# Properties

 Label 180.3.o.a Level $180$ Weight $3$ Character orbit 180.o Analytic conductor $4.905$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,3,Mod(41,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, 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("180.41");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.o (of order $$6$$, degree $$2$$, 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: $$4.90464475849$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \beta_{2} - 3) q^{3} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{7} - 9 \beta_{2} q^{9}+O(q^{10})$$ q + (3*b2 - 3) * q^3 + (b3 - b1) * q^5 + (4*b3 - 4*b2 - 2*b1 + 4) * q^7 - 9*b2 * q^9 $$q + (3 \beta_{2} - 3) q^{3} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{7} - 9 \beta_{2} q^{9} + (\beta_{2} - 6 \beta_1 + 1) q^{11} + ( - 2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{13} - 3 \beta_{3} q^{15} + ( - 6 \beta_{3} + 2 \beta_{2} - 1) q^{17} + (2 \beta_{3} - 4 \beta_1 - 19) q^{19} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{21} + (12 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 8) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + 27 q^{27} + ( - 10 \beta_{2} - 12 \beta_1 - 10) q^{29} + (14 \beta_{3} - 2 \beta_{2} + 14 \beta_1) q^{31} + ( - 18 \beta_{3} + 3 \beta_{2} + 18 \beta_1 - 6) q^{33} + (4 \beta_{3} - 20 \beta_{2} + 10) q^{35} + 14 q^{37} + ( - 6 \beta_{3} + 12 \beta_1 - 30) q^{39} + (15 \beta_{2} - 30) q^{41} + (16 \beta_{3} + 11 \beta_{2} - 8 \beta_1 - 11) q^{43} + 9 \beta_1 q^{45} + (34 \beta_{2} - 6 \beta_1 + 34) q^{47} + (16 \beta_{3} - 27 \beta_{2} + 16 \beta_1) q^{49} + ( - 3 \beta_{2} + 18 \beta_1 - 3) q^{51} + ( - 6 \beta_{3} + 28 \beta_{2} - 14) q^{53} + (\beta_{3} - 2 \beta_1 + 30) q^{55} + ( - 12 \beta_{3} - 57 \beta_{2} + 6 \beta_1 + 57) q^{57} + ( - 6 \beta_{3} + 29 \beta_{2} + 6 \beta_1 - 58) q^{59} + ( - 12 \beta_{3} - 22 \beta_{2} + 6 \beta_1 + 22) q^{61} + ( - 18 \beta_{3} + 36 \beta_1 - 36) q^{63} + (10 \beta_{2} - 10 \beta_1 + 10) q^{65} + (24 \beta_{3} + 7 \beta_{2} + 24 \beta_1) q^{67} + ( - 36 \beta_{3} + 24 \beta_{2} - 12) q^{69} + ( - 18 \beta_{3} - 72 \beta_{2} + 36) q^{71} + ( - 6 \beta_{3} + 12 \beta_1 - 37) q^{73} + 15 \beta_{2} q^{75} + (30 \beta_{3} - 64 \beta_{2} - 30 \beta_1 + 128) q^{77} + ( - 12 \beta_{3} + 80 \beta_{2} + 6 \beta_1 - 80) q^{79} + (81 \beta_{2} - 81) q^{81} + ( - 52 \beta_{2} - 52) q^{83} + ( - \beta_{3} + 30 \beta_{2} - \beta_1) q^{85} + ( - 36 \beta_{3} - 30 \beta_{2} + 36 \beta_1 + 60) q^{87} + (24 \beta_{3} + 48 \beta_{2} - 24) q^{89} + (28 \beta_{3} - 56 \beta_1 + 100) q^{91} + (42 \beta_{3} - 84 \beta_1 + 6) q^{93} + ( - 19 \beta_{3} - 10 \beta_{2} + 19 \beta_1 + 20) q^{95} + ( - 52 \beta_{3} - \beta_{2} + 26 \beta_1 + 1) q^{97} + (54 \beta_{3} - 18 \beta_{2} + 9) q^{99}+O(q^{100})$$ q + (3*b2 - 3) * q^3 + (b3 - b1) * q^5 + (4*b3 - 4*b2 - 2*b1 + 4) * q^7 - 9*b2 * q^9 + (b2 - 6*b1 + 1) * q^11 + (-2*b3 + 10*b2 - 2*b1) * q^13 - 3*b3 * q^15 + (-6*b3 + 2*b2 - 1) * q^17 + (2*b3 - 4*b1 - 19) * q^19 + (-6*b3 + 12*b2 - 6*b1) * q^21 + (12*b3 - 4*b2 - 12*b1 + 8) * q^23 + (-5*b2 + 5) * q^25 + 27 * q^27 + (-10*b2 - 12*b1 - 10) * q^29 + (14*b3 - 2*b2 + 14*b1) * q^31 + (-18*b3 + 3*b2 + 18*b1 - 6) * q^33 + (4*b3 - 20*b2 + 10) * q^35 + 14 * q^37 + (-6*b3 + 12*b1 - 30) * q^39 + (15*b2 - 30) * q^41 + (16*b3 + 11*b2 - 8*b1 - 11) * q^43 + 9*b1 * q^45 + (34*b2 - 6*b1 + 34) * q^47 + (16*b3 - 27*b2 + 16*b1) * q^49 + (-3*b2 + 18*b1 - 3) * q^51 + (-6*b3 + 28*b2 - 14) * q^53 + (b3 - 2*b1 + 30) * q^55 + (-12*b3 - 57*b2 + 6*b1 + 57) * q^57 + (-6*b3 + 29*b2 + 6*b1 - 58) * q^59 + (-12*b3 - 22*b2 + 6*b1 + 22) * q^61 + (-18*b3 + 36*b1 - 36) * q^63 + (10*b2 - 10*b1 + 10) * q^65 + (24*b3 + 7*b2 + 24*b1) * q^67 + (-36*b3 + 24*b2 - 12) * q^69 + (-18*b3 - 72*b2 + 36) * q^71 + (-6*b3 + 12*b1 - 37) * q^73 + 15*b2 * q^75 + (30*b3 - 64*b2 - 30*b1 + 128) * q^77 + (-12*b3 + 80*b2 + 6*b1 - 80) * q^79 + (81*b2 - 81) * q^81 + (-52*b2 - 52) * q^83 + (-b3 + 30*b2 - b1) * q^85 + (-36*b3 - 30*b2 + 36*b1 + 60) * q^87 + (24*b3 + 48*b2 - 24) * q^89 + (28*b3 - 56*b1 + 100) * q^91 + (42*b3 - 84*b1 + 6) * q^93 + (-19*b3 - 10*b2 + 19*b1 + 20) * q^95 + (-52*b3 - b2 + 26*b1 + 1) * q^97 + (54*b3 - 18*b2 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} + 8 q^{7} - 18 q^{9}+O(q^{10})$$ 4 * q - 6 * q^3 + 8 * q^7 - 18 * q^9 $$4 q - 6 q^{3} + 8 q^{7} - 18 q^{9} + 6 q^{11} + 20 q^{13} - 76 q^{19} + 24 q^{21} + 24 q^{23} + 10 q^{25} + 108 q^{27} - 60 q^{29} - 4 q^{31} - 18 q^{33} + 56 q^{37} - 120 q^{39} - 90 q^{41} - 22 q^{43} + 204 q^{47} - 54 q^{49} - 18 q^{51} + 120 q^{55} + 114 q^{57} - 174 q^{59} + 44 q^{61} - 144 q^{63} + 60 q^{65} + 14 q^{67} - 148 q^{73} + 30 q^{75} + 384 q^{77} - 160 q^{79} - 162 q^{81} - 312 q^{83} + 60 q^{85} + 180 q^{87} + 400 q^{91} + 24 q^{93} + 60 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q - 6 * q^3 + 8 * q^7 - 18 * q^9 + 6 * q^11 + 20 * q^13 - 76 * q^19 + 24 * q^21 + 24 * q^23 + 10 * q^25 + 108 * q^27 - 60 * q^29 - 4 * q^31 - 18 * q^33 + 56 * q^37 - 120 * q^39 - 90 * q^41 - 22 * q^43 + 204 * q^47 - 54 * q^49 - 18 * q^51 + 120 * q^55 + 114 * q^57 - 174 * q^59 + 44 * q^61 - 144 * q^63 + 60 * q^65 + 14 * q^67 - 148 * q^73 + 30 * q^75 + 384 * q^77 - 160 * q^79 - 162 * q^81 - 312 * q^83 + 60 * q^85 + 180 * q^87 + 400 * q^91 + 24 * q^93 + 60 * q^95 + 2 * 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 $$\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 1.93649 − 1.11803i −1.93649 + 1.11803i 1.93649 + 1.11803i −1.93649 − 1.11803i
0 −1.50000 2.59808i 0 −1.93649 1.11803i 0 −1.87298 3.24410i 0 −4.50000 + 7.79423i 0
41.2 0 −1.50000 2.59808i 0 1.93649 + 1.11803i 0 5.87298 + 10.1723i 0 −4.50000 + 7.79423i 0
101.1 0 −1.50000 + 2.59808i 0 −1.93649 + 1.11803i 0 −1.87298 + 3.24410i 0 −4.50000 7.79423i 0
101.2 0 −1.50000 + 2.59808i 0 1.93649 1.11803i 0 5.87298 10.1723i 0 −4.50000 7.79423i 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
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 180.3.o.a 4
3.b odd 2 1 540.3.o.a 4
4.b odd 2 1 720.3.bs.a 4
5.b even 2 1 900.3.p.b 4
5.c odd 4 2 900.3.u.b 8
9.c even 3 1 540.3.o.a 4
9.c even 3 1 1620.3.g.a 4
9.d odd 6 1 inner 180.3.o.a 4
9.d odd 6 1 1620.3.g.a 4
12.b even 2 1 2160.3.bs.a 4
15.d odd 2 1 2700.3.p.a 4
15.e even 4 2 2700.3.u.a 8
36.f odd 6 1 2160.3.bs.a 4
36.h even 6 1 720.3.bs.a 4
45.h odd 6 1 900.3.p.b 4
45.j even 6 1 2700.3.p.a 4
45.k odd 12 2 2700.3.u.a 8
45.l even 12 2 900.3.u.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 1.a even 1 1 trivial
180.3.o.a 4 9.d odd 6 1 inner
540.3.o.a 4 3.b odd 2 1
540.3.o.a 4 9.c even 3 1
720.3.bs.a 4 4.b odd 2 1
720.3.bs.a 4 36.h even 6 1
900.3.p.b 4 5.b even 2 1
900.3.p.b 4 45.h odd 6 1
900.3.u.b 8 5.c odd 4 2
900.3.u.b 8 45.l even 12 2
1620.3.g.a 4 9.c even 3 1
1620.3.g.a 4 9.d odd 6 1
2160.3.bs.a 4 12.b even 2 1
2160.3.bs.a 4 36.f odd 6 1
2700.3.p.a 4 15.d odd 2 1
2700.3.p.a 4 45.j even 6 1
2700.3.u.a 8 15.e even 4 2
2700.3.u.a 8 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 8T_{7}^{3} + 108T_{7}^{2} + 352T_{7} + 1936$$ acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 3 T + 9)^{2}$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$T^{4} - 8 T^{3} + 108 T^{2} + \cdots + 1936$$
$11$ $$T^{4} - 6 T^{3} - 165 T^{2} + \cdots + 31329$$
$13$ $$T^{4} - 20 T^{3} + 360 T^{2} + \cdots + 1600$$
$17$ $$T^{4} + 366 T^{2} + 31329$$
$19$ $$(T^{2} + 38 T + 301)^{2}$$
$23$ $$T^{4} - 24 T^{3} - 480 T^{2} + \cdots + 451584$$
$29$ $$T^{4} + 60 T^{3} + 780 T^{2} + \cdots + 176400$$
$31$ $$T^{4} + 4 T^{3} + 2952 T^{2} + \cdots + 8620096$$
$37$ $$(T - 14)^{4}$$
$41$ $$(T^{2} + 45 T + 675)^{2}$$
$43$ $$T^{4} + 22 T^{3} + 1323 T^{2} + \cdots + 703921$$
$47$ $$T^{4} - 204 T^{3} + \cdots + 10810944$$
$53$ $$T^{4} + 1536 T^{2} + 166464$$
$59$ $$T^{4} + 174 T^{3} + 12435 T^{2} + \cdots + 5489649$$
$61$ $$T^{4} - 44 T^{3} + 1992 T^{2} + \cdots + 3136$$
$67$ $$T^{4} - 14 T^{3} + 8787 T^{2} + \cdots + 73805281$$
$71$ $$T^{4} + 11016 T^{2} + \cdots + 5143824$$
$73$ $$(T^{2} + 74 T + 829)^{2}$$
$79$ $$T^{4} + 160 T^{3} + \cdots + 34339600$$
$83$ $$(T^{2} + 156 T + 8112)^{2}$$
$89$ $$T^{4} + 9216 T^{2} + \cdots + 1327104$$
$97$ $$T^{4} - 2 T^{3} + 10143 T^{2} + \cdots + 102799321$$