# Properties

 Label 180.3.c.c Level $180$ Weight $3$ Character orbit 180.c Analytic conductor $4.905$ Analytic rank $0$ Dimension $8$ 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(91,180)]

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, 0]))

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

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

chi := DirichletCharacter("180.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.c (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: $$8$$ Coefficient field: 8.0.15012375625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256$$ x^8 + 7*x^6 + 28*x^4 + 112*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{4} - 2) q^{4} + \beta_{3} q^{5} + 2 \beta_{6} q^{7} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b4 - 2) * q^4 + b3 * q^5 + 2*b6 * q^7 + (b5 + b2 - 2*b1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{4} - 2) q^{4} + \beta_{3} q^{5} + 2 \beta_{6} q^{7} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{4} + 1) q^{10} + (\beta_{5} + 2 \beta_{2}) q^{11} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + 1) q^{13} + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} - 1) q^{16} + (3 \beta_{5} + 4 \beta_{3} - 6 \beta_1) q^{17} + (\beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 1) q^{19} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{20} - 2 \beta_{7} q^{22} + (\beta_{5} - 2 \beta_{3} + 6 \beta_1) q^{23} + 5 q^{25} + (8 \beta_{3} + 4 \beta_{2}) q^{26} + ( - 8 \beta_{6} - 2 \beta_{4} - 12) q^{28} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{29} + (\beta_{7} + 7 \beta_{6} + 5 \beta_{4} + 1) q^{31} + ( - \beta_{5} - 12 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{32} + ( - 10 \beta_{6} + 2 \beta_{4} - 14) q^{34} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{35} + ( - \beta_{7} + \beta_{6} + 3 \beta_{4} + 35) q^{37} + ( - 4 \beta_{5} + 16 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{38} + ( - \beta_{7} + 3 \beta_{6} - 1) q^{40} + ( - 4 \beta_{5} + 8 \beta_1) q^{41} + (2 \beta_{7} + 6 \beta_{6} + 10 \beta_{4} + 2) q^{43} + (4 \beta_{5} - 20 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{44} + ( - 4 \beta_{4} - 24) q^{46} + ( - 3 \beta_{5} - 2 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{47} + (4 \beta_{7} - 4 \beta_{6} - 12 \beta_{4} - 23) q^{49} + 5 \beta_1 q^{50} + ( - 4 \beta_{7} - 4 \beta_{6} - 8 \beta_{4} + 28) q^{52} + (4 \beta_{5} - 14 \beta_{3} - 8 \beta_1) q^{53} + ( - \beta_{7} + 3 \beta_{6} - 5 \beta_{4} - 1) q^{55} + ( - 6 \beta_{5} - 16 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{56} + (6 \beta_{6} - 2 \beta_{4} + 10) q^{58} + ( - 9 \beta_{5} + 12 \beta_{3} - 6 \beta_{2} - 36 \beta_1) q^{59} + (2 \beta_{7} - 2 \beta_{6} - 6 \beta_{4}) q^{61} + (24 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{62} + (3 \beta_{7} + 11 \beta_{6} + 10 \beta_{4} - 21) q^{64} + ( - 5 \beta_{5} + 2 \beta_{3} + 10 \beta_1) q^{65} + ( - 4 \beta_{7} + 8 \beta_{6} - 20 \beta_{4} - 4) q^{67} + ( - 12 \beta_{5} - 20 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{68} + ( - 2 \beta_{7} + 4 \beta_{4} + 24) q^{70} + (4 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 12 \beta_1) q^{71} + ( - 2 \beta_{7} + 2 \beta_{6} + 6 \beta_{4} - 28) q^{73} + ( - 8 \beta_{3} - 4 \beta_{2} + 36 \beta_1) q^{74} + (4 \beta_{7} - 12 \beta_{6} - 12 \beta_{4} + 44) q^{76} + ( - 6 \beta_{5} - 32 \beta_{3} + 12 \beta_1) q^{77} + (3 \beta_{7} - 11 \beta_{6} + 15 \beta_{4} + 3) q^{79} + (5 \beta_{5} - 4 \beta_{3} - \beta_{2} - 2 \beta_1) q^{80} + (8 \beta_{6} - 8 \beta_{4} + 24) q^{82} + (6 \beta_{5} - 16 \beta_{3} - 4 \beta_{2} + 48 \beta_1) q^{83} + ( - 3 \beta_{7} + 3 \beta_{6} + 9 \beta_{4} + 23) q^{85} + ( - 8 \beta_{5} + 32 \beta_{3} - 8 \beta_{2} - 8 \beta_1) q^{86} + (2 \beta_{7} + 10 \beta_{6} + 16 \beta_{4} - 78) q^{88} + ( - 4 \beta_{5} + 52 \beta_{3} + 8 \beta_1) q^{89} + ( - 2 \beta_{7} - 30 \beta_{6} - 10 \beta_{4} - 2) q^{91} + (4 \beta_{5} + 4 \beta_{2} - 24 \beta_1) q^{92} + (8 \beta_{7} - 4 \beta_{4} - 24) q^{94} + ( - 5 \beta_{5} + 6 \beta_{3} - 4 \beta_{2} - 18 \beta_1) q^{95} + ( - 6 \beta_{7} + 6 \beta_{6} + 18 \beta_{4} + 36) q^{97} + (32 \beta_{3} + 16 \beta_{2} - 27 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b4 - 2) * q^4 + b3 * q^5 + 2*b6 * q^7 + (b5 + b2 - 2*b1) * q^8 + (-b6 - b4 + 1) * q^10 + (b5 + 2*b2) * q^11 + (b7 - b6 - 3*b4 + 1) * q^13 + (2*b5 + 4*b3 - 2*b1) * q^14 + (-b7 - b6 + 2*b4 - 1) * q^16 + (3*b5 + 4*b3 - 6*b1) * q^17 + (b7 + 3*b6 + 5*b4 + 1) * q^19 + (-2*b3 + b2 + 2*b1) * q^20 - 2*b7 * q^22 + (b5 - 2*b3 + 6*b1) * q^23 + 5 * q^25 + (8*b3 + 4*b2) * q^26 + (-8*b6 - 2*b4 - 12) * q^28 + (-2*b5 - 2*b3 + 4*b1) * q^29 + (b7 + 7*b6 + 5*b4 + 1) * q^31 + (-b5 - 12*b3 - 3*b2 + 2*b1) * q^32 + (-10*b6 + 2*b4 - 14) * q^34 + (2*b3 + 2*b2 - 6*b1) * q^35 + (-b7 + b6 + 3*b4 + 35) * q^37 + (-4*b5 + 16*b3 - 4*b2 - 4*b1) * q^38 + (-b7 + 3*b6 - 1) * q^40 + (-4*b5 + 8*b1) * q^41 + (2*b7 + 6*b6 + 10*b4 + 2) * q^43 + (4*b5 - 20*b3 - 2*b2 + 4*b1) * q^44 + (-4*b4 - 24) * q^46 + (-3*b5 - 2*b3 - 8*b2 + 6*b1) * q^47 + (4*b7 - 4*b6 - 12*b4 - 23) * q^49 + 5*b1 * q^50 + (-4*b7 - 4*b6 - 8*b4 + 28) * q^52 + (4*b5 - 14*b3 - 8*b1) * q^53 + (-b7 + 3*b6 - 5*b4 - 1) * q^55 + (-6*b5 - 16*b3 + 2*b2 - 4*b1) * q^56 + (6*b6 - 2*b4 + 10) * q^58 + (-9*b5 + 12*b3 - 6*b2 - 36*b1) * q^59 + (2*b7 - 2*b6 - 6*b4) * q^61 + (24*b3 - 4*b2 - 8*b1) * q^62 + (3*b7 + 11*b6 + 10*b4 - 21) * q^64 + (-5*b5 + 2*b3 + 10*b1) * q^65 + (-4*b7 + 8*b6 - 20*b4 - 4) * q^67 + (-12*b5 - 20*b3 - 2*b2 - 4*b1) * q^68 + (-2*b7 + 4*b4 + 24) * q^70 + (4*b5 - 4*b3 + 4*b2 + 12*b1) * q^71 + (-2*b7 + 2*b6 + 6*b4 - 28) * q^73 + (-8*b3 - 4*b2 + 36*b1) * q^74 + (4*b7 - 12*b6 - 12*b4 + 44) * q^76 + (-6*b5 - 32*b3 + 12*b1) * q^77 + (3*b7 - 11*b6 + 15*b4 + 3) * q^79 + (5*b5 - 4*b3 - b2 - 2*b1) * q^80 + (8*b6 - 8*b4 + 24) * q^82 + (6*b5 - 16*b3 - 4*b2 + 48*b1) * q^83 + (-3*b7 + 3*b6 + 9*b4 + 23) * q^85 + (-8*b5 + 32*b3 - 8*b2 - 8*b1) * q^86 + (2*b7 + 10*b6 + 16*b4 - 78) * q^88 + (-4*b5 + 52*b3 + 8*b1) * q^89 + (-2*b7 - 30*b6 - 10*b4 - 2) * q^91 + (4*b5 + 4*b2 - 24*b1) * q^92 + (8*b7 - 4*b4 - 24) * q^94 + (-5*b5 + 6*b3 - 4*b2 - 18*b1) * q^95 + (-6*b7 + 6*b6 + 18*b4 + 36) * q^97 + (32*b3 + 16*b2 - 27*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 14 q^{4}+O(q^{10})$$ 8 * q - 14 * q^4 $$8 q - 14 q^{4} + 10 q^{10} + 16 q^{13} - 14 q^{16} - 4 q^{22} + 40 q^{25} - 92 q^{28} - 116 q^{34} + 272 q^{37} - 10 q^{40} - 184 q^{46} - 152 q^{49} + 232 q^{52} + 84 q^{58} + 16 q^{61} - 182 q^{64} + 180 q^{70} - 240 q^{73} + 384 q^{76} + 208 q^{82} + 160 q^{85} - 652 q^{88} - 168 q^{94} + 240 q^{97}+O(q^{100})$$ 8 * q - 14 * q^4 + 10 * q^10 + 16 * q^13 - 14 * q^16 - 4 * q^22 + 40 * q^25 - 92 * q^28 - 116 * q^34 + 272 * q^37 - 10 * q^40 - 184 * q^46 - 152 * q^49 + 232 * q^52 + 84 * q^58 + 16 * q^61 - 182 * q^64 + 180 * q^70 - 240 * q^73 + 384 * q^76 + 208 * q^82 + 160 * q^85 - 652 * q^88 - 168 * q^94 + 240 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 7\nu^{5} + 28\nu^{3} + 112\nu ) / 64$$ (v^7 + 7*v^5 + 28*v^3 + 112*v) / 64 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 7\nu^{5} + 36\nu^{3} - 48\nu ) / 64$$ (-v^7 - 7*v^5 + 36*v^3 - 48*v) / 64 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + \nu^{5} - 4\nu^{3} - 48\nu ) / 64$$ (-v^7 + v^5 - 4*v^3 - 48*v) / 64 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 7\nu^{4} + 28\nu^{2} + 80 ) / 16$$ (v^6 + 7*v^4 + 28*v^2 + 80) / 16 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} - 3\nu^{5} - 16\nu^{3} - 16\nu ) / 16$$ (-v^7 - 3*v^5 - 16*v^3 - 16*v) / 16 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 3\nu^{4} - 16\nu^{2} - 56 ) / 8$$ (-v^6 - 3*v^4 - 16*v^2 - 56) / 8 $$\beta_{7}$$ $$=$$ $$( -3\nu^{6} - 13\nu^{4} + 4\nu^{2} - 80 ) / 16$$ (-3*v^6 - 13*v^4 + 4*v^2 - 80) / 16
 $$\nu$$ $$=$$ $$( \beta_{5} - 2\beta_{3} + 2\beta_1 ) / 4$$ (b5 - 2*b3 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{4} - 7 ) / 4$$ (b7 - b6 + b4 - 7) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{5} + 2\beta_{3} + 4\beta_{2} + 2\beta_1 ) / 4$$ (-b5 + 2*b3 + 4*b2 + 2*b1) / 4 $$\nu^{4}$$ $$=$$ $$( -3\beta_{7} + 11\beta_{6} + 13\beta_{4} - 3 ) / 4$$ (-3*b7 + 11*b6 + 13*b4 - 3) / 4 $$\nu^{5}$$ $$=$$ $$( -5\beta_{5} + 42\beta_{3} - 12\beta_{2} + 10\beta_1 ) / 4$$ (-5*b5 + 42*b3 - 12*b2 + 10*b1) / 4 $$\nu^{6}$$ $$=$$ $$( -7\beta_{7} - 49\beta_{6} - 55\beta_{4} - 103 ) / 4$$ (-7*b7 - 49*b6 - 55*b4 - 103) / 4 $$\nu^{7}$$ $$=$$ $$( -49\beta_{5} - 126\beta_{3} - 28\beta_{2} - 94\beta_1 ) / 4$$ (-49*b5 - 126*b3 - 28*b2 - 94*b1) / 4

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.46040 − 1.36646i 1.46040 + 1.36646i 0.342371 − 1.97048i 0.342371 + 1.97048i −0.342371 − 1.97048i −0.342371 + 1.97048i −1.46040 − 1.36646i −1.46040 + 1.36646i
−1.46040 1.36646i 0 0.265564 + 3.99117i −2.23607 0 1.87135i 5.06596 6.19161i 0 3.26556 + 3.05550i
91.2 −1.46040 + 1.36646i 0 0.265564 3.99117i −2.23607 0 1.87135i 5.06596 + 6.19161i 0 3.26556 3.05550i
91.3 −0.342371 1.97048i 0 −3.76556 + 1.34927i 2.23607 0 11.5108i 3.94792 + 6.95801i 0 −0.765564 4.40612i
91.4 −0.342371 + 1.97048i 0 −3.76556 1.34927i 2.23607 0 11.5108i 3.94792 6.95801i 0 −0.765564 + 4.40612i
91.5 0.342371 1.97048i 0 −3.76556 1.34927i −2.23607 0 11.5108i −3.94792 + 6.95801i 0 −0.765564 + 4.40612i
91.6 0.342371 + 1.97048i 0 −3.76556 + 1.34927i −2.23607 0 11.5108i −3.94792 6.95801i 0 −0.765564 4.40612i
91.7 1.46040 1.36646i 0 0.265564 3.99117i 2.23607 0 1.87135i −5.06596 6.19161i 0 3.26556 3.05550i
91.8 1.46040 + 1.36646i 0 0.265564 + 3.99117i 2.23607 0 1.87135i −5.06596 + 6.19161i 0 3.26556 + 3.05550i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.8 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.c 8
3.b odd 2 1 inner 180.3.c.c 8
4.b odd 2 1 inner 180.3.c.c 8
5.b even 2 1 900.3.c.o 8
5.c odd 4 2 900.3.f.i 16
8.b even 2 1 2880.3.e.i 8
8.d odd 2 1 2880.3.e.i 8
12.b even 2 1 inner 180.3.c.c 8
15.d odd 2 1 900.3.c.o 8
15.e even 4 2 900.3.f.i 16
20.d odd 2 1 900.3.c.o 8
20.e even 4 2 900.3.f.i 16
24.f even 2 1 2880.3.e.i 8
24.h odd 2 1 2880.3.e.i 8
60.h even 2 1 900.3.c.o 8
60.l odd 4 2 900.3.f.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c 8 1.a even 1 1 trivial
180.3.c.c 8 3.b odd 2 1 inner
180.3.c.c 8 4.b odd 2 1 inner
180.3.c.c 8 12.b even 2 1 inner
900.3.c.o 8 5.b even 2 1
900.3.c.o 8 15.d odd 2 1
900.3.c.o 8 20.d odd 2 1
900.3.c.o 8 60.h even 2 1
900.3.f.i 16 5.c odd 4 2
900.3.f.i 16 15.e even 4 2
900.3.f.i 16 20.e even 4 2
900.3.f.i 16 60.l odd 4 2
2880.3.e.i 8 8.b even 2 1
2880.3.e.i 8 8.d odd 2 1
2880.3.e.i 8 24.f even 2 1
2880.3.e.i 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 136T_{7}^{2} + 464$$ acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 7 T^{6} + 28 T^{4} + 112 T^{2} + \cdots + 256$$
$3$ $$T^{8}$$
$5$ $$(T^{2} - 5)^{4}$$
$7$ $$(T^{4} + 136 T^{2} + 464)^{2}$$
$11$ $$(T^{4} + 328 T^{2} + 22736)^{2}$$
$13$ $$(T^{2} - 4 T - 256)^{4}$$
$17$ $$(T^{4} - 1096 T^{2} + 150544)^{2}$$
$19$ $$(T^{4} + 944 T^{2} + 118784)^{2}$$
$23$ $$(T^{4} + 368 T^{2} + 29696)^{2}$$
$29$ $$(T^{4} - 456 T^{2} + 35344)^{2}$$
$31$ $$(T^{4} + 1840 T^{2} + 742400)^{2}$$
$37$ $$(T^{2} - 68 T + 896)^{4}$$
$41$ $$(T^{2} - 832)^{4}$$
$43$ $$(T^{4} + 3776 T^{2} + 1900544)^{2}$$
$47$ $$(T^{4} + 5552 T^{2} + 7602176)^{2}$$
$53$ $$(T^{4} - 3624 T^{2} + 21904)^{2}$$
$59$ $$(T^{4} + 16488 T^{2} + 51452496)^{2}$$
$61$ $$(T^{2} - 4 T - 1036)^{4}$$
$67$ $$(T^{4} + 21664 T^{2} + 75732224)^{2}$$
$71$ $$(T^{4} + 2848 T^{2} + 363776)^{2}$$
$73$ $$(T^{2} + 60 T - 140)^{4}$$
$79$ $$(T^{4} + 16816 T^{2} + 65598464)^{2}$$
$83$ $$(T^{4} + 24608 T^{2} + 69852416)^{2}$$
$89$ $$(T^{4} - 28704 T^{2} + \cdots + 160985344)^{2}$$
$97$ $$(T^{2} - 60 T - 8460)^{4}$$