# Properties

 Label 162.9.d.f Level $162$ Weight $9$ Character orbit 162.d Analytic conductor $65.995$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.206763233181696.28 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441$$ x^8 - 80*x^6 + 4879*x^4 - 121680*x^2 + 2313441 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{18}\cdot 3^{12}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8}+O(q^{10})$$ q + b3 * q^2 - 128*b1 * q^4 + (-b4 + 21*b2) * q^5 + (b6 - b5 - 41*b1 - 41) * q^7 + (128*b3 - 128*b2) * q^8 $$q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8} + ( - 2 \beta_{5} + 2688) q^{10} + ( - 29 \beta_{7} + 21 \beta_{3}) q^{11} + (\beta_{6} + 24911 \beta_1) q^{13} + ( - 64 \beta_{4} - 41 \beta_{2}) q^{14} + ( - 16384 \beta_1 - 16384) q^{16} + ( - 53 \beta_{7} + 53 \beta_{4} - 8379 \beta_{3} + 8379 \beta_{2}) q^{17} + ( - 22 \beta_{5} + 128009) q^{19} + ( - 128 \beta_{7} + 2688 \beta_{3}) q^{20} + ( - 58 \beta_{6} - 2688 \beta_1) q^{22} + (259 \beta_{4} - 17583 \beta_{2}) q^{23} + (84 \beta_{6} - 84 \beta_{5} + 126551 \beta_1 + 126551) q^{25} + (64 \beta_{7} - 64 \beta_{4} - 24911 \beta_{3} + 24911 \beta_{2}) q^{26} + ( - 128 \beta_{5} - 5248) q^{28} + (1306 \beta_{7} + 38838 \beta_{3}) q^{29} + (204 \beta_{6} + 148886 \beta_1) q^{31} - 16384 \beta_{2} q^{32} + ( - 106 \beta_{6} + 106 \beta_{5} + 1072512 \beta_1 + 1072512) q^{34} + (1303 \beta_{7} - 1303 \beta_{4} - 229503 \beta_{3} + 229503 \beta_{2}) q^{35} + (505 \beta_{5} - 252745) q^{37} + ( - 1408 \beta_{7} + 128009 \beta_{3}) q^{38} + ( - 256 \beta_{6} - 344064 \beta_1) q^{40} + (338 \beta_{4} - 204162 \beta_{2}) q^{41} + ( - 588 \beta_{6} + 588 \beta_{5} + 2023546 \beta_1 + 2023546) q^{43} + ( - 3712 \beta_{7} + 3712 \beta_{4} + 2688 \beta_{3} - 2688 \beta_{2}) q^{44} + (518 \beta_{5} - 2250624) q^{46} + ( - 2253 \beta_{7} + 296697 \beta_{3}) q^{47} + ( - 82 \beta_{6} + 8980176 \beta_1) q^{49} + ( - 5376 \beta_{4} + 126551 \beta_{2}) q^{50} + (128 \beta_{6} - 128 \beta_{5} + 3188608 \beta_1 + 3188608) q^{52} + ( - 1990 \beta_{7} + 1990 \beta_{4} + 229086 \beta_{3} - 229086 \beta_{2}) q^{53} + ( - 1260 \beta_{5} + 13417560) q^{55} + ( - 8192 \beta_{7} - 5248 \beta_{3}) q^{56} + (2612 \beta_{6} - 4971264 \beta_1) q^{58} + (5527 \beta_{4} + 1402905 \beta_{2}) q^{59} + (1761 \beta_{6} - 1761 \beta_{5} - 1196159 \beta_1 - 1196159) q^{61} + (13056 \beta_{7} - 13056 \beta_{4} - 148886 \beta_{3} + 148886 \beta_{2}) q^{62} - 2097152 q^{64} + (26255 \beta_{7} - 753495 \beta_{3}) q^{65} + ( - 5748 \beta_{6} + 14289425 \beta_1) q^{67} + (6784 \beta_{4} + 1072512 \beta_{2}) q^{68} + (2606 \beta_{6} - 2606 \beta_{5} + 29376384 \beta_1 + 29376384) q^{70} + (1300 \beta_{7} - 1300 \beta_{4} - 387324 \beta_{3} + 387324 \beta_{2}) q^{71} + ( - 6066 \beta_{5} + 3309023) q^{73} + (32320 \beta_{7} - 252745 \beta_{3}) q^{74} + ( - 2816 \beta_{6} - 16385152 \beta_1) q^{76} + ( - 155 \beta_{4} + 6679695 \beta_{2}) q^{77} + ( - 4537 \beta_{6} + 4537 \beta_{5} - 12395921 \beta_1 - 12395921) q^{79} + ( - 16384 \beta_{7} + 16384 \beta_{4} + 344064 \beta_{3} - 344064 \beta_{2}) q^{80} + (676 \beta_{5} - 26132736) q^{82} + ( - 101538 \beta_{7} - 7206 \beta_{3}) q^{83} + (14532 \beta_{6} - 1895832 \beta_1) q^{85} + (37632 \beta_{4} + 2023546 \beta_{2}) q^{86} + ( - 7424 \beta_{6} + 7424 \beta_{5} - 344064 \beta_1 - 344064) q^{88} + ( - 14585 \beta_{7} + 14585 \beta_{4} - 6311223 \beta_{3} + 6311223 \beta_{2}) q^{89} + (24870 \beta_{5} - 13721945) q^{91} + (33152 \beta_{7} - 2250624 \beta_{3}) q^{92} + ( - 4506 \beta_{6} - 37977216 \beta_1) q^{94} + ( - 157577 \beta_{4} + 7756197 \beta_{2}) q^{95} + (14114 \beta_{6} - 14114 \beta_{5} - 102086231 \beta_1 - 102086231) q^{97} + ( - 5248 \beta_{7} + 5248 \beta_{4} - 8980176 \beta_{3} + 8980176 \beta_{2}) q^{98}+O(q^{100})$$ q + b3 * q^2 - 128*b1 * q^4 + (-b4 + 21*b2) * q^5 + (b6 - b5 - 41*b1 - 41) * q^7 + (128*b3 - 128*b2) * q^8 + (-2*b5 + 2688) * q^10 + (-29*b7 + 21*b3) * q^11 + (b6 + 24911*b1) * q^13 + (-64*b4 - 41*b2) * q^14 + (-16384*b1 - 16384) * q^16 + (-53*b7 + 53*b4 - 8379*b3 + 8379*b2) * q^17 + (-22*b5 + 128009) * q^19 + (-128*b7 + 2688*b3) * q^20 + (-58*b6 - 2688*b1) * q^22 + (259*b4 - 17583*b2) * q^23 + (84*b6 - 84*b5 + 126551*b1 + 126551) * q^25 + (64*b7 - 64*b4 - 24911*b3 + 24911*b2) * q^26 + (-128*b5 - 5248) * q^28 + (1306*b7 + 38838*b3) * q^29 + (204*b6 + 148886*b1) * q^31 - 16384*b2 * q^32 + (-106*b6 + 106*b5 + 1072512*b1 + 1072512) * q^34 + (1303*b7 - 1303*b4 - 229503*b3 + 229503*b2) * q^35 + (505*b5 - 252745) * q^37 + (-1408*b7 + 128009*b3) * q^38 + (-256*b6 - 344064*b1) * q^40 + (338*b4 - 204162*b2) * q^41 + (-588*b6 + 588*b5 + 2023546*b1 + 2023546) * q^43 + (-3712*b7 + 3712*b4 + 2688*b3 - 2688*b2) * q^44 + (518*b5 - 2250624) * q^46 + (-2253*b7 + 296697*b3) * q^47 + (-82*b6 + 8980176*b1) * q^49 + (-5376*b4 + 126551*b2) * q^50 + (128*b6 - 128*b5 + 3188608*b1 + 3188608) * q^52 + (-1990*b7 + 1990*b4 + 229086*b3 - 229086*b2) * q^53 + (-1260*b5 + 13417560) * q^55 + (-8192*b7 - 5248*b3) * q^56 + (2612*b6 - 4971264*b1) * q^58 + (5527*b4 + 1402905*b2) * q^59 + (1761*b6 - 1761*b5 - 1196159*b1 - 1196159) * q^61 + (13056*b7 - 13056*b4 - 148886*b3 + 148886*b2) * q^62 - 2097152 * q^64 + (26255*b7 - 753495*b3) * q^65 + (-5748*b6 + 14289425*b1) * q^67 + (6784*b4 + 1072512*b2) * q^68 + (2606*b6 - 2606*b5 + 29376384*b1 + 29376384) * q^70 + (1300*b7 - 1300*b4 - 387324*b3 + 387324*b2) * q^71 + (-6066*b5 + 3309023) * q^73 + (32320*b7 - 252745*b3) * q^74 + (-2816*b6 - 16385152*b1) * q^76 + (-155*b4 + 6679695*b2) * q^77 + (-4537*b6 + 4537*b5 - 12395921*b1 - 12395921) * q^79 + (-16384*b7 + 16384*b4 + 344064*b3 - 344064*b2) * q^80 + (676*b5 - 26132736) * q^82 + (-101538*b7 - 7206*b3) * q^83 + (14532*b6 - 1895832*b1) * q^85 + (37632*b4 + 2023546*b2) * q^86 + (-7424*b6 + 7424*b5 - 344064*b1 - 344064) * q^88 + (-14585*b7 + 14585*b4 - 6311223*b3 + 6311223*b2) * q^89 + (24870*b5 - 13721945) * q^91 + (33152*b7 - 2250624*b3) * q^92 + (-4506*b6 - 37977216*b1) * q^94 + (-157577*b4 + 7756197*b2) * q^95 + (14114*b6 - 14114*b5 - 102086231*b1 - 102086231) * q^97 + (-5248*b7 + 5248*b4 - 8980176*b3 + 8980176*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 512 q^{4} - 164 q^{7}+O(q^{10})$$ 8 * q + 512 * q^4 - 164 * q^7 $$8 q + 512 q^{4} - 164 q^{7} + 21504 q^{10} - 99644 q^{13} - 65536 q^{16} + 1024072 q^{19} + 10752 q^{22} + 506204 q^{25} - 41984 q^{28} - 595544 q^{31} + 4290048 q^{34} - 2021960 q^{37} + 1376256 q^{40} + 8094184 q^{43} - 18004992 q^{46} - 35920704 q^{49} + 12754432 q^{52} + 107340480 q^{55} + 19885056 q^{58} - 4784636 q^{61} - 16777216 q^{64} - 57157700 q^{67} + 117505536 q^{70} + 26472184 q^{73} + 65540608 q^{76} - 49583684 q^{79} - 209061888 q^{82} + 7583328 q^{85} - 1376256 q^{88} - 109775560 q^{91} + 151908864 q^{94} - 408344924 q^{97}+O(q^{100})$$ 8 * q + 512 * q^4 - 164 * q^7 + 21504 * q^10 - 99644 * q^13 - 65536 * q^16 + 1024072 * q^19 + 10752 * q^22 + 506204 * q^25 - 41984 * q^28 - 595544 * q^31 + 4290048 * q^34 - 2021960 * q^37 + 1376256 * q^40 + 8094184 * q^43 - 18004992 * q^46 - 35920704 * q^49 + 12754432 * q^52 + 107340480 * q^55 + 19885056 * q^58 - 4784636 * q^61 - 16777216 * q^64 - 57157700 * q^67 + 117505536 * q^70 + 26472184 * q^73 + 65540608 * q^76 - 49583684 * q^79 - 209061888 * q^82 + 7583328 * q^85 - 1376256 * q^88 - 109775560 * q^91 + 151908864 * q^94 - 408344924 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441$$ :

 $$\beta_{1}$$ $$=$$ $$( 80\nu^{6} - 4879\nu^{4} + 390320\nu^{2} - 9734400 ) / 7420959$$ (80*v^6 - 4879*v^4 + 390320*v^2 - 9734400) / 7420959 $$\beta_{2}$$ $$=$$ $$( -8\nu^{7} - 626872\nu ) / 190281$$ (-8*v^7 - 626872*v) / 190281 $$\beta_{3}$$ $$=$$ $$( 14072\nu^{7} - 1600312\nu^{5} + 68657288\nu^{3} - 1712280960\nu ) / 289417401$$ (14072*v^7 - 1600312*v^5 + 68657288*v^3 - 1712280960*v) / 289417401 $$\beta_{4}$$ $$=$$ $$( -18\nu^{7} - 8260578\nu ) / 63427$$ (-18*v^7 - 8260578*v) / 63427 $$\beta_{5}$$ $$=$$ $$( 432\nu^{6} + 31743360 ) / 4879$$ (432*v^6 + 31743360) / 4879 $$\beta_{6}$$ $$=$$ $$( -80592\nu^{6} + 9367680\nu^{4} - 393208368\nu^{2} + 9806434560 ) / 824551$$ (-80592*v^6 + 9367680*v^4 - 393208368*v^2 + 9806434560) / 824551 $$\beta_{7}$$ $$=$$ $$( 15998\nu^{7} - 1161202\nu^{5} + 78054242\nu^{3} - 1946636640\nu ) / 10719163$$ (15998*v^7 - 1161202*v^5 + 78054242*v^3 - 1946636640*v) / 10719163
 $$\nu$$ $$=$$ $$( -4\beta_{4} + 27\beta_{2} ) / 432$$ (-4*b4 + 27*b2) / 432 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + 17280\beta _1 + 17280 ) / 432$$ (b6 - b5 + 17280*b1 + 17280) / 432 $$\nu^{3}$$ $$=$$ $$( 164\beta_{7} - 164\beta_{4} - 3213\beta_{3} + 3213\beta_{2} ) / 432$$ (164*b7 - 164*b4 - 3213*b3 + 3213*b2) / 432 $$\nu^{4}$$ $$=$$ $$( 5\beta_{6} + 45333\beta_1 ) / 27$$ (5*b6 + 45333*b1) / 27 $$\nu^{5}$$ $$=$$ $$( 7036\beta_{7} - 215973\beta_{3} ) / 432$$ (7036*b7 - 215973*b3) / 432 $$\nu^{6}$$ $$=$$ $$( 4879\beta_{5} - 31743360 ) / 432$$ (4879*b5 - 31743360) / 432 $$\nu^{7}$$ $$=$$ $$( 313436\beta_{4} - 12390867\beta_{2} ) / 432$$ (313436*b4 - 12390867*b2) / 432

## Character values

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

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

## 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
 −6.05526 − 3.49600i 4.83051 + 2.78890i −4.83051 − 2.78890i 6.05526 + 3.49600i −6.05526 + 3.49600i 4.83051 − 2.78890i −4.83051 + 2.78890i 6.05526 − 3.49600i
−9.79796 + 5.65685i 0 64.0000 110.851i −793.589 458.179i 0 1899.35 + 3289.77i 1448.15i 0 10367.4
53.2 −9.79796 + 5.65685i 0 64.0000 110.851i 382.074 + 220.591i 0 −1940.35 3360.78i 1448.15i 0 −4991.40
53.3 9.79796 5.65685i 0 64.0000 110.851i −382.074 220.591i 0 −1940.35 3360.78i 1448.15i 0 −4991.40
53.4 9.79796 5.65685i 0 64.0000 110.851i 793.589 + 458.179i 0 1899.35 + 3289.77i 1448.15i 0 10367.4
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i −793.589 + 458.179i 0 1899.35 3289.77i 1448.15i 0 10367.4
107.2 −9.79796 5.65685i 0 64.0000 + 110.851i 382.074 220.591i 0 −1940.35 + 3360.78i 1448.15i 0 −4991.40
107.3 9.79796 + 5.65685i 0 64.0000 + 110.851i −382.074 + 220.591i 0 −1940.35 + 3360.78i 1448.15i 0 −4991.40
107.4 9.79796 + 5.65685i 0 64.0000 + 110.851i 793.589 458.179i 0 1899.35 3289.77i 1448.15i 0 10367.4
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.4 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.f 8
3.b odd 2 1 inner 162.9.d.f 8
9.c even 3 1 54.9.b.b 4
9.c even 3 1 inner 162.9.d.f 8
9.d odd 6 1 54.9.b.b 4
9.d odd 6 1 inner 162.9.d.f 8
36.f odd 6 1 432.9.e.i 4
36.h even 6 1 432.9.e.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.b 4 9.c even 3 1
54.9.b.b 4 9.d odd 6 1
162.9.d.f 8 1.a even 1 1 trivial
162.9.d.f 8 3.b odd 2 1 inner
162.9.d.f 8 9.c even 3 1 inner
162.9.d.f 8 9.d odd 6 1 inner
432.9.e.i 4 36.f odd 6 1
432.9.e.i 4 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 1034352T_{5}^{6} + 906441741504T_{5}^{4} - 169056888921676800T_{5}^{2} + 26713391443966978560000$$ acting on $$S_{9}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 128 T^{2} + 16384)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 1034352 T^{6} + \cdots + 26\!\cdots\!00$$
$7$ $$(T^{4} + 82 T^{3} + \cdots + 217315212808225)^{2}$$
$11$ $$T^{8} - 775057392 T^{6} + \cdots + 22\!\cdots\!00$$
$13$ $$(T^{4} + 49822 T^{3} + \cdots + 36\!\cdots\!25)^{2}$$
$17$ $$(T^{4} + 20561526000 T^{2} + \cdots + 59\!\cdots\!16)^{2}$$
$19$ $$(T^{2} - 256018 T + 9250548817)^{4}$$
$23$ $$T^{8} - 140957633520 T^{6} + \cdots + 56\!\cdots\!76$$
$29$ $$T^{8} - 1957816428480 T^{6} + \cdots + 12\!\cdots\!76$$
$31$ $$(T^{4} + 297772 T^{3} + \cdots + 34\!\cdots\!00)^{2}$$
$37$ $$(T^{2} + 505490 T - 3696029027375)^{4}$$
$41$ $$T^{8} - 10775894113728 T^{6} + \cdots + 77\!\cdots\!00$$
$43$ $$(T^{4} - 4047092 T^{3} + \cdots + 10\!\cdots\!64)^{2}$$
$47$ $$T^{8} - 27212771060208 T^{6} + \cdots + 63\!\cdots\!00$$
$53$ $$(T^{4} + 17084039126976 T^{2} + \cdots + 23\!\cdots\!44)^{2}$$
$59$ $$T^{8} - 531992852563824 T^{6} + \cdots + 32\!\cdots\!36$$
$61$ $$(T^{4} + 2392318 T^{3} + \cdots + 19\!\cdots\!25)^{2}$$
$67$ $$(T^{4} + 28578850 T^{3} + \cdots + 80\!\cdots\!81)^{2}$$
$71$ $$(T^{4} + 39962350169856 T^{2} + \cdots + 33\!\cdots\!84)^{2}$$
$73$ $$(T^{2} - 6618046 T - 531549935014847)^{4}$$
$79$ $$(T^{4} + 24791842 T^{3} + \cdots + 22\!\cdots\!89)^{2}$$
$83$ $$T^{8} + \cdots + 50\!\cdots\!76$$
$89$ $$(T^{4} + \cdots + 25\!\cdots\!44)^{2}$$
$97$ $$(T^{4} + 204172462 T^{3} + \cdots + 56\!\cdots\!25)^{2}$$