# Properties

 Label 162.9.d.c Level $162$ Weight $9$ Character orbit 162.d Analytic conductor $65.995$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,9,Mod(53,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$65.9953348299$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 60 \beta_1 q^{5} + 2065 \beta_{2} q^{7} + 128 \beta_{3} q^{8}+O(q^{10})$$ q + (b3 - b1) * q^2 + (-128*b2 + 128) * q^4 + 60*b1 * q^5 + 2065*b2 * q^7 + 128*b3 * q^8 $$q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 60 \beta_1 q^{5} + 2065 \beta_{2} q^{7} + 128 \beta_{3} q^{8} - 7680 q^{10} + (588 \beta_{3} - 588 \beta_1) q^{11} + (8063 \beta_{2} - 8063) q^{13} - 2065 \beta_1 q^{14} - 16384 \beta_{2} q^{16} + 1908 \beta_{3} q^{17} - 226609 q^{19} + ( - 7680 \beta_{3} + 7680 \beta_1) q^{20} + ( - 75264 \beta_{2} + 75264) q^{22} - 32556 \beta_1 q^{23} + 70175 \beta_{2} q^{25} - 8063 \beta_{3} q^{26} + 264320 q^{28} + ( - 82824 \beta_{3} + 82824 \beta_1) q^{29} + (826370 \beta_{2} - 826370) q^{31} + 16384 \beta_1 q^{32} - 244224 \beta_{2} q^{34} + 123900 \beta_{3} q^{35} + 1344575 q^{37} + ( - 226609 \beta_{3} + 226609 \beta_1) q^{38} + (983040 \beta_{2} - 983040) q^{40} - 458904 \beta_1 q^{41} + 6147742 \beta_{2} q^{43} + 75264 \beta_{3} q^{44} + 4167168 q^{46} + (522444 \beta_{3} - 522444 \beta_1) q^{47} + ( - 1500576 \beta_{2} + 1500576) q^{49} - 70175 \beta_1 q^{50} + 1032064 \beta_{2} q^{52} + 67896 \beta_{3} q^{53} - 4515840 q^{55} + (264320 \beta_{3} - 264320 \beta_1) q^{56} + (10601472 \beta_{2} - 10601472) q^{58} + 41892 \beta_1 q^{59} + 14985697 \beta_{2} q^{61} - 826370 \beta_{3} q^{62} - 2097152 q^{64} + (483780 \beta_{3} - 483780 \beta_1) q^{65} + ( - 10023697 \beta_{2} + 10023697) q^{67} + 244224 \beta_1 q^{68} - 15859200 \beta_{2} q^{70} - 4020336 \beta_{3} q^{71} - 23261569 q^{73} + (1344575 \beta_{3} - 1344575 \beta_1) q^{74} + (29005952 \beta_{2} - 29005952) q^{76} - 1214220 \beta_1 q^{77} - 14267183 \beta_{2} q^{79} - 983040 \beta_{3} q^{80} + 58739712 q^{82} + (3198936 \beta_{3} - 3198936 \beta_1) q^{83} + (14653440 \beta_{2} - 14653440) q^{85} - 6147742 \beta_1 q^{86} - 9633792 \beta_{2} q^{88} - 10172412 \beta_{3} q^{89} - 16650095 q^{91} + (4167168 \beta_{3} - 4167168 \beta_1) q^{92} + ( - 66872832 \beta_{2} + 66872832) q^{94} - 13596540 \beta_1 q^{95} + 40571617 \beta_{2} q^{97} + 1500576 \beta_{3} q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 + (-128*b2 + 128) * q^4 + 60*b1 * q^5 + 2065*b2 * q^7 + 128*b3 * q^8 - 7680 * q^10 + (588*b3 - 588*b1) * q^11 + (8063*b2 - 8063) * q^13 - 2065*b1 * q^14 - 16384*b2 * q^16 + 1908*b3 * q^17 - 226609 * q^19 + (-7680*b3 + 7680*b1) * q^20 + (-75264*b2 + 75264) * q^22 - 32556*b1 * q^23 + 70175*b2 * q^25 - 8063*b3 * q^26 + 264320 * q^28 + (-82824*b3 + 82824*b1) * q^29 + (826370*b2 - 826370) * q^31 + 16384*b1 * q^32 - 244224*b2 * q^34 + 123900*b3 * q^35 + 1344575 * q^37 + (-226609*b3 + 226609*b1) * q^38 + (983040*b2 - 983040) * q^40 - 458904*b1 * q^41 + 6147742*b2 * q^43 + 75264*b3 * q^44 + 4167168 * q^46 + (522444*b3 - 522444*b1) * q^47 + (-1500576*b2 + 1500576) * q^49 - 70175*b1 * q^50 + 1032064*b2 * q^52 + 67896*b3 * q^53 - 4515840 * q^55 + (264320*b3 - 264320*b1) * q^56 + (10601472*b2 - 10601472) * q^58 + 41892*b1 * q^59 + 14985697*b2 * q^61 - 826370*b3 * q^62 - 2097152 * q^64 + (483780*b3 - 483780*b1) * q^65 + (-10023697*b2 + 10023697) * q^67 + 244224*b1 * q^68 - 15859200*b2 * q^70 - 4020336*b3 * q^71 - 23261569 * q^73 + (1344575*b3 - 1344575*b1) * q^74 + (29005952*b2 - 29005952) * q^76 - 1214220*b1 * q^77 - 14267183*b2 * q^79 - 983040*b3 * q^80 + 58739712 * q^82 + (3198936*b3 - 3198936*b1) * q^83 + (14653440*b2 - 14653440) * q^85 - 6147742*b1 * q^86 - 9633792*b2 * q^88 - 10172412*b3 * q^89 - 16650095 * q^91 + (4167168*b3 - 4167168*b1) * q^92 + (-66872832*b2 + 66872832) * q^94 - 13596540*b1 * q^95 + 40571617*b2 * q^97 + 1500576*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 256 q^{4} + 4130 q^{7}+O(q^{10})$$ 4 * q + 256 * q^4 + 4130 * q^7 $$4 q + 256 q^{4} + 4130 q^{7} - 30720 q^{10} - 16126 q^{13} - 32768 q^{16} - 906436 q^{19} + 150528 q^{22} + 140350 q^{25} + 1057280 q^{28} - 1652740 q^{31} - 488448 q^{34} + 5378300 q^{37} - 1966080 q^{40} + 12295484 q^{43} + 16668672 q^{46} + 3001152 q^{49} + 2064128 q^{52} - 18063360 q^{55} - 21202944 q^{58} + 29971394 q^{61} - 8388608 q^{64} + 20047394 q^{67} - 31718400 q^{70} - 93046276 q^{73} - 58011904 q^{76} - 28534366 q^{79} + 234958848 q^{82} - 29306880 q^{85} - 19267584 q^{88} - 66600380 q^{91} + 133745664 q^{94} + 81143234 q^{97}+O(q^{100})$$ 4 * q + 256 * q^4 + 4130 * q^7 - 30720 * q^10 - 16126 * q^13 - 32768 * q^16 - 906436 * q^19 + 150528 * q^22 + 140350 * q^25 + 1057280 * q^28 - 1652740 * q^31 - 488448 * q^34 + 5378300 * q^37 - 1966080 * q^40 + 12295484 * q^43 + 16668672 * q^46 + 3001152 * q^49 + 2064128 * q^52 - 18063360 * q^55 - 21202944 * q^58 + 29971394 * q^61 - 8388608 * q^64 + 20047394 * q^67 - 31718400 * q^70 - 93046276 * q^73 - 58011904 * q^76 - 28534366 * q^79 + 234958848 * q^82 - 29306880 * q^85 - 19267584 * q^88 - 66600380 * q^91 + 133745664 * q^94 + 81143234 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$8\nu$$ 8*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$4\nu^{3}$$ 4*v^3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 8$$ (b1) / 8 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} ) / 4$$ (b3) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$1 - \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
−9.79796 + 5.65685i 0 64.0000 110.851i 587.878 + 339.411i 0 1032.50 + 1788.34i 1448.15i 0 −7680.00
53.2 9.79796 5.65685i 0 64.0000 110.851i −587.878 339.411i 0 1032.50 + 1788.34i 1448.15i 0 −7680.00
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i 587.878 339.411i 0 1032.50 1788.34i 1448.15i 0 −7680.00
107.2 9.79796 + 5.65685i 0 64.0000 + 110.851i −587.878 + 339.411i 0 1032.50 1788.34i 1448.15i 0 −7680.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.d.c 4
3.b odd 2 1 inner 162.9.d.c 4
9.c even 3 1 54.9.b.a 2
9.c even 3 1 inner 162.9.d.c 4
9.d odd 6 1 54.9.b.a 2
9.d odd 6 1 inner 162.9.d.c 4
36.f odd 6 1 432.9.e.g 2
36.h even 6 1 432.9.e.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.a 2 9.c even 3 1
54.9.b.a 2 9.d odd 6 1
162.9.d.c 4 1.a even 1 1 trivial
162.9.d.c 4 3.b odd 2 1 inner
162.9.d.c 4 9.c even 3 1 inner
162.9.d.c 4 9.d odd 6 1 inner
432.9.e.g 2 36.f odd 6 1
432.9.e.g 2 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 460800T_{5}^{2} + 212336640000$$ acting on $$S_{9}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 128 T^{2} + 16384$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 460800 T^{2} + \cdots + 212336640000$$
$7$ $$(T^{2} - 2065 T + 4264225)^{2}$$
$11$ $$T^{4} - 44255232 T^{2} + \cdots + 19\!\cdots\!24$$
$13$ $$(T^{2} + 8063 T + 65011969)^{2}$$
$17$ $$(T^{2} + 465979392)^{2}$$
$19$ $$(T + 226609)^{4}$$
$23$ $$T^{4} - 135666321408 T^{2} + \cdots + 18\!\cdots\!64$$
$29$ $$T^{4} - 878056316928 T^{2} + \cdots + 77\!\cdots\!84$$
$31$ $$(T^{2} + 826370 T + 682887376900)^{2}$$
$37$ $$(T - 1344575)^{4}$$
$41$ $$T^{4} - 26955888795648 T^{2} + \cdots + 72\!\cdots\!04$$
$43$ $$(T^{2} - 6147742 T + 37794731698564)^{2}$$
$47$ $$T^{4} - 34937309841408 T^{2} + \cdots + 12\!\cdots\!64$$
$53$ $$(T^{2} + 590062952448)^{2}$$
$59$ $$T^{4} - 224632276992 T^{2} + \cdots + 50\!\cdots\!64$$
$61$ $$(T^{2} - 14985697 T + 224571114575809)^{2}$$
$67$ $$(T^{2} - 10023697 T + 100474501547809)^{2}$$
$71$ $$(T^{2} + 20\!\cdots\!88)^{2}$$
$73$ $$(T + 23261569)^{4}$$
$79$ $$(T^{2} + 14267183 T + 203552510755489)^{2}$$
$83$ $$T^{4} + \cdots + 17\!\cdots\!44$$
$89$ $$(T^{2} + 13\!\cdots\!32)^{2}$$
$97$ $$(T^{2} - 40571617 T + 16\!\cdots\!89)^{2}$$