# Properties

 Label 1620.2.x.b Level $1620$ Weight $2$ Character orbit 1620.x Analytic conductor $12.936$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(53,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 10, 9]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1620.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.x (of order $$12$$, degree $$4$$, 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: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.12960000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ x^8 - 3*x^6 + 8*x^4 - 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{4} - \beta_1) q^{7}+O(q^{10})$$ q + (-b6 + b3) * q^5 + (-b4 - b1) * q^7 $$q + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} - 2 \beta_{2} q^{11} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1 + 3) q^{13} + ( - \beta_{7} - \beta_{3}) q^{17} - 2 \beta_{5} q^{19} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2}) q^{23} + 5 \beta_{4} q^{25} - 2 \beta_{6} q^{29} + ( - 4 \beta_{4} - 4) q^{31} + ( - \beta_{7} + \beta_{3}) q^{35} + ( - 3 \beta_{5} - 3) q^{37} + (4 \beta_{7} - 4 \beta_{2}) q^{41} + ( - 3 \beta_{4} + 3 \beta_1) q^{43} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{47} + (5 \beta_{5} + 5 \beta_1) q^{49} + (\beta_{7} - \beta_{3}) q^{53} + 10 \beta_{5} q^{55} + ( - 4 \beta_{6} + 4 \beta_{3}) q^{59} - 6 \beta_{4} q^{61} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{65} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{67} - 2 \beta_{7} q^{71} + ( - \beta_{5} + 1) q^{73} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2}) q^{77} - 6 \beta_1 q^{79} + (3 \beta_{6} - 3 \beta_{2}) q^{83} + (5 \beta_{5} - 5 \beta_{4} + 5 \beta_1 - 5) q^{85} - 2 \beta_{3} q^{89} + 6 q^{91} + (2 \beta_{7} - 2 \beta_{2}) q^{95} + (9 \beta_{4} + 9 \beta_1) q^{97}+O(q^{100})$$ q + (-b6 + b3) * q^5 + (-b4 - b1) * q^7 - 2*b2 * q^11 + (-3*b5 + 3*b4 - 3*b1 + 3) * q^13 + (-b7 - b3) * q^17 - 2*b5 * q^19 + (b7 + b6 - b3 - b2) * q^23 + 5*b4 * q^25 - 2*b6 * q^29 + (-4*b4 - 4) * q^31 + (-b7 + b3) * q^35 + (-3*b5 - 3) * q^37 + (4*b7 - 4*b2) * q^41 + (-3*b4 + 3*b1) * q^43 + (-3*b6 - 3*b2) * q^47 + (5*b5 + 5*b1) * q^49 + (b7 - b3) * q^53 + 10*b5 * q^55 + (-4*b6 + 4*b3) * q^59 - 6*b4 * q^61 + (-3*b6 - 3*b2) * q^65 + (b5 + b4 + b1 + 1) * q^67 - 2*b7 * q^71 + (-b5 + 1) * q^73 + (2*b7 - 2*b6 + 2*b3 - 2*b2) * q^77 - 6*b1 * q^79 + (3*b6 - 3*b2) * q^83 + (5*b5 - 5*b4 + 5*b1 - 5) * q^85 - 2*b3 * q^89 + 6 * q^91 + (2*b7 - 2*b2) * q^95 + (9*b4 + 9*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{7}+O(q^{10})$$ 8 * q + 4 * q^7 $$8 q + 4 q^{7} + 12 q^{13} - 20 q^{25} - 16 q^{31} - 24 q^{37} + 12 q^{43} + 24 q^{61} + 4 q^{67} + 8 q^{73} - 20 q^{85} + 48 q^{91} - 36 q^{97}+O(q^{100})$$ 8 * q + 4 * q^7 + 12 * q^13 - 20 * q^25 - 16 * q^31 - 24 * q^37 + 12 * q^43 + 24 * q^61 + 4 * q^67 + 8 * q^73 - 20 * q^85 + 48 * q^91 - 36 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 8$$ (v^7 + 13*v) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 29\nu ) / 8$$ (v^7 + 29*v) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 9 ) / 4$$ (v^6 + 9) / 4 $$\beta_{4}$$ $$=$$ $$( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 9 ) / 8$$ (3*v^6 - 8*v^4 + 24*v^2 - 9) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4$$ (-3*v^7 + 8*v^5 - 20*v^3 + v) / 4 $$\beta_{6}$$ $$=$$ $$( -7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 21 ) / 8$$ (-7*v^6 + 24*v^4 - 56*v^2 + 21) / 8 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2$$ (3*v^7 - 8*v^5 + 22*v^3 - v) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{6} + 3\beta_{4} - \beta_{3} + 3 ) / 2$$ (b6 + 3*b4 - b3 + 3) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{5}$$ b7 + 2*b5 $$\nu^{4}$$ $$=$$ $$( 3\beta_{6} + 7\beta_{4} ) / 2$$ (3*b6 + 7*b4) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} + 11\beta_{5} - 5\beta_{2} + 11\beta_1 ) / 2$$ (5*b7 + 11*b5 - 5*b2 + 11*b1) / 2 $$\nu^{6}$$ $$=$$ $$4\beta_{3} - 9$$ 4*b3 - 9 $$\nu^{7}$$ $$=$$ $$( -13\beta_{2} + 29\beta_1 ) / 2$$ (-13*b2 + 29*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$ $$1 + \beta_{4}$$

## 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
 1.40126 − 0.809017i −0.535233 + 0.309017i −1.40126 + 0.809017i 0.535233 − 0.309017i −1.40126 − 0.809017i 0.535233 + 0.309017i 1.40126 + 0.809017i −0.535233 − 0.309017i
0 0 0 −1.11803 + 1.93649i 0 1.36603 + 0.366025i 0 0 0
53.2 0 0 0 1.11803 1.93649i 0 1.36603 + 0.366025i 0 0 0
377.1 0 0 0 −1.11803 + 1.93649i 0 −0.366025 + 1.36603i 0 0 0
377.2 0 0 0 1.11803 1.93649i 0 −0.366025 + 1.36603i 0 0 0
593.1 0 0 0 −1.11803 1.93649i 0 −0.366025 1.36603i 0 0 0
593.2 0 0 0 1.11803 + 1.93649i 0 −0.366025 1.36603i 0 0 0
917.1 0 0 0 −1.11803 1.93649i 0 1.36603 0.366025i 0 0 0
917.2 0 0 0 1.11803 + 1.93649i 0 1.36603 0.366025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.2 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
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.b 8
3.b odd 2 1 inner 1620.2.x.b 8
5.c odd 4 1 inner 1620.2.x.b 8
9.c even 3 1 60.2.i.a 4
9.c even 3 1 inner 1620.2.x.b 8
9.d odd 6 1 60.2.i.a 4
9.d odd 6 1 inner 1620.2.x.b 8
15.e even 4 1 inner 1620.2.x.b 8
36.f odd 6 1 240.2.v.b 4
36.h even 6 1 240.2.v.b 4
45.h odd 6 1 300.2.i.a 4
45.j even 6 1 300.2.i.a 4
45.k odd 12 1 60.2.i.a 4
45.k odd 12 1 300.2.i.a 4
45.k odd 12 1 inner 1620.2.x.b 8
45.l even 12 1 60.2.i.a 4
45.l even 12 1 300.2.i.a 4
45.l even 12 1 inner 1620.2.x.b 8
72.j odd 6 1 960.2.v.e 4
72.l even 6 1 960.2.v.h 4
72.n even 6 1 960.2.v.e 4
72.p odd 6 1 960.2.v.h 4
180.n even 6 1 1200.2.v.i 4
180.p odd 6 1 1200.2.v.i 4
180.v odd 12 1 240.2.v.b 4
180.v odd 12 1 1200.2.v.i 4
180.x even 12 1 240.2.v.b 4
180.x even 12 1 1200.2.v.i 4
360.bo even 12 1 960.2.v.h 4
360.br even 12 1 960.2.v.e 4
360.bt odd 12 1 960.2.v.h 4
360.bu odd 12 1 960.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 9.c even 3 1
60.2.i.a 4 9.d odd 6 1
60.2.i.a 4 45.k odd 12 1
60.2.i.a 4 45.l even 12 1
240.2.v.b 4 36.f odd 6 1
240.2.v.b 4 36.h even 6 1
240.2.v.b 4 180.v odd 12 1
240.2.v.b 4 180.x even 12 1
300.2.i.a 4 45.h odd 6 1
300.2.i.a 4 45.j even 6 1
300.2.i.a 4 45.k odd 12 1
300.2.i.a 4 45.l even 12 1
960.2.v.e 4 72.j odd 6 1
960.2.v.e 4 72.n even 6 1
960.2.v.e 4 360.br even 12 1
960.2.v.e 4 360.bu odd 12 1
960.2.v.h 4 72.l even 6 1
960.2.v.h 4 72.p odd 6 1
960.2.v.h 4 360.bo even 12 1
960.2.v.h 4 360.bt odd 12 1
1200.2.v.i 4 180.n even 6 1
1200.2.v.i 4 180.p odd 6 1
1200.2.v.i 4 180.v odd 12 1
1200.2.v.i 4 180.x even 12 1
1620.2.x.b 8 1.a even 1 1 trivial
1620.2.x.b 8 3.b odd 2 1 inner
1620.2.x.b 8 5.c odd 4 1 inner
1620.2.x.b 8 9.c even 3 1 inner
1620.2.x.b 8 9.d odd 6 1 inner
1620.2.x.b 8 15.e even 4 1 inner
1620.2.x.b 8 45.k odd 12 1 inner
1620.2.x.b 8 45.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 5 T^{2} + 25)^{2}$$
$7$ $$(T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2}$$
$11$ $$(T^{4} - 20 T^{2} + 400)^{2}$$
$13$ $$(T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2}$$
$17$ $$(T^{4} + 100)^{2}$$
$19$ $$(T^{2} + 4)^{4}$$
$23$ $$T^{8} - 100 T^{4} + 10000$$
$29$ $$(T^{4} + 20 T^{2} + 400)^{2}$$
$31$ $$(T^{2} + 4 T + 16)^{4}$$
$37$ $$(T^{2} + 6 T + 18)^{4}$$
$41$ $$(T^{4} - 80 T^{2} + 6400)^{2}$$
$43$ $$(T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2}$$
$47$ $$T^{8} - 8100 T^{4} + \cdots + 65610000$$
$53$ $$(T^{4} + 100)^{2}$$
$59$ $$(T^{4} + 80 T^{2} + 6400)^{2}$$
$61$ $$(T^{2} - 6 T + 36)^{4}$$
$67$ $$(T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2}$$
$71$ $$(T^{2} + 20)^{4}$$
$73$ $$(T^{2} - 2 T + 2)^{4}$$
$79$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$83$ $$T^{8} - 8100 T^{4} + \cdots + 65610000$$
$89$ $$(T^{2} - 20)^{4}$$
$97$ $$(T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 26244)^{2}$$