# Properties

 Label 1620.3.o.b Level $1620$ Weight $3$ Character orbit 1620.o Analytic conductor $44.142$ 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] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1620.o (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: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{5} + (2 \beta_{2} - 2) q^{7}+O(q^{10})$$ q + (b3 - b1) * q^5 + (2*b2 - 2) * q^7 $$q + (\beta_{3} - \beta_1) q^{5} + (2 \beta_{2} - 2) q^{7} - 6 \beta_1 q^{11} - 8 \beta_{2} q^{13} + 6 \beta_{3} q^{17} - 34 q^{19} + ( - 18 \beta_{3} + 18 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + 18 \beta_1 q^{29} - 14 \beta_{2} q^{31} - 2 \beta_{3} q^{35} + 56 q^{37} + (12 \beta_{3} - 12 \beta_1) q^{41} + (8 \beta_{2} - 8) q^{43} + 18 \beta_1 q^{47} + 45 \beta_{2} q^{49} - 18 \beta_{3} q^{53} + 30 q^{55} + ( - 6 \beta_{3} + 6 \beta_1) q^{59} + ( - 46 \beta_{2} + 46) q^{61} + 8 \beta_1 q^{65} - 32 \beta_{2} q^{67} + 24 \beta_{3} q^{71} - 106 q^{73} + ( - 12 \beta_{3} + 12 \beta_1) q^{77} + ( - 22 \beta_{2} + 22) q^{79} + 54 \beta_1 q^{83} - 30 \beta_{2} q^{85} + 48 \beta_{3} q^{89} + 16 q^{91} + ( - 34 \beta_{3} + 34 \beta_1) q^{95} + (122 \beta_{2} - 122) q^{97}+O(q^{100})$$ q + (b3 - b1) * q^5 + (2*b2 - 2) * q^7 - 6*b1 * q^11 - 8*b2 * q^13 + 6*b3 * q^17 - 34 * q^19 + (-18*b3 + 18*b1) * q^23 + (-5*b2 + 5) * q^25 + 18*b1 * q^29 - 14*b2 * q^31 - 2*b3 * q^35 + 56 * q^37 + (12*b3 - 12*b1) * q^41 + (8*b2 - 8) * q^43 + 18*b1 * q^47 + 45*b2 * q^49 - 18*b3 * q^53 + 30 * q^55 + (-6*b3 + 6*b1) * q^59 + (-46*b2 + 46) * q^61 + 8*b1 * q^65 - 32*b2 * q^67 + 24*b3 * q^71 - 106 * q^73 + (-12*b3 + 12*b1) * q^77 + (-22*b2 + 22) * q^79 + 54*b1 * q^83 - 30*b2 * q^85 + 48*b3 * q^89 + 16 * q^91 + (-34*b3 + 34*b1) * q^95 + (122*b2 - 122) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7}+O(q^{10})$$ 4 * q - 4 * q^7 $$4 q - 4 q^{7} - 16 q^{13} - 136 q^{19} + 10 q^{25} - 28 q^{31} + 224 q^{37} - 16 q^{43} + 90 q^{49} + 120 q^{55} + 92 q^{61} - 64 q^{67} - 424 q^{73} + 44 q^{79} - 60 q^{85} + 64 q^{91} - 244 q^{97}+O(q^{100})$$ 4 * q - 4 * q^7 - 16 * q^13 - 136 * q^19 + 10 * q^25 - 28 * q^31 + 224 * q^37 - 16 * q^43 + 90 * q^49 + 120 * q^55 + 92 * q^61 - 64 * q^67 - 424 * q^73 + 44 * q^79 - 60 * q^85 + 64 * q^91 - 244 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 1.93649 − 1.11803i −1.93649 + 1.11803i 1.93649 + 1.11803i −1.93649 − 1.11803i
0 0 0 −1.93649 1.11803i 0 −1.00000 1.73205i 0 0 0
701.2 0 0 0 1.93649 + 1.11803i 0 −1.00000 1.73205i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −1.00000 + 1.73205i 0 0 0
1241.2 0 0 0 1.93649 1.11803i 0 −1.00000 + 1.73205i 0 0 0
 $$n$$: 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 1620.3.o.b 4
3.b odd 2 1 inner 1620.3.o.b 4
9.c even 3 1 60.3.g.a 2
9.c even 3 1 inner 1620.3.o.b 4
9.d odd 6 1 60.3.g.a 2
9.d odd 6 1 inner 1620.3.o.b 4
36.f odd 6 1 240.3.l.a 2
36.h even 6 1 240.3.l.a 2
45.h odd 6 1 300.3.g.d 2
45.j even 6 1 300.3.g.d 2
45.k odd 12 2 300.3.b.c 4
45.l even 12 2 300.3.b.c 4
72.j odd 6 1 960.3.l.a 2
72.l even 6 1 960.3.l.d 2
72.n even 6 1 960.3.l.a 2
72.p odd 6 1 960.3.l.d 2
180.n even 6 1 1200.3.l.r 2
180.p odd 6 1 1200.3.l.r 2
180.v odd 12 2 1200.3.c.e 4
180.x even 12 2 1200.3.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 9.c even 3 1
60.3.g.a 2 9.d odd 6 1
240.3.l.a 2 36.f odd 6 1
240.3.l.a 2 36.h even 6 1
300.3.b.c 4 45.k odd 12 2
300.3.b.c 4 45.l even 12 2
300.3.g.d 2 45.h odd 6 1
300.3.g.d 2 45.j even 6 1
960.3.l.a 2 72.j odd 6 1
960.3.l.a 2 72.n even 6 1
960.3.l.d 2 72.l even 6 1
960.3.l.d 2 72.p odd 6 1
1200.3.c.e 4 180.v odd 12 2
1200.3.c.e 4 180.x even 12 2
1200.3.l.r 2 180.n even 6 1
1200.3.l.r 2 180.p odd 6 1
1620.3.o.b 4 1.a even 1 1 trivial
1620.3.o.b 4 3.b odd 2 1 inner
1620.3.o.b 4 9.c even 3 1 inner
1620.3.o.b 4 9.d odd 6 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$(T^{2} + 2 T + 4)^{2}$$
$11$ $$T^{4} - 180 T^{2} + 32400$$
$13$ $$(T^{2} + 8 T + 64)^{2}$$
$17$ $$(T^{2} + 180)^{2}$$
$19$ $$(T + 34)^{4}$$
$23$ $$T^{4} - 1620 T^{2} + \cdots + 2624400$$
$29$ $$T^{4} - 1620 T^{2} + \cdots + 2624400$$
$31$ $$(T^{2} + 14 T + 196)^{2}$$
$37$ $$(T - 56)^{4}$$
$41$ $$T^{4} - 720 T^{2} + 518400$$
$43$ $$(T^{2} + 8 T + 64)^{2}$$
$47$ $$T^{4} - 1620 T^{2} + \cdots + 2624400$$
$53$ $$(T^{2} + 1620)^{2}$$
$59$ $$T^{4} - 180 T^{2} + 32400$$
$61$ $$(T^{2} - 46 T + 2116)^{2}$$
$67$ $$(T^{2} + 32 T + 1024)^{2}$$
$71$ $$(T^{2} + 2880)^{2}$$
$73$ $$(T + 106)^{4}$$
$79$ $$(T^{2} - 22 T + 484)^{2}$$
$83$ $$T^{4} - 14580 T^{2} + \cdots + 212576400$$
$89$ $$(T^{2} + 11520)^{2}$$
$97$ $$(T^{2} + 122 T + 14884)^{2}$$