# Properties

 Label 162.6.a.j Level $162$ Weight $6$ Character orbit 162.a Self dual yes Analytic conductor $25.982$ Analytic rank $0$ 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] = [162,6,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 162.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: $$25.9821788097$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.125628.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 63x + 159$$ x^3 - x^2 - 63*x + 159 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + ( - \beta_1 + 18) q^{5} + ( - \beta_{2} + 44) q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + (-b1 + 18) * q^5 + (-b2 + 44) * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} + ( - \beta_1 + 18) q^{5} + ( - \beta_{2} + 44) q^{7} + 64 q^{8} + ( - 4 \beta_1 + 72) q^{10} + (\beta_{2} - 5 \beta_1 + 105) q^{11} + (2 \beta_{2} + 11 \beta_1 + 248) q^{13} + ( - 4 \beta_{2} + 176) q^{14} + 256 q^{16} + (4 \beta_{2} + 9 \beta_1 + 483) q^{17} + (6 \beta_{2} - 5 \beta_1 + 377) q^{19} + ( - 16 \beta_1 + 288) q^{20} + (4 \beta_{2} - 20 \beta_1 + 420) q^{22} + ( - \beta_{2} + 14 \beta_1 + 1056) q^{23} + ( - 10 \beta_{2} - 5 \beta_1 + 961) q^{25} + (8 \beta_{2} + 44 \beta_1 + 992) q^{26} + ( - 16 \beta_{2} + 704) q^{28} + ( - 6 \beta_{2} + 59 \beta_1 + 1716) q^{29} + ( - 17 \beta_{2} + 44 \beta_1 + 2870) q^{31} + 1024 q^{32} + (16 \beta_{2} + 36 \beta_1 + 1932) q^{34} + ( - 49 \beta_{2} - 136 \beta_1 + 450) q^{35} + (10 \beta_{2} + 40 \beta_1 + 6656) q^{37} + (24 \beta_{2} - 20 \beta_1 + 1508) q^{38} + ( - 64 \beta_1 + 1152) q^{40} + (70 \beta_{2} - 80 \beta_1 - 1683) q^{41} + (39 \beta_{2} - 117 \beta_1 + 10463) q^{43} + (16 \beta_{2} - 80 \beta_1 + 1680) q^{44} + ( - 4 \beta_{2} + 56 \beta_1 + 4224) q^{46} + ( - 49 \beta_{2} + 40 \beta_1 - 4308) q^{47} + (4 \beta_{2} - 251 \beta_1 + 17619) q^{49} + ( - 40 \beta_{2} - 20 \beta_1 + 3844) q^{50} + (32 \beta_{2} + 176 \beta_1 + 3968) q^{52} + (54 \beta_{2} + 116 \beta_1 - 16008) q^{53} + ( - \beta_{2} + 52 \beta_1 + 21042) q^{55} + ( - 64 \beta_{2} + 2816) q^{56} + ( - 24 \beta_{2} + 236 \beta_1 + 6864) q^{58} + ( - 71 \beta_{2} + 255 \beta_1 - 20985) q^{59} + ( - 82 \beta_{2} + 37 \beta_1 + 25322) q^{61} + ( - 68 \beta_{2} + 176 \beta_1 + 11480) q^{62} + 4096 q^{64} + (208 \beta_{2} - 207 \beta_1 - 36234) q^{65} + (15 \beta_{2} + 519 \beta_1 + 10997) q^{67} + (64 \beta_{2} + 144 \beta_1 + 7728) q^{68} + ( - 196 \beta_{2} - 544 \beta_1 + 1800) q^{70} + (130 \beta_{2} + 354 \beta_1 - 21612) q^{71} + ( - 68 \beta_{2} - 901 \beta_1 - 1411) q^{73} + (40 \beta_{2} + 160 \beta_1 + 26624) q^{74} + (96 \beta_{2} - 80 \beta_1 + 6032) q^{76} + ( - 308 \beta_{2} - 429 \beta_1 - 29580) q^{77} + ( - 15 \beta_{2} + 724 \beta_1 - 29734) q^{79} + ( - 256 \beta_1 + 4608) q^{80} + (280 \beta_{2} - 320 \beta_1 - 6732) q^{82} + ( - 211 \beta_{2} - 476 \beta_1 - 10878) q^{83} + (286 \beta_{2} - 232 \beta_1 - 23796) q^{85} + (156 \beta_{2} - 468 \beta_1 + 41852) q^{86} + (64 \beta_{2} - 320 \beta_1 + 6720) q^{88} + ( - 98 \beta_{2} + 294 \beta_1 + 11022) q^{89} + ( - 3 \beta_{2} + 1998 \beta_1 - 50306) q^{91} + ( - 16 \beta_{2} + 224 \beta_1 + 16896) q^{92} + ( - 196 \beta_{2} + 160 \beta_1 - 17232) q^{94} + (244 \beta_{2} + 240 \beta_1 + 27648) q^{95} + (386 \beta_{2} - 16 \beta_1 - 15415) q^{97} + (16 \beta_{2} - 1004 \beta_1 + 70476) q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + (-b1 + 18) * q^5 + (-b2 + 44) * q^7 + 64 * q^8 + (-4*b1 + 72) * q^10 + (b2 - 5*b1 + 105) * q^11 + (2*b2 + 11*b1 + 248) * q^13 + (-4*b2 + 176) * q^14 + 256 * q^16 + (4*b2 + 9*b1 + 483) * q^17 + (6*b2 - 5*b1 + 377) * q^19 + (-16*b1 + 288) * q^20 + (4*b2 - 20*b1 + 420) * q^22 + (-b2 + 14*b1 + 1056) * q^23 + (-10*b2 - 5*b1 + 961) * q^25 + (8*b2 + 44*b1 + 992) * q^26 + (-16*b2 + 704) * q^28 + (-6*b2 + 59*b1 + 1716) * q^29 + (-17*b2 + 44*b1 + 2870) * q^31 + 1024 * q^32 + (16*b2 + 36*b1 + 1932) * q^34 + (-49*b2 - 136*b1 + 450) * q^35 + (10*b2 + 40*b1 + 6656) * q^37 + (24*b2 - 20*b1 + 1508) * q^38 + (-64*b1 + 1152) * q^40 + (70*b2 - 80*b1 - 1683) * q^41 + (39*b2 - 117*b1 + 10463) * q^43 + (16*b2 - 80*b1 + 1680) * q^44 + (-4*b2 + 56*b1 + 4224) * q^46 + (-49*b2 + 40*b1 - 4308) * q^47 + (4*b2 - 251*b1 + 17619) * q^49 + (-40*b2 - 20*b1 + 3844) * q^50 + (32*b2 + 176*b1 + 3968) * q^52 + (54*b2 + 116*b1 - 16008) * q^53 + (-b2 + 52*b1 + 21042) * q^55 + (-64*b2 + 2816) * q^56 + (-24*b2 + 236*b1 + 6864) * q^58 + (-71*b2 + 255*b1 - 20985) * q^59 + (-82*b2 + 37*b1 + 25322) * q^61 + (-68*b2 + 176*b1 + 11480) * q^62 + 4096 * q^64 + (208*b2 - 207*b1 - 36234) * q^65 + (15*b2 + 519*b1 + 10997) * q^67 + (64*b2 + 144*b1 + 7728) * q^68 + (-196*b2 - 544*b1 + 1800) * q^70 + (130*b2 + 354*b1 - 21612) * q^71 + (-68*b2 - 901*b1 - 1411) * q^73 + (40*b2 + 160*b1 + 26624) * q^74 + (96*b2 - 80*b1 + 6032) * q^76 + (-308*b2 - 429*b1 - 29580) * q^77 + (-15*b2 + 724*b1 - 29734) * q^79 + (-256*b1 + 4608) * q^80 + (280*b2 - 320*b1 - 6732) * q^82 + (-211*b2 - 476*b1 - 10878) * q^83 + (286*b2 - 232*b1 - 23796) * q^85 + (156*b2 - 468*b1 + 41852) * q^86 + (64*b2 - 320*b1 + 6720) * q^88 + (-98*b2 + 294*b1 + 11022) * q^89 + (-3*b2 + 1998*b1 - 50306) * q^91 + (-16*b2 + 224*b1 + 16896) * q^92 + (-196*b2 + 160*b1 - 17232) * q^94 + (244*b2 + 240*b1 + 27648) * q^95 + (386*b2 - 16*b1 - 15415) * q^97 + (16*b2 - 1004*b1 + 70476) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8}+O(q^{10})$$ 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8 $$3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8} + 216 q^{10} + 315 q^{11} + 744 q^{13} + 528 q^{14} + 768 q^{16} + 1449 q^{17} + 1131 q^{19} + 864 q^{20} + 1260 q^{22} + 3168 q^{23} + 2883 q^{25} + 2976 q^{26} + 2112 q^{28} + 5148 q^{29} + 8610 q^{31} + 3072 q^{32} + 5796 q^{34} + 1350 q^{35} + 19968 q^{37} + 4524 q^{38} + 3456 q^{40} - 5049 q^{41} + 31389 q^{43} + 5040 q^{44} + 12672 q^{46} - 12924 q^{47} + 52857 q^{49} + 11532 q^{50} + 11904 q^{52} - 48024 q^{53} + 63126 q^{55} + 8448 q^{56} + 20592 q^{58} - 62955 q^{59} + 75966 q^{61} + 34440 q^{62} + 12288 q^{64} - 108702 q^{65} + 32991 q^{67} + 23184 q^{68} + 5400 q^{70} - 64836 q^{71} - 4233 q^{73} + 79872 q^{74} + 18096 q^{76} - 88740 q^{77} - 89202 q^{79} + 13824 q^{80} - 20196 q^{82} - 32634 q^{83} - 71388 q^{85} + 125556 q^{86} + 20160 q^{88} + 33066 q^{89} - 150918 q^{91} + 50688 q^{92} - 51696 q^{94} + 82944 q^{95} - 46245 q^{97} + 211428 q^{98}+O(q^{100})$$ 3 * q + 12 * q^2 + 48 * q^4 + 54 * q^5 + 132 * q^7 + 192 * q^8 + 216 * q^10 + 315 * q^11 + 744 * q^13 + 528 * q^14 + 768 * q^16 + 1449 * q^17 + 1131 * q^19 + 864 * q^20 + 1260 * q^22 + 3168 * q^23 + 2883 * q^25 + 2976 * q^26 + 2112 * q^28 + 5148 * q^29 + 8610 * q^31 + 3072 * q^32 + 5796 * q^34 + 1350 * q^35 + 19968 * q^37 + 4524 * q^38 + 3456 * q^40 - 5049 * q^41 + 31389 * q^43 + 5040 * q^44 + 12672 * q^46 - 12924 * q^47 + 52857 * q^49 + 11532 * q^50 + 11904 * q^52 - 48024 * q^53 + 63126 * q^55 + 8448 * q^56 + 20592 * q^58 - 62955 * q^59 + 75966 * q^61 + 34440 * q^62 + 12288 * q^64 - 108702 * q^65 + 32991 * q^67 + 23184 * q^68 + 5400 * q^70 - 64836 * q^71 - 4233 * q^73 + 79872 * q^74 + 18096 * q^76 - 88740 * q^77 - 89202 * q^79 + 13824 * q^80 - 20196 * q^82 - 32634 * q^83 - 71388 * q^85 + 125556 * q^86 + 20160 * q^88 + 33066 * q^89 - 150918 * q^91 + 50688 * q^92 - 51696 * q^94 + 82944 * q^95 - 46245 * q^97 + 211428 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 63x + 159$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{2} + 24\nu - 135 ) / 2$$ (3*v^2 + 24*v - 135) / 2 $$\beta_{2}$$ $$=$$ $$( -15\nu^{2} - 12\nu + 639 ) / 2$$ (-15*v^2 - 12*v + 639) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + 5\beta _1 + 18 ) / 54$$ (b2 + 5*b1 + 18) / 54 $$\nu^{2}$$ $$=$$ $$( -4\beta_{2} - 2\beta _1 + 1143 ) / 27$$ (-4*b2 - 2*b1 + 1143) / 27

## 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
 6.81933 2.72791 −8.54724
4.00000 0 16.0000 −66.0868 0 114.190 64.0000 0 −264.347
1.2 4.00000 0 16.0000 41.6029 0 −203.321 64.0000 0 166.412
1.3 4.00000 0 16.0000 78.4839 0 221.131 64.0000 0 313.936
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## 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 162.6.a.j 3
3.b odd 2 1 162.6.a.i 3
9.c even 3 2 18.6.c.b 6
9.d odd 6 2 54.6.c.b 6
36.f odd 6 2 144.6.i.b 6
36.h even 6 2 432.6.i.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b 6 9.c even 3 2
54.6.c.b 6 9.d odd 6 2
144.6.i.b 6 36.f odd 6 2
162.6.a.i 3 3.b odd 2 1
162.6.a.j 3 1.a even 1 1 trivial
432.6.i.b 6 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 54T_{5}^{2} - 4671T_{5} + 215784$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(162))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 4)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 54 T^{2} - 4671 T + 215784$$
$7$ $$T^{3} - 132 T^{2} - 42927 T + 5134078$$
$11$ $$T^{3} - 315 T^{2} + \cdots + 41768163$$
$13$ $$T^{3} - 744 T^{2} + \cdots + 384824422$$
$17$ $$T^{3} - 1449 T^{2} + \cdots + 930192444$$
$19$ $$T^{3} - 1131 T^{2} + \cdots - 352455920$$
$23$ $$T^{3} - 3168 T^{2} + \cdots - 425600514$$
$29$ $$T^{3} - 5148 T^{2} + \cdots - 6505725654$$
$31$ $$T^{3} - 8610 T^{2} + \cdots + 59316561604$$
$37$ $$T^{3} - 19968 T^{2} + \cdots - 188019064016$$
$41$ $$T^{3} + 5049 T^{2} + \cdots - 2157363401913$$
$43$ $$T^{3} - 31389 T^{2} + \cdots + 513661035961$$
$47$ $$T^{3} + 12924 T^{2} + \cdots + 84596352750$$
$53$ $$T^{3} + 48024 T^{2} + \cdots + 1764512817552$$
$59$ $$T^{3} + 62955 T^{2} + \cdots - 5778215946243$$
$61$ $$T^{3} - 75966 T^{2} + \cdots - 5359653497960$$
$67$ $$T^{3} - 32991 T^{2} + \cdots + 3034793402875$$
$71$ $$T^{3} + 64836 T^{2} + \cdots - 139951336896$$
$73$ $$T^{3} + 4233 T^{2} + \cdots + 14322358753732$$
$79$ $$T^{3} + \cdots - 115303078320596$$
$83$ $$T^{3} + \cdots - 103421607911604$$
$89$ $$T^{3} - 33066 T^{2} + \cdots + 9104584153608$$
$97$ $$T^{3} + \cdots - 292114819729997$$