# Properties

 Label 539.6.a.e Level $539$ Weight $6$ Character orbit 539.a Self dual yes Analytic conductor $86.447$ 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] = [539,6,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("539.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 539.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: $$86.4468788792$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (\beta_{2} - 9 \beta_1 - 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (26 \beta_{2} - 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 - b1 - 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (b2 - 9*b1 - 5) * q^5 + (-13*b2 - 10*b1 + 72) * q^6 + (26*b2 - 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (\beta_{2} - 9 \beta_1 - 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (26 \beta_{2} - 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9} + (9 \beta_{2} - 42 \beta_1 + 152) q^{10} + 121 q^{11} + (112 \beta_{2} + 70 \beta_1 - 354) q^{12} + ( - 70 \beta_{2} - 18 \beta_1 - 156) q^{13} + (27 \beta_{2} + 85 \beta_1 + 523) q^{15} + ( - 164 \beta_{2} + 36 \beta_1 + 652) q^{16} + ( - 132 \beta_{2} + 60 \beta_1 - 382) q^{17} + (48 \beta_{2} + 158 \beta_1 - 1288) q^{18} + (60 \beta_{2} + 60 \beta_1 - 480) q^{19} + (168 \beta_{2} + 66 \beta_1 + 1138) q^{20} + 121 \beta_{2} q^{22} + ( - 21 \beta_{2} + 501 \beta_1 - 1189) q^{23} + ( - 526 \beta_{2} - 72 \beta_1 + 3940) q^{24} + (265 \beta_{2} + 255 \beta_1 - 104) q^{25} + (160 \beta_{2} + 348 \beta_1 - 4160) q^{26} + ( - 57 \beta_{2} + 329 \beta_1 + 887) q^{27} + ( - 182 \beta_{2} - 138 \beta_1 - 1096) q^{29} + (245 \beta_{2} + 178 \beta_1 + 824) q^{30} + (101 \beta_{2} - 69 \beta_1 + 1389) q^{31} + (404 \beta_{2} + 1128 \beta_1 - 4512) q^{32} + (121 \beta_{2} - 121 \beta_1 - 1331) q^{33} + (26 \beta_{2} + 1032 \beta_1 - 8784) q^{34} + ( - 1188 \beta_{2} - 8 \beta_1 + 1588) q^{36} + (937 \beta_{2} + 591 \beta_1 + 5711) q^{37} + ( - 840 \beta_{2} - 120 \beta_1 + 3120) q^{38} + (844 \beta_{2} + 1036 \beta_1 - 2532) q^{39} + (46 \beta_{2} + 600 \beta_1 + 4892) q^{40} + (1378 \beta_{2} - 282 \beta_1 - 1904) q^{41} + (1190 \beta_{2} - 1110 \beta_1 - 8366) q^{43} + ( - 484 \beta_{2} - 726 \beta_1 + 3630) q^{44} + ( - 496 \beta_{2} + 544 \beta_1 - 6334) q^{45} + ( - 2107 \beta_{2} + 2130 \beta_1 - 6312) q^{46} + (600 \beta_{2} + 1272 \beta_1 + 5320) q^{47} + (2604 \beta_{2} + 628 \beta_1 - 20564) q^{48} + ( - 1674 \beta_{2} - 570 \beta_1 + 13880) q^{50} + (1034 \beta_{2} + 1102 \beta_1 - 7942) q^{51} + ( - 3256 \beta_{2} + 1008 \beta_1 + 11432) q^{52} + ( - 476 \beta_{2} - 1620 \beta_1 + 17402) q^{53} + (457 \beta_{2} + 1658 \beta_1 - 6824) q^{54} + (121 \beta_{2} - 1089 \beta_1 - 605) q^{55} + ( - 1560 \beta_{2} - 720 \beta_1 + 6960) q^{57} + ( - 92 \beta_{2} + 540 \beta_1 - 9904) q^{58} + ( - 3141 \beta_{2} - 747 \beta_1 + 1495) q^{59} + ( - 1376 \beta_{2} - 3478 \beta_1 - 3326) q^{60} + (1466 \beta_{2} + 4038 \beta_1 - 7508) q^{61} + (1123 \beta_{2} - 882 \beta_1 + 6952) q^{62} + ( - 3136 \beta_{2} + 936 \beta_1 - 7096) q^{64} + ( - 264 \beta_{2} + 4848 \beta_1 - 4172) q^{65} + ( - 1573 \beta_{2} - 1210 \beta_1 + 8712) q^{66} + ( - 6575 \beta_{2} - 2721 \beta_1 - 15011) q^{67} + ( - 6728 \beta_{2} + 2052 \beta_1 + 3516) q^{68} + ( - 3421 \beta_{2} - 3611 \beta_1 - 10477) q^{69} + ( - 3935 \beta_{2} - 2673 \beta_1 + 13985) q^{71} + (4820 \beta_{2} + 2040 \beta_1 - 32360) q^{72} + ( - 5370 \beta_{2} + 3522 \beta_1 - 6316) q^{73} + (781 \beta_{2} - 3258 \beta_1 + 52184) q^{74} + ( - 4824 \beta_{2} - 5096 \beta_1 + 9004) q^{75} + (4800 \beta_{2} + 2640 \beta_1 - 35520) q^{76} + ( - 7980 \beta_{2} - 920 \beta_1 + 41968) q^{78} + (3902 \beta_{2} + 1362 \beta_1 + 41262) q^{79} + ( - 1868 \beta_{2} + 12 \beta_1 - 39564) q^{80} + (4600 \beta_{2} - 6280 \beta_1 - 26879) q^{81} + ( - 6852 \beta_{2} - 9396 \beta_1 + 88256) q^{82} + ( - 3150 \beta_{2} - 11250 \beta_1 + 51726) q^{83} + ( - 3310 \beta_{2} + 7302 \beta_1 - 37114) q^{85} + ( - 10906 \beta_{2} - 11580 \beta_1 + 84880) q^{86} + (1960 \beta_{2} + 4296 \beta_1 + 5024) q^{87} + (3146 \beta_{2} - 22748) q^{88} + (5167 \beta_{2} + 4569 \beta_1 + 34085) q^{89} + ( - 5438 \beta_{2} + 5152 \beta_1 - 36192) q^{90} + ( - 1472 \beta_{2} + 5130 \beta_1 - 113886) q^{92} + (421 \beta_{2} - 1709 \beta_1 - 4971) q^{93} + (376 \beta_{2} + 1488 \beta_1 + 24480) q^{94} + ( - 1680 \beta_{2} + 120 \beta_1 - 7440) q^{95} + ( - 15404 \beta_{2} - 10808 \beta_1 + 29088) q^{96} + ( - 123 \beta_{2} - 6429 \beta_1 - 1085) q^{97} + ( - 2299 \beta_{2} + 1331 \beta_1 - 726) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 - b1 - 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (b2 - 9*b1 - 5) * q^5 + (-13*b2 - 10*b1 + 72) * q^6 + (26*b2 - 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 + (9*b2 - 42*b1 + 152) * q^10 + 121 * q^11 + (112*b2 + 70*b1 - 354) * q^12 + (-70*b2 - 18*b1 - 156) * q^13 + (27*b2 + 85*b1 + 523) * q^15 + (-164*b2 + 36*b1 + 652) * q^16 + (-132*b2 + 60*b1 - 382) * q^17 + (48*b2 + 158*b1 - 1288) * q^18 + (60*b2 + 60*b1 - 480) * q^19 + (168*b2 + 66*b1 + 1138) * q^20 + 121*b2 * q^22 + (-21*b2 + 501*b1 - 1189) * q^23 + (-526*b2 - 72*b1 + 3940) * q^24 + (265*b2 + 255*b1 - 104) * q^25 + (160*b2 + 348*b1 - 4160) * q^26 + (-57*b2 + 329*b1 + 887) * q^27 + (-182*b2 - 138*b1 - 1096) * q^29 + (245*b2 + 178*b1 + 824) * q^30 + (101*b2 - 69*b1 + 1389) * q^31 + (404*b2 + 1128*b1 - 4512) * q^32 + (121*b2 - 121*b1 - 1331) * q^33 + (26*b2 + 1032*b1 - 8784) * q^34 + (-1188*b2 - 8*b1 + 1588) * q^36 + (937*b2 + 591*b1 + 5711) * q^37 + (-840*b2 - 120*b1 + 3120) * q^38 + (844*b2 + 1036*b1 - 2532) * q^39 + (46*b2 + 600*b1 + 4892) * q^40 + (1378*b2 - 282*b1 - 1904) * q^41 + (1190*b2 - 1110*b1 - 8366) * q^43 + (-484*b2 - 726*b1 + 3630) * q^44 + (-496*b2 + 544*b1 - 6334) * q^45 + (-2107*b2 + 2130*b1 - 6312) * q^46 + (600*b2 + 1272*b1 + 5320) * q^47 + (2604*b2 + 628*b1 - 20564) * q^48 + (-1674*b2 - 570*b1 + 13880) * q^50 + (1034*b2 + 1102*b1 - 7942) * q^51 + (-3256*b2 + 1008*b1 + 11432) * q^52 + (-476*b2 - 1620*b1 + 17402) * q^53 + (457*b2 + 1658*b1 - 6824) * q^54 + (121*b2 - 1089*b1 - 605) * q^55 + (-1560*b2 - 720*b1 + 6960) * q^57 + (-92*b2 + 540*b1 - 9904) * q^58 + (-3141*b2 - 747*b1 + 1495) * q^59 + (-1376*b2 - 3478*b1 - 3326) * q^60 + (1466*b2 + 4038*b1 - 7508) * q^61 + (1123*b2 - 882*b1 + 6952) * q^62 + (-3136*b2 + 936*b1 - 7096) * q^64 + (-264*b2 + 4848*b1 - 4172) * q^65 + (-1573*b2 - 1210*b1 + 8712) * q^66 + (-6575*b2 - 2721*b1 - 15011) * q^67 + (-6728*b2 + 2052*b1 + 3516) * q^68 + (-3421*b2 - 3611*b1 - 10477) * q^69 + (-3935*b2 - 2673*b1 + 13985) * q^71 + (4820*b2 + 2040*b1 - 32360) * q^72 + (-5370*b2 + 3522*b1 - 6316) * q^73 + (781*b2 - 3258*b1 + 52184) * q^74 + (-4824*b2 - 5096*b1 + 9004) * q^75 + (4800*b2 + 2640*b1 - 35520) * q^76 + (-7980*b2 - 920*b1 + 41968) * q^78 + (3902*b2 + 1362*b1 + 41262) * q^79 + (-1868*b2 + 12*b1 - 39564) * q^80 + (4600*b2 - 6280*b1 - 26879) * q^81 + (-6852*b2 - 9396*b1 + 88256) * q^82 + (-3150*b2 - 11250*b1 + 51726) * q^83 + (-3310*b2 + 7302*b1 - 37114) * q^85 + (-10906*b2 - 11580*b1 + 84880) * q^86 + (1960*b2 + 4296*b1 + 5024) * q^87 + (3146*b2 - 22748) * q^88 + (5167*b2 + 4569*b1 + 34085) * q^89 + (-5438*b2 + 5152*b1 - 36192) * q^90 + (-1472*b2 + 5130*b1 - 113886) * q^92 + (421*b2 - 1709*b1 - 4971) * q^93 + (376*b2 + 1488*b1 + 24480) * q^94 + (-1680*b2 + 120*b1 - 7440) * q^95 + (-15404*b2 - 10808*b1 + 29088) * q^96 + (-123*b2 - 6429*b1 - 1085) * q^97 + (-2299*b2 + 1331*b1 - 726) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9}+O(q^{10})$$ 3 * q - 34 * q^3 + 84 * q^4 - 24 * q^5 + 206 * q^6 - 564 * q^8 - 7 * q^9 $$3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9} + 414 q^{10} + 363 q^{11} - 992 q^{12} - 486 q^{13} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} - 3706 q^{18} - 1380 q^{19} + 3480 q^{20} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} - 12132 q^{26} + 2990 q^{27} - 3426 q^{29} + 2650 q^{30} + 4098 q^{31} - 12408 q^{32} - 4114 q^{33} - 25320 q^{34} + 4756 q^{36} + 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 26208 q^{43} + 10164 q^{44} - 18458 q^{45} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 41070 q^{50} - 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 2904 q^{55} + 20160 q^{57} - 29172 q^{58} + 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} - 20352 q^{64} - 7668 q^{65} + 24926 q^{66} - 47754 q^{67} + 12600 q^{68} - 35042 q^{69} + 39282 q^{71} - 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 21916 q^{75} - 103920 q^{76} + 124984 q^{78} + 125148 q^{79} - 118680 q^{80} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} - 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 68244 q^{88} + 106824 q^{89} - 103424 q^{90} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} - 22200 q^{95} + 76456 q^{96} - 9684 q^{97} - 847 q^{99}+O(q^{100})$$ 3 * q - 34 * q^3 + 84 * q^4 - 24 * q^5 + 206 * q^6 - 564 * q^8 - 7 * q^9 + 414 * q^10 + 363 * q^11 - 992 * q^12 - 486 * q^13 + 1654 * q^15 + 1992 * q^16 - 1086 * q^17 - 3706 * q^18 - 1380 * q^19 + 3480 * q^20 - 3066 * q^23 + 11748 * q^24 - 57 * q^25 - 12132 * q^26 + 2990 * q^27 - 3426 * q^29 + 2650 * q^30 + 4098 * q^31 - 12408 * q^32 - 4114 * q^33 - 25320 * q^34 + 4756 * q^36 + 17724 * q^37 + 9240 * q^38 - 6560 * q^39 + 15276 * q^40 - 5994 * q^41 - 26208 * q^43 + 10164 * q^44 - 18458 * q^45 - 16806 * q^46 + 17232 * q^47 - 61064 * q^48 + 41070 * q^50 - 22724 * q^51 + 35304 * q^52 + 50586 * q^53 - 18814 * q^54 - 2904 * q^55 + 20160 * q^57 - 29172 * q^58 + 3738 * q^59 - 13456 * q^60 - 18486 * q^61 + 19974 * q^62 - 20352 * q^64 - 7668 * q^65 + 24926 * q^66 - 47754 * q^67 + 12600 * q^68 - 35042 * q^69 + 39282 * q^71 - 95040 * q^72 - 15426 * q^73 + 153294 * q^74 + 21916 * q^75 - 103920 * q^76 + 124984 * q^78 + 125148 * q^79 - 118680 * q^80 - 86917 * q^81 + 255372 * q^82 + 143928 * q^83 - 104040 * q^85 + 243060 * q^86 + 19368 * q^87 - 68244 * q^88 + 106824 * q^89 - 103424 * q^90 - 336528 * q^92 - 16622 * q^93 + 74928 * q^94 - 22200 * q^95 + 76456 * q^96 - 9684 * q^97 - 847 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x - 38$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 3\nu - 34 ) / 3$$ (v^2 - 3*v - 34) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 3\beta _1 + 34$$ 3*b2 + 3*b1 + 34

## 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
 −0.749680 8.04796 −6.29828
−10.3963 −20.6466 76.0833 −8.64919 214.649 0 −458.304 183.283 89.9197
1.2 2.20859 −16.8394 −27.1221 −75.2230 −37.1913 0 −130.577 40.5643 −166.137
1.3 8.18772 3.48600 35.0388 59.8722 28.5424 0 24.8808 −230.848 490.217
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 539.6.a.e 3
7.b odd 2 1 11.6.a.b 3
21.c even 2 1 99.6.a.g 3
28.d even 2 1 176.6.a.i 3
35.c odd 2 1 275.6.a.b 3
35.f even 4 2 275.6.b.b 6
56.e even 2 1 704.6.a.t 3
56.h odd 2 1 704.6.a.q 3
77.b even 2 1 121.6.a.d 3
231.h odd 2 1 1089.6.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 7.b odd 2 1
99.6.a.g 3 21.c even 2 1
121.6.a.d 3 77.b even 2 1
176.6.a.i 3 28.d even 2 1
275.6.a.b 3 35.c odd 2 1
275.6.b.b 6 35.f even 4 2
539.6.a.e 3 1.a even 1 1 trivial
704.6.a.q 3 56.h odd 2 1
704.6.a.t 3 56.e even 2 1
1089.6.a.r 3 231.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(539))$$:

 $$T_{2}^{3} - 90T_{2} + 188$$ T2^3 - 90*T2 + 188 $$T_{3}^{3} + 34T_{3}^{2} + 217T_{3} - 1212$$ T3^3 + 34*T3^2 + 217*T3 - 1212

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 90T + 188$$
$3$ $$T^{3} + 34 T^{2} + \cdots - 1212$$
$5$ $$T^{3} + 24 T^{2} + \cdots - 38954$$
$7$ $$T^{3}$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} + 486 T^{2} + \cdots - 164136608$$
$17$ $$T^{3} + 1086 T^{2} + \cdots - 331752056$$
$19$ $$T^{3} + 1380 T^{2} + \cdots - 57024000$$
$23$ $$T^{3} + \cdots - 17004325928$$
$29$ $$T^{3} + \cdots - 4029189120$$
$31$ $$T^{3} + \cdots - 1094344400$$
$37$ $$T^{3} + \cdots + 541788167034$$
$41$ $$T^{3} + \cdots - 201929821568$$
$43$ $$T^{3} + \cdots - 2443875098544$$
$47$ $$T^{3} + \cdots + 70174939136$$
$53$ $$T^{3} + \cdots - 1850911309656$$
$59$ $$T^{3} + \cdots - 7759637437060$$
$61$ $$T^{3} + \cdots - 15233874751008$$
$67$ $$T^{3} + \cdots - 147288561330212$$
$71$ $$T^{3} + \cdots - 1290398551704$$
$73$ $$T^{3} + \cdots + 34539701265952$$
$79$ $$T^{3} + \cdots + 1279883216320$$
$83$ $$T^{3} + \cdots + 411597824719824$$
$89$ $$T^{3} + \cdots + 90320980174650$$
$97$ $$T^{3} + \cdots + 10221902527106$$