# Properties

 Label 162.7.d.f Level $162$ Weight $7$ Character orbit 162.d Analytic conductor $37.269$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,7,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, 7, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$37.2687615464$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 4 \beta_{3} q^{2} + ( - 32 \beta_1 + 32) q^{4} + ( - 25 \beta_{6} + 95 \beta_{5}) q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 128 \beta_{5} + 128 \beta_{3}) q^{8}+O(q^{10})$$ q + 4*b3 * q^2 + (-32*b1 + 32) * q^4 + (-25*b6 + 95*b5) * q^5 + (b2 + b1) * q^7 + (-128*b5 + 128*b3) * q^8 $$q + 4 \beta_{3} q^{2} + ( - 32 \beta_1 + 32) q^{4} + ( - 25 \beta_{6} + 95 \beta_{5}) q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 128 \beta_{5} + 128 \beta_{3}) q^{8} + ( - 100 \beta_{4} + 660) q^{10} + (290 \beta_{7} - 290 \beta_{6} + 991 \beta_{3}) q^{11} + (362 \beta_{4} - 362 \beta_{2} + \cdots - 959) q^{13}+ \cdots + (16 \beta_{7} + \cdots + 470492 \beta_{3}) q^{98}+O(q^{100})$$ q + 4*b3 * q^2 + (-32*b1 + 32) * q^4 + (-25*b6 + 95*b5) * q^5 + (b2 + b1) * q^7 + (-128*b5 + 128*b3) * q^8 + (-100*b4 + 660) * q^10 + (290*b7 - 290*b6 + 991*b3) * q^11 + (362*b4 - 362*b2 + 959*b1 - 959) * q^13 + 8*b6 * q^14 - 1024*b1 * q^16 + (-193*b7 - 1795*b5 + 1795*b3) * q^17 + (547*b4 + 2561) * q^19 + (800*b7 - 800*b6 + 3040*b3) * q^20 + (-1160*b4 + 1160*b2 - 6768*b1 + 6768) * q^22 + (-2600*b6 + 6385*b5) * q^23 + (-4125*b2 + 6425*b1) * q^25 + (-2896*b7 + 5284*b5 - 5284*b3) * q^26 + (32*b4 + 32) * q^28 + (-9805*b7 + 9805*b6 - 5672*b3) * q^29 + (-2826*b4 + 2826*b2 - 10024*b1 + 10024) * q^31 - 4096*b5 * q^32 + (-772*b2 - 15132*b1) * q^34 + (140*b7 - 325*b5 + 325*b3) * q^35 + (2303*b4 - 5050) * q^37 + (-4376*b7 + 4376*b6 + 8056*b3) * q^38 + (-3200*b4 + 3200*b2 - 21120*b1 + 21120) * q^40 + (-23146*b6 + 33629*b5) * q^41 + (-14799*b2 - 46235*b1) * q^43 + (9280*b7 - 31712*b5 + 31712*b3) * q^44 + (-10400*b4 + 40680) * q^46 + (-2502*b7 + 2502*b6 + 60168*b3) * q^47 + (-2*b4 + 2*b2 - 117621*b1 + 117621) * q^49 + (-33000*b6 + 42200*b5) * q^50 + (-11584*b2 + 30688*b1) * q^52 + (36874*b7 - 20681*b5 + 20681*b3) * q^53 + (-45075*b4 + 237465) * q^55 + (-256*b7 + 256*b6) * q^56 + (39220*b4 - 39220*b2 + 6156*b1 - 6156) * q^58 + (16858*b6 - 72788*b5) * q^59 + (-32409*b2 - 152264*b1) * q^61 + (22608*b7 - 51400*b5 + 51400*b3) * q^62 - 32768 * q^64 + (-83705*b7 + 83705*b6 - 243145*b3) * q^65 + (5307*b4 - 5307*b2 - 502243*b1 + 502243) * q^67 + (-6176*b6 - 57440*b5) * q^68 + (560*b2 - 2040*b1) * q^70 + (65072*b7 + 242831*b5 - 242831*b3) * q^71 + (-38535*b4 + 131456) * q^73 + (-18424*b7 + 18424*b6 - 29412*b3) * q^74 + (17504*b4 - 17504*b2 - 81952*b1 + 81952) * q^76 + (1402*b6 - 3770*b5) * q^77 + (-131623*b2 + 212179*b1) * q^79 + (25600*b7 - 97280*b5 + 97280*b3) * q^80 + (-92584*b4 + 176448) * q^82 + (-133176*b7 + 133176*b6 - 388866*b3) * q^83 + (-31365*b4 + 31365*b2 - 246960*b1 + 246960) * q^85 + (-118392*b6 - 125744*b5) * q^86 + (37120*b2 - 216576*b1) * q^88 + (245729*b7 - 43270*b5 + 43270*b3) * q^89 + (-597*b4 + 8815) * q^91 + (83200*b7 - 83200*b6 + 204320*b3) * q^92 + (10008*b4 - 10008*b2 - 491352*b1 + 491352) * q^94 + (26230*b6 + 13555*b5) * q^95 + (93770*b2 - 905432*b1) * q^97 + (16*b7 - 470492*b5 + 470492*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 128 q^{4} + 4 q^{7}+O(q^{10})$$ 8 * q + 128 * q^4 + 4 * q^7 $$8 q + 128 q^{4} + 4 q^{7} + 5280 q^{10} - 3836 q^{13} - 4096 q^{16} + 20488 q^{19} + 27072 q^{22} + 25700 q^{25} + 256 q^{28} + 40096 q^{31} - 60528 q^{34} - 40400 q^{37} + 84480 q^{40} - 184940 q^{43} + 325440 q^{46} + 470484 q^{49} + 122752 q^{52} + 1899720 q^{55} - 24624 q^{58} - 609056 q^{61} - 262144 q^{64} + 2008972 q^{67} - 8160 q^{70} + 1051648 q^{73} + 327808 q^{76} + 848716 q^{79} + 1411584 q^{82} + 987840 q^{85} - 866304 q^{88} + 70520 q^{91} + 1965408 q^{94} - 3621728 q^{97}+O(q^{100})$$ 8 * q + 128 * q^4 + 4 * q^7 + 5280 * q^10 - 3836 * q^13 - 4096 * q^16 + 20488 * q^19 + 27072 * q^22 + 25700 * q^25 + 256 * q^28 + 40096 * q^31 - 60528 * q^34 - 40400 * q^37 + 84480 * q^40 - 184940 * q^43 + 325440 * q^46 + 470484 * q^49 + 122752 * q^52 + 1899720 * q^55 - 24624 * q^58 - 609056 * q^61 - 262144 * q^64 + 2008972 * q^67 - 8160 * q^70 + 1051648 * q^73 + 327808 * q^76 + 848716 * q^79 + 1411584 * q^82 + 987840 * q^85 - 866304 * q^88 + 70520 * q^91 + 1965408 * q^94 - 3621728 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$3\zeta_{24}^{6} + 3\zeta_{24}^{2}$$ 3*v^6 + 3*v^2 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{4}$$ $$=$$ $$-3\zeta_{24}^{6} + 6\zeta_{24}^{2}$$ -3*v^6 + 6*v^2 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{6}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24}$$ v^7 - v^5 + 2*v^3 + 3*v $$\beta_{7}$$ $$=$$ $$3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 3*v^7 + 2*v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9$$ (b7 + b6 - b5 + 5*b3) / 9 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{2} ) / 9$$ (b4 + b2) / 9 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9$$ (-b7 + 2*b6 + 4*b5 - 5*b3) / 9 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9$$ (2*b7 - b6 + 4*b5 + b3) / 9 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{2} ) / 9$$ (-b4 + 2*b2) / 9 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9$$ (b7 + b6 - b5 - 4*b3) / 9

## Character values

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

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

## 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
 −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i
−4.89898 + 2.82843i 0 16.0000 27.7128i −180.591 104.264i 0 −2.09808 3.63397i 181.019i 0 1179.62
53.2 −4.89898 + 2.82843i 0 16.0000 27.7128i −21.4919 12.4084i 0 3.09808 + 5.36603i 181.019i 0 140.385
53.3 4.89898 2.82843i 0 16.0000 27.7128i 21.4919 + 12.4084i 0 3.09808 + 5.36603i 181.019i 0 140.385
53.4 4.89898 2.82843i 0 16.0000 27.7128i 180.591 + 104.264i 0 −2.09808 3.63397i 181.019i 0 1179.62
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −180.591 + 104.264i 0 −2.09808 + 3.63397i 181.019i 0 1179.62
107.2 −4.89898 2.82843i 0 16.0000 + 27.7128i −21.4919 + 12.4084i 0 3.09808 5.36603i 181.019i 0 140.385
107.3 4.89898 + 2.82843i 0 16.0000 + 27.7128i 21.4919 12.4084i 0 3.09808 5.36603i 181.019i 0 140.385
107.4 4.89898 + 2.82843i 0 16.0000 + 27.7128i 180.591 104.264i 0 −2.09808 + 3.63397i 181.019i 0 1179.62
 $$n$$: 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 162.7.d.f 8
3.b odd 2 1 inner 162.7.d.f 8
9.c even 3 1 162.7.b.a 4
9.c even 3 1 inner 162.7.d.f 8
9.d odd 6 1 162.7.b.a 4
9.d odd 6 1 inner 162.7.d.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.7.b.a 4 9.c even 3 1
162.7.b.a 4 9.d odd 6 1
162.7.d.f 8 1.a even 1 1 trivial
162.7.d.f 8 3.b odd 2 1 inner
162.7.d.f 8 9.c even 3 1 inner
162.7.d.f 8 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 44100T_{5}^{6} + 1918029375T_{5}^{4} - 1181025562500T_{5}^{2} + 717201875390625$$ acting on $$S_{7}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 32 T^{2} + 1024)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + \cdots + 717201875390625$$
$7$ $$(T^{4} - 2 T^{3} + \cdots + 676)^{2}$$
$11$ $$T^{8} + \cdots + 76\!\cdots\!76$$
$13$ $$(T^{4} + \cdots + 6856578909049)^{2}$$
$17$ $$(T^{4} + \cdots + 44258191098489)^{2}$$
$19$ $$(T^{2} - 5122 T - 1519922)^{4}$$
$23$ $$T^{8} + \cdots + 24\!\cdots\!00$$
$29$ $$T^{8} + \cdots + 28\!\cdots\!41$$
$31$ $$(T^{4} + \cdots + 13\!\cdots\!76)^{2}$$
$37$ $$(T^{2} + 10100 T - 117700343)^{4}$$
$41$ $$T^{8} + \cdots + 15\!\cdots\!96$$
$43$ $$(T^{4} + \cdots + 14\!\cdots\!04)^{2}$$
$47$ $$T^{8} + \cdots + 30\!\cdots\!76$$
$53$ $$(T^{4} + \cdots + 33\!\cdots\!16)^{2}$$
$59$ $$T^{8} + \cdots + 39\!\cdots\!16$$
$61$ $$(T^{4} + \cdots + 26\!\cdots\!81)^{2}$$
$67$ $$(T^{4} + \cdots + 63\!\cdots\!76)^{2}$$
$71$ $$(T^{4} + \cdots + 89\!\cdots\!36)^{2}$$
$73$ $$(T^{2} - 262912 T - 22812868139)^{4}$$
$79$ $$(T^{4} + \cdots + 17\!\cdots\!64)^{2}$$
$83$ $$T^{8} + \cdots + 10\!\cdots\!96$$
$89$ $$(T^{4} + \cdots + 64\!\cdots\!49)^{2}$$
$97$ $$(T^{4} + \cdots + 33\!\cdots\!76)^{2}$$