# Properties

 Label 162.7.d.b Level $162$ Weight $7$ Character orbit 162.d Analytic conductor $37.269$ 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] = [162,7,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: $$4$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 30 \beta_1 q^{5} - 2 \beta_{2} q^{7} - 32 \beta_{3} q^{8}+O(q^{10})$$ q + (-b3 + b1) * q^2 + (-32*b2 + 32) * q^4 + 30*b1 * q^5 - 2*b2 * q^7 - 32*b3 * q^8 $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 30 \beta_1 q^{5} - 2 \beta_{2} q^{7} - 32 \beta_{3} q^{8} + 960 q^{10} + (6 \beta_{3} - 6 \beta_1) q^{11} + ( - 2950 \beta_{2} + 2950) q^{13} - 2 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 792 \beta_{3} q^{17} + 5258 q^{19} + ( - 960 \beta_{3} + 960 \beta_1) q^{20} + (192 \beta_{2} - 192) q^{22} + 1812 \beta_1 q^{23} + 13175 \beta_{2} q^{25} - 2950 \beta_{3} q^{26} - 64 q^{28} + ( - 390 \beta_{3} + 390 \beta_1) q^{29} + (22898 \beta_{2} - 22898) q^{31} - 1024 \beta_1 q^{32} + 25344 \beta_{2} q^{34} - 60 \beta_{3} q^{35} + 34058 q^{37} + ( - 5258 \beta_{3} + 5258 \beta_1) q^{38} + ( - 30720 \beta_{2} + 30720) q^{40} + 2964 \beta_1 q^{41} + 6406 \beta_{2} q^{43} + 192 \beta_{3} q^{44} + 57984 q^{46} + ( - 31800 \beta_{3} + 31800 \beta_1) q^{47} + ( - 117645 \beta_{2} + 117645) q^{49} + 13175 \beta_1 q^{50} - 94400 \beta_{2} q^{52} - 34038 \beta_{3} q^{53} - 5760 q^{55} + (64 \beta_{3} - 64 \beta_1) q^{56} + ( - 12480 \beta_{2} + 12480) q^{58} + 57774 \beta_1 q^{59} + 62566 \beta_{2} q^{61} + 22898 \beta_{3} q^{62} - 32768 q^{64} + ( - 88500 \beta_{3} + 88500 \beta_1) q^{65} + (438698 \beta_{2} - 438698) q^{67} + 25344 \beta_1 q^{68} - 1920 \beta_{2} q^{70} - 12060 \beta_{3} q^{71} - 730510 q^{73} + ( - 34058 \beta_{3} + 34058 \beta_1) q^{74} + ( - 168256 \beta_{2} + 168256) q^{76} + 12 \beta_1 q^{77} - 340562 \beta_{2} q^{79} - 30720 \beta_{3} q^{80} + 94848 q^{82} + ( - 87726 \beta_{3} + 87726 \beta_1) q^{83} + (760320 \beta_{2} - 760320) q^{85} + 6406 \beta_1 q^{86} + 6144 \beta_{2} q^{88} - 68364 \beta_{3} q^{89} - 5900 q^{91} + ( - 57984 \beta_{3} + 57984 \beta_1) q^{92} + ( - 1017600 \beta_{2} + 1017600) q^{94} + 157740 \beta_1 q^{95} + 281086 \beta_{2} q^{97} - 117645 \beta_{3} q^{98}+O(q^{100})$$ q + (-b3 + b1) * q^2 + (-32*b2 + 32) * q^4 + 30*b1 * q^5 - 2*b2 * q^7 - 32*b3 * q^8 + 960 * q^10 + (6*b3 - 6*b1) * q^11 + (-2950*b2 + 2950) * q^13 - 2*b1 * q^14 - 1024*b2 * q^16 + 792*b3 * q^17 + 5258 * q^19 + (-960*b3 + 960*b1) * q^20 + (192*b2 - 192) * q^22 + 1812*b1 * q^23 + 13175*b2 * q^25 - 2950*b3 * q^26 - 64 * q^28 + (-390*b3 + 390*b1) * q^29 + (22898*b2 - 22898) * q^31 - 1024*b1 * q^32 + 25344*b2 * q^34 - 60*b3 * q^35 + 34058 * q^37 + (-5258*b3 + 5258*b1) * q^38 + (-30720*b2 + 30720) * q^40 + 2964*b1 * q^41 + 6406*b2 * q^43 + 192*b3 * q^44 + 57984 * q^46 + (-31800*b3 + 31800*b1) * q^47 + (-117645*b2 + 117645) * q^49 + 13175*b1 * q^50 - 94400*b2 * q^52 - 34038*b3 * q^53 - 5760 * q^55 + (64*b3 - 64*b1) * q^56 + (-12480*b2 + 12480) * q^58 + 57774*b1 * q^59 + 62566*b2 * q^61 + 22898*b3 * q^62 - 32768 * q^64 + (-88500*b3 + 88500*b1) * q^65 + (438698*b2 - 438698) * q^67 + 25344*b1 * q^68 - 1920*b2 * q^70 - 12060*b3 * q^71 - 730510 * q^73 + (-34058*b3 + 34058*b1) * q^74 + (-168256*b2 + 168256) * q^76 + 12*b1 * q^77 - 340562*b2 * q^79 - 30720*b3 * q^80 + 94848 * q^82 + (-87726*b3 + 87726*b1) * q^83 + (760320*b2 - 760320) * q^85 + 6406*b1 * q^86 + 6144*b2 * q^88 - 68364*b3 * q^89 - 5900 * q^91 + (-57984*b3 + 57984*b1) * q^92 + (-1017600*b2 + 1017600) * q^94 + 157740*b1 * q^95 + 281086*b2 * q^97 - 117645*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 64 q^{4} - 4 q^{7}+O(q^{10})$$ 4 * q + 64 * q^4 - 4 * q^7 $$4 q + 64 q^{4} - 4 q^{7} + 3840 q^{10} + 5900 q^{13} - 2048 q^{16} + 21032 q^{19} - 384 q^{22} + 26350 q^{25} - 256 q^{28} - 45796 q^{31} + 50688 q^{34} + 136232 q^{37} + 61440 q^{40} + 12812 q^{43} + 231936 q^{46} + 235290 q^{49} - 188800 q^{52} - 23040 q^{55} + 24960 q^{58} + 125132 q^{61} - 131072 q^{64} - 877396 q^{67} - 3840 q^{70} - 2922040 q^{73} + 336512 q^{76} - 681124 q^{79} + 379392 q^{82} - 1520640 q^{85} + 12288 q^{88} - 23600 q^{91} + 2035200 q^{94} + 562172 q^{97}+O(q^{100})$$ 4 * q + 64 * q^4 - 4 * q^7 + 3840 * q^10 + 5900 * q^13 - 2048 * q^16 + 21032 * q^19 - 384 * q^22 + 26350 * q^25 - 256 * q^28 - 45796 * q^31 + 50688 * q^34 + 136232 * q^37 + 61440 * q^40 + 12812 * q^43 + 231936 * q^46 + 235290 * q^49 - 188800 * q^52 - 23040 * q^55 + 24960 * q^58 + 125132 * q^61 - 131072 * q^64 - 877396 * q^67 - 3840 * q^70 - 2922040 * q^73 + 336512 * q^76 - 681124 * q^79 + 379392 * q^82 - 1520640 * q^85 + 12288 * q^88 - 23600 * q^91 + 2035200 * q^94 + 562172 * q^97

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

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

## 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
−4.89898 + 2.82843i 0 16.0000 27.7128i −146.969 84.8528i 0 −1.00000 1.73205i 181.019i 0 960.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i 146.969 + 84.8528i 0 −1.00000 1.73205i 181.019i 0 960.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −146.969 + 84.8528i 0 −1.00000 + 1.73205i 181.019i 0 960.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i 146.969 84.8528i 0 −1.00000 + 1.73205i 181.019i 0 960.000
 $$n$$: 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 162.7.d.b 4
3.b odd 2 1 inner 162.7.d.b 4
9.c even 3 1 6.7.b.a 2
9.c even 3 1 inner 162.7.d.b 4
9.d odd 6 1 6.7.b.a 2
9.d odd 6 1 inner 162.7.d.b 4
36.f odd 6 1 48.7.e.b 2
36.h even 6 1 48.7.e.b 2
45.h odd 6 1 150.7.d.a 2
45.j even 6 1 150.7.d.a 2
45.k odd 12 2 150.7.b.a 4
45.l even 12 2 150.7.b.a 4
63.l odd 6 1 294.7.b.a 2
63.o even 6 1 294.7.b.a 2
72.j odd 6 1 192.7.e.c 2
72.l even 6 1 192.7.e.f 2
72.n even 6 1 192.7.e.c 2
72.p odd 6 1 192.7.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 9.c even 3 1
6.7.b.a 2 9.d odd 6 1
48.7.e.b 2 36.f odd 6 1
48.7.e.b 2 36.h even 6 1
150.7.b.a 4 45.k odd 12 2
150.7.b.a 4 45.l even 12 2
150.7.d.a 2 45.h odd 6 1
150.7.d.a 2 45.j even 6 1
162.7.d.b 4 1.a even 1 1 trivial
162.7.d.b 4 3.b odd 2 1 inner
162.7.d.b 4 9.c even 3 1 inner
162.7.d.b 4 9.d odd 6 1 inner
192.7.e.c 2 72.j odd 6 1
192.7.e.c 2 72.n even 6 1
192.7.e.f 2 72.l even 6 1
192.7.e.f 2 72.p odd 6 1
294.7.b.a 2 63.l odd 6 1
294.7.b.a 2 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 28800T_{5}^{2} + 829440000$$ acting on $$S_{7}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 32T^{2} + 1024$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 28800 T^{2} + \cdots + 829440000$$
$7$ $$(T^{2} + 2 T + 4)^{2}$$
$11$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$13$ $$(T^{2} - 2950 T + 8702500)^{2}$$
$17$ $$(T^{2} + 20072448)^{2}$$
$19$ $$(T - 5258)^{4}$$
$23$ $$T^{4} - 105067008 T^{2} + \cdots + 11\!\cdots\!64$$
$29$ $$T^{4} - 4867200 T^{2} + \cdots + 23689635840000$$
$31$ $$(T^{2} + 22898 T + 524318404)^{2}$$
$37$ $$(T - 34058)^{4}$$
$41$ $$T^{4} - 281129472 T^{2} + \cdots + 79\!\cdots\!84$$
$43$ $$(T^{2} - 6406 T + 41036836)^{2}$$
$47$ $$T^{4} - 32359680000 T^{2} + \cdots + 10\!\cdots\!00$$
$53$ $$(T^{2} + 37074734208)^{2}$$
$59$ $$T^{4} - 106810722432 T^{2} + \cdots + 11\!\cdots\!24$$
$61$ $$(T^{2} - 62566 T + 3914504356)^{2}$$
$67$ $$(T^{2} + 438698 T + 192455935204)^{2}$$
$71$ $$(T^{2} + 4654195200)^{2}$$
$73$ $$(T + 730510)^{4}$$
$79$ $$(T^{2} + 340562 T + 115982475844)^{2}$$
$83$ $$T^{4} - 246267234432 T^{2} + \cdots + 60\!\cdots\!24$$
$89$ $$(T^{2} + 149556367872)^{2}$$
$97$ $$(T^{2} - 281086 T + 79009339396)^{2}$$