*b2 + 2997*b1 - 1701) * q^72 + (1028*b2 - 1656*b1 + 846) * q^73 + (1096*b2 - 3702*b1 + 10670) * q^74 - 5625 * q^75 + (-154*b2 + 744*b1 + 1842) * q^76 + (-726*b2 - 2420*b1 - 1694) * q^77 + (-612*b2 + 4374*b1 - 7650) * q^78 + (-4202*b2 - 8252*b1 - 8130) * q^79 + (-1675*b2 + 875*b1 - 4650) * q^80 + 6561 * q^81 + (1474*b2 + 5234*b1 + 25764) * q^82 + (2800*b2 - 5744*b1 - 15284) * q^83 + (-1242*b2 - 2124*b1 + 2862) * q^84 + (-2550*b2 + 5900*b1 + 2500) * q^85 + (-722*b2 + 6288*b1 - 29686) * q^86 + (-2970*b2 - 5868*b1 + 16560) * q^87 + (242*b2 + 4477*b1 - 2541) * q^88 + (2640*b2 + 8896*b1 - 66838) * q^89 + (-2025*b1 + 2025) * q^90 + (1436*b2 - 5496*b1 - 29716) * q^91 + (-2560*b2 - 4752*b1 + 112) * q^92 + (396*b2 + 504*b1 + 44604) * q^93 + (1908*b2 + 4544*b1 + 21340) * q^94 + (-1150*b2 + 4300*b1 - 24850) * q^95 + (1494*b2 + 3555*b1 + 5805) * q^96 + (-48*b2 - 13696*b1 + 87090) * q^97 + (-2552*b2 + 3415*b1 - 51839) * q^98 + 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 27 * q^3 - 42 * q^4 + 75 * q^5 - 18 * q^6 - 68 * q^7 - 24 * q^8 + 243 * q^9 $$3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 378 q^{12} + 290 q^{13} + 916 q^{14} - 675 q^{15} - 590 q^{16} + 434 q^{17} + 162 q^{18} - 2856 q^{19} - 1050 q^{20} + 612 q^{21} + 242 q^{22} - 640 q^{23} + 216 q^{24} + 1875 q^{25} + 2132 q^{26} - 2187 q^{27} - 580 q^{28} - 4538 q^{29} - 450 q^{30} - 14968 q^{31} - 2496 q^{32} - 3267 q^{33} - 13704 q^{34} - 1700 q^{35} - 3402 q^{36} - 6190 q^{37} - 11668 q^{38} - 2610 q^{39} - 600 q^{40} - 8926 q^{41} - 8244 q^{42} - 33592 q^{43} - 5082 q^{44} + 6075 q^{45} - 35680 q^{46} - 24640 q^{47} + 5310 q^{48} - 14693 q^{49} + 1250 q^{50} - 3906 q^{51} + 18780 q^{52} - 22934 q^{53} - 1458 q^{54} + 9075 q^{55} - 40012 q^{56} + 25704 q^{57} - 32304 q^{58} - 13756 q^{59} + 9450 q^{60} + 24602 q^{61} - 7704 q^{62} - 5508 q^{63} + 35474 q^{64} + 7250 q^{65} - 2178 q^{66} + 16868 q^{67} - 71288 q^{68} + 5760 q^{69} + 22900 q^{70} + 4856 q^{71} - 1944 q^{72} + 1910 q^{73} + 29404 q^{74} - 16875 q^{75} + 6116 q^{76} - 8228 q^{77} - 19188 q^{78} - 36844 q^{79} - 14750 q^{80} + 19683 q^{81} + 84000 q^{82} - 48796 q^{83} + 5220 q^{84} + 10850 q^{85} - 83492 q^{86} + 40842 q^{87} - 2904 q^{88} - 188978 q^{89} + 4050 q^{90} - 93208 q^{91} - 6976 q^{92} + 134712 q^{93} + 70472 q^{94} - 71400 q^{95} + 22464 q^{96} + 247526 q^{97} - 154654 q^{98} + 29403 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 27 * q^3 - 42 * q^4 + 75 * q^5 - 18 * q^6 - 68 * q^7 - 24 * q^8 + 243 * q^9 + 50 * q^10 + 363 * q^11 + 378 * q^12 + 290 * q^13 + 916 * q^14 - 675 * q^15 - 590 * q^16 + 434 * q^17 + 162 * q^18 - 2856 * q^19 - 1050 * q^20 + 612 * q^21 + 242 * q^22 - 640 * q^23 + 216 * q^24 + 1875 * q^25 + 2132 * q^26 - 2187 * q^27 - 580 * q^28 - 4538 * q^29 - 450 * q^30 - 14968 * q^31 - 2496 * q^32 - 3267 * q^33 - 13704 * q^34 - 1700 * q^35 - 3402 * q^36 - 6190 * q^37 - 11668 * q^38 - 2610 * q^39 - 600 * q^40 - 8926 * q^41 - 8244 * q^42 - 33592 * q^43 - 5082 * q^44 + 6075 * q^45 - 35680 * q^46 - 24640 * q^47 + 5310 * q^48 - 14693 * q^49 + 1250 * q^50 - 3906 * q^51 + 18780 * q^52 - 22934 * q^53 - 1458 * q^54 + 9075 * q^55 - 40012 * q^56 + 25704 * q^57 - 32304 * q^58 - 13756 * q^59 + 9450 * q^60 + 24602 * q^61 - 7704 * q^62 - 5508 * q^63 + 35474 * q^64 + 7250 * q^65 - 2178 * q^66 + 16868 * q^67 - 71288 * q^68 + 5760 * q^69 + 22900 * q^70 + 4856 * q^71 - 1944 * q^72 + 1910 * q^73 + 29404 * q^74 - 16875 * q^75 + 6116 * q^76 - 8228 * q^77 - 19188 * q^78 - 36844 * q^79 - 14750 * q^80 + 19683 * q^81 + 84000 * q^82 - 48796 * q^83 + 5220 * q^84 + 10850 * q^85 - 83492 * q^86 + 40842 * q^87 - 2904 * q^88 - 188978 * q^89 + 4050 * q^90 - 93208 * q^91 - 6976 * q^92 + 134712 * q^93 + 70472 * q^94 - 71400 * q^95 + 22464 * q^96 + 247526 * q^97 - 154654 * q^98 + 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 26x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 17$$ v^2 - v - 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 17$$ b2 + b1 + 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.

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
 5.47894 0.305203 −4.78415
−4.47894 −9.00000 −11.9391 25.0000 40.3105 −168.818 196.801 81.0000 −111.974
1.2 0.694797 −9.00000 −31.5173 25.0000 −6.25317 83.1683 −44.1316 81.0000 17.3699
1.3 5.78415 −9.00000 1.45634 25.0000 −52.0573 17.6498 −176.669 81.0000 144.604
 $$n$$: 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.c 3
3.b odd 2 1 495.6.a.b 3
5.b even 2 1 825.6.a.g 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 25T_{2} + 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} - 25 T + 18$$
$3$ $$(T + 9)^{3}$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} + 68 T^{2} - 15552 T + 247808$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} - 290 T^{2} + \cdots + 132063592$$
$17$ $$T^{3} - 434 T^{2} + \cdots + 2547052488$$
$19$ $$T^{3} + 2856 T^{2} + \cdots + 137703680$$
$23$ $$T^{3} + 640 T^{2} + \cdots - 15777349632$$
$29$ $$T^{3} + 4538 T^{2} + \cdots - 44413548456$$
$31$ $$T^{3} + 14968 T^{2} + \cdots + 121645522944$$
$37$ $$T^{3} + 6190 T^{2} + \cdots + 28013661736$$
$41$ $$T^{3} + 8926 T^{2} + \cdots + 119305168392$$
$43$ $$T^{3} + 33592 T^{2} + \cdots - 38997547520$$
$47$ $$T^{3} + 24640 T^{2} + \cdots - 679997104128$$
$53$ $$T^{3} + 22934 T^{2} + \cdots - 4393759072056$$
$59$ $$T^{3} + 13756 T^{2} + \cdots - 20798004639936$$
$61$ $$T^{3} - 24602 T^{2} + \cdots + 43064794504$$
$67$ $$T^{3} - 16868 T^{2} + \cdots + 1826752720192$$
$71$ $$T^{3} - 4856 T^{2} + \cdots + 34155066048000$$
$73$ $$T^{3} - 1910 T^{2} + \cdots - 163103734088$$
$79$ $$T^{3} + 36844 T^{2} + \cdots - 70772253539328$$
$83$ $$T^{3} + 48796 T^{2} + \cdots - 70386077185728$$
$89$ $$T^{3} + 188978 T^{2} + \cdots - 16088649675432$$
$97$ $$T^{3} + \cdots - 148869121092488$$
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