# Properties

 Label 180.3.f.g Level $180$ Weight $3$ Character orbit 180.f Analytic conductor $4.905$ 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] = [180,3,Mod(19,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("180.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.f (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 22x^{2} + 484$$ x^4 + 22*x^2 + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 8 q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + (2*b2 - 2) * q^4 + (-b2 - b1) * q^5 - 2*b1 * q^7 - 8 * q^8 $$q + (\beta_{2} + 1) q^{2} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 8 q^{8} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{10} + 2 \beta_{3} q^{11} - 2 \beta_{3} q^{13} + (2 \beta_{3} - 2 \beta_1) q^{14} + ( - 8 \beta_{2} - 8) q^{16} + 6 \beta_{2} q^{17} + 12 \beta_{2} q^{19} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{20} + (2 \beta_{3} + 6 \beta_1) q^{22} - 28 q^{23} + ( - 2 \beta_{3} + 19) q^{25} + ( - 2 \beta_{3} - 6 \beta_1) q^{26} + (4 \beta_{3} + 4 \beta_1) q^{28} - 2 \beta_1 q^{29} - 20 \beta_{2} q^{31} + ( - 16 \beta_{2} + 16) q^{32} + (6 \beta_{2} - 18) q^{34} + ( - 2 \beta_{3} + 44) q^{35} - 6 \beta_{3} q^{37} + (12 \beta_{2} - 36) q^{38} + (8 \beta_{2} + 8 \beta_1) q^{40} + 4 \beta_1 q^{41} - 8 \beta_1 q^{43} + ( - 4 \beta_{3} + 12 \beta_1) q^{44} + ( - 28 \beta_{2} - 28) q^{46} - 4 q^{47} + 39 q^{49} + ( - 2 \beta_{3} + 19 \beta_{2} - 6 \beta_1 + 19) q^{50} + (4 \beta_{3} - 12 \beta_1) q^{52} - 18 \beta_{2} q^{53} + (44 \beta_{2} - 6 \beta_1) q^{55} + 16 \beta_1 q^{56} + (2 \beta_{3} - 2 \beta_1) q^{58} - 2 \beta_{3} q^{59} - 58 q^{61} + ( - 20 \beta_{2} + 60) q^{62} + 64 q^{64} + ( - 44 \beta_{2} + 6 \beta_1) q^{65} + 4 \beta_1 q^{67} + ( - 12 \beta_{2} - 36) q^{68} + ( - 2 \beta_{3} + 44 \beta_{2} - 6 \beta_1 + 44) q^{70} - 12 \beta_{3} q^{71} + ( - 6 \beta_{3} - 18 \beta_1) q^{74} + ( - 24 \beta_{2} - 72) q^{76} + 88 \beta_{2} q^{77} - 4 \beta_{2} q^{79} + ( - 8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 24) q^{80} + ( - 4 \beta_{3} + 4 \beta_1) q^{82} + 32 q^{83} + (6 \beta_{3} + 18) q^{85} + (8 \beta_{3} - 8 \beta_1) q^{86} - 16 \beta_{3} q^{88} + 16 \beta_1 q^{89} - 88 \beta_{2} q^{91} + ( - 56 \beta_{2} + 56) q^{92} + ( - 4 \beta_{2} - 4) q^{94} + (12 \beta_{3} + 36) q^{95} + 20 \beta_{3} q^{97} + (39 \beta_{2} + 39) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + (2*b2 - 2) * q^4 + (-b2 - b1) * q^5 - 2*b1 * q^7 - 8 * q^8 + (b3 - b2 - b1 + 3) * q^10 + 2*b3 * q^11 - 2*b3 * q^13 + (2*b3 - 2*b1) * q^14 + (-8*b2 - 8) * q^16 + 6*b2 * q^17 + 12*b2 * q^19 + (2*b3 + 2*b2 + 2*b1 + 6) * q^20 + (2*b3 + 6*b1) * q^22 - 28 * q^23 + (-2*b3 + 19) * q^25 + (-2*b3 - 6*b1) * q^26 + (4*b3 + 4*b1) * q^28 - 2*b1 * q^29 - 20*b2 * q^31 + (-16*b2 + 16) * q^32 + (6*b2 - 18) * q^34 + (-2*b3 + 44) * q^35 - 6*b3 * q^37 + (12*b2 - 36) * q^38 + (8*b2 + 8*b1) * q^40 + 4*b1 * q^41 - 8*b1 * q^43 + (-4*b3 + 12*b1) * q^44 + (-28*b2 - 28) * q^46 - 4 * q^47 + 39 * q^49 + (-2*b3 + 19*b2 - 6*b1 + 19) * q^50 + (4*b3 - 12*b1) * q^52 - 18*b2 * q^53 + (44*b2 - 6*b1) * q^55 + 16*b1 * q^56 + (2*b3 - 2*b1) * q^58 - 2*b3 * q^59 - 58 * q^61 + (-20*b2 + 60) * q^62 + 64 * q^64 + (-44*b2 + 6*b1) * q^65 + 4*b1 * q^67 + (-12*b2 - 36) * q^68 + (-2*b3 + 44*b2 - 6*b1 + 44) * q^70 - 12*b3 * q^71 + (-6*b3 - 18*b1) * q^74 + (-24*b2 - 72) * q^76 + 88*b2 * q^77 - 4*b2 * q^79 + (-8*b3 + 8*b2 + 8*b1 - 24) * q^80 + (-4*b3 + 4*b1) * q^82 + 32 * q^83 + (6*b3 + 18) * q^85 + (8*b3 - 8*b1) * q^86 - 16*b3 * q^88 + 16*b1 * q^89 - 88*b2 * q^91 + (-56*b2 + 56) * q^92 + (-4*b2 - 4) * q^94 + (12*b3 + 36) * q^95 + 20*b3 * q^97 + (39*b2 + 39) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} + 12 q^{10} - 32 q^{16} + 24 q^{20} - 112 q^{23} + 76 q^{25} + 64 q^{32} - 72 q^{34} + 176 q^{35} - 144 q^{38} - 112 q^{46} - 16 q^{47} + 156 q^{49} + 76 q^{50} - 232 q^{61} + 240 q^{62} + 256 q^{64} - 144 q^{68} + 176 q^{70} - 288 q^{76} - 96 q^{80} + 128 q^{83} + 72 q^{85} + 224 q^{92} - 16 q^{94} + 144 q^{95} + 156 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 + 12 * q^10 - 32 * q^16 + 24 * q^20 - 112 * q^23 + 76 * q^25 + 64 * q^32 - 72 * q^34 + 176 * q^35 - 144 * q^38 - 112 * q^46 - 16 * q^47 + 156 * q^49 + 76 * q^50 - 232 * q^61 + 240 * q^62 + 256 * q^64 - 144 * q^68 + 176 * q^70 - 288 * q^76 - 96 * q^80 + 128 * q^83 + 72 * q^85 + 224 * q^92 - 16 * q^94 + 144 * q^95 + 156 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 22x^{2} + 484$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 22$$ (v^3) / 22 $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 11 ) / 11$$ (v^2 + 11) / 11 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 44\nu ) / 22$$ (v^3 + 44*v) / 22
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$11\beta_{2} - 11$$ 11*b2 - 11 $$\nu^{3}$$ $$=$$ $$22\beta_1$$ 22*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −2.34521 + 4.06202i 2.34521 − 4.06202i −2.34521 − 4.06202i 2.34521 + 4.06202i
1.00000 1.73205i 0 −2.00000 3.46410i −4.69042 + 1.73205i 0 −9.38083 −8.00000 0 −1.69042 + 9.85609i
19.2 1.00000 1.73205i 0 −2.00000 3.46410i 4.69042 + 1.73205i 0 9.38083 −8.00000 0 7.69042 6.39199i
19.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.69042 1.73205i 0 −9.38083 −8.00000 0 −1.69042 9.85609i
19.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.69042 1.73205i 0 9.38083 −8.00000 0 7.69042 + 6.39199i
 $$n$$: 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
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.g yes 4
3.b odd 2 1 180.3.f.d 4
4.b odd 2 1 180.3.f.d 4
5.b even 2 1 180.3.f.d 4
5.c odd 4 2 900.3.c.s 8
12.b even 2 1 inner 180.3.f.g yes 4
15.d odd 2 1 inner 180.3.f.g yes 4
15.e even 4 2 900.3.c.s 8
20.d odd 2 1 inner 180.3.f.g yes 4
20.e even 4 2 900.3.c.s 8
60.h even 2 1 180.3.f.d 4
60.l odd 4 2 900.3.c.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.d 4 3.b odd 2 1
180.3.f.d 4 4.b odd 2 1
180.3.f.d 4 5.b even 2 1
180.3.f.d 4 60.h even 2 1
180.3.f.g yes 4 1.a even 1 1 trivial
180.3.f.g yes 4 12.b even 2 1 inner
180.3.f.g yes 4 15.d odd 2 1 inner
180.3.f.g yes 4 20.d odd 2 1 inner
900.3.c.s 8 5.c odd 4 2
900.3.c.s 8 15.e even 4 2
900.3.c.s 8 20.e even 4 2
900.3.c.s 8 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$:

 $$T_{7}^{2} - 88$$ T7^2 - 88 $$T_{13}^{2} + 264$$ T13^2 + 264 $$T_{23} + 28$$ T23 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 38T^{2} + 625$$
$7$ $$(T^{2} - 88)^{2}$$
$11$ $$(T^{2} + 264)^{2}$$
$13$ $$(T^{2} + 264)^{2}$$
$17$ $$(T^{2} + 108)^{2}$$
$19$ $$(T^{2} + 432)^{2}$$
$23$ $$(T + 28)^{4}$$
$29$ $$(T^{2} - 88)^{2}$$
$31$ $$(T^{2} + 1200)^{2}$$
$37$ $$(T^{2} + 2376)^{2}$$
$41$ $$(T^{2} - 352)^{2}$$
$43$ $$(T^{2} - 1408)^{2}$$
$47$ $$(T + 4)^{4}$$
$53$ $$(T^{2} + 972)^{2}$$
$59$ $$(T^{2} + 264)^{2}$$
$61$ $$(T + 58)^{4}$$
$67$ $$(T^{2} - 352)^{2}$$
$71$ $$(T^{2} + 9504)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 48)^{2}$$
$83$ $$(T - 32)^{4}$$
$89$ $$(T^{2} - 5632)^{2}$$
$97$ $$(T^{2} + 26400)^{2}$$