# Properties

 Label 180.3.c.a Level $180$ Weight $3$ Character orbit 180.c 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(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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 20) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + b2 + b1 - 1) * q^4 + (-b3 - b1) * q^5 + (-2*b3 + b2 + 2*b1 - 2) * q^7 + (-2*b3 + 2*b2 - 2*b1 + 2) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - \beta_{3} - \beta_{2} - 3) q^{10} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{11} + (2 \beta_{3} + 2 \beta_1 - 4) q^{13} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 11) q^{14} + ( - 8 \beta_{3} - 8) q^{16} + (8 \beta_{3} + 8 \beta_1 + 6) q^{17} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{19} + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{20} + (10 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 18) q^{22} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{23} + 5 q^{25} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{26} + ( - 8 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 8) q^{28} + ( - 4 \beta_{3} - 4 \beta_1 + 2) q^{29} + (4 \beta_{3} - 10 \beta_{2} - 4 \beta_1 + 4) q^{31} - 32 q^{32} + (8 \beta_{3} + 8 \beta_{2} + 6 \beta_1 + 24) q^{34} - 5 \beta_{2} q^{35} + (10 \beta_{3} + 10 \beta_1 + 4) q^{37} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 20) q^{38} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 14) q^{40} + (6 \beta_{3} + 6 \beta_1 + 28) q^{41} + (4 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{43} + (8 \beta_{2} - 24 \beta_1 + 32) q^{44} + (7 \beta_{3} - 5 \beta_{2} + \beta_1 + 13) q^{46} + ( - 6 \beta_{3} - 13 \beta_{2} + 6 \beta_1 - 6) q^{47} + ( - 10 \beta_{3} - 10 \beta_1 - 1) q^{49} + 5 \beta_1 q^{50} + ( - 10 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 10) q^{52} + ( - 10 \beta_{3} - 10 \beta_1 + 44) q^{53} + ( - 8 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 8) q^{55} + ( - 24 \beta_{3} - 8 \beta_{2} - 16 \beta_1 - 24) q^{56} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 12) q^{58} + ( - 12 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 12) q^{59} + ( - 26 \beta_{3} - 26 \beta_1 + 32) q^{61} + (26 \beta_{3} - 14 \beta_{2} + 6 \beta_1 + 30) q^{62} - 32 \beta_1 q^{64} + (4 \beta_{3} + 4 \beta_1 - 10) q^{65} + (20 \beta_{3} - 11 \beta_{2} - 20 \beta_1 + 20) q^{67} + ( - 18 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 18) q^{68} + (15 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 5) q^{70} + (20 \beta_{3} + 2 \beta_{2} - 20 \beta_1 + 20) q^{71} + (32 \beta_{3} + 32 \beta_1 + 66) q^{73} + (10 \beta_{3} + 10 \beta_{2} + 4 \beta_1 + 30) q^{74} + ( - 8 \beta_{3} + 8 \beta_{2} - 24 \beta_1 + 8) q^{76} + (20 \beta_{3} + 20 \beta_1 - 60) q^{77} + ( - 16 \beta_{3} + 20 \beta_{2} + 16 \beta_1 - 16) q^{79} + ( - 8 \beta_{2} + 8 \beta_1 + 16) q^{80} + (6 \beta_{3} + 6 \beta_{2} + 28 \beta_1 + 18) q^{82} + (12 \beta_{3} + 13 \beta_{2} - 12 \beta_1 + 12) q^{83} + ( - 6 \beta_{3} - 6 \beta_1 - 40) q^{85} + ( - 13 \beta_{3} - \beta_{2} - 7 \beta_1 + 17) q^{86} + ( - 48 \beta_{3} - 16 \beta_{2} + 16) q^{88} + ( - 40 \beta_{3} - 40 \beta_1 + 22) q^{89} + (8 \beta_{3} + 6 \beta_{2} - 8 \beta_1 + 8) q^{91} + (16 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 32) q^{92} + (45 \beta_{3} - 7 \beta_{2} + 19 \beta_1 - 17) q^{94} + ( - 4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 4) q^{95} + (12 \beta_{3} + 12 \beta_1 - 66) q^{97} + ( - 10 \beta_{3} - 10 \beta_{2} - \beta_1 - 30) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + b2 + b1 - 1) * q^4 + (-b3 - b1) * q^5 + (-2*b3 + b2 + 2*b1 - 2) * q^7 + (-2*b3 + 2*b2 - 2*b1 + 2) * q^8 + (-b3 - b2 - 3) * q^10 + (-4*b3 - 2*b2 + 4*b1 - 4) * q^11 + (2*b3 + 2*b1 - 4) * q^13 + (-b3 + 3*b2 + b1 - 11) * q^14 + (-8*b3 - 8) * q^16 + (8*b3 + 8*b1 + 6) * q^17 + (-4*b3 + 4*b1 - 4) * q^19 + (3*b3 - b2 - b1 - 3) * q^20 + (10*b3 + 2*b2 + 6*b1 - 18) * q^22 + (2*b3 - 3*b2 - 2*b1 + 2) * q^23 + 5 * q^25 + (2*b3 + 2*b2 - 4*b1 + 6) * q^26 + (-8*b3 + 4*b2 - 12*b1 - 8) * q^28 + (-4*b3 - 4*b1 + 2) * q^29 + (4*b3 - 10*b2 - 4*b1 + 4) * q^31 - 32 * q^32 + (8*b3 + 8*b2 + 6*b1 + 24) * q^34 - 5*b2 * q^35 + (10*b3 + 10*b1 + 4) * q^37 + (4*b3 + 4*b2 + 4*b1 - 20) * q^38 + (2*b3 - 2*b2 - 6*b1 + 14) * q^40 + (6*b3 + 6*b1 + 28) * q^41 + (4*b3 + 3*b2 - 4*b1 + 4) * q^43 + (8*b2 - 24*b1 + 32) * q^44 + (7*b3 - 5*b2 + b1 + 13) * q^46 + (-6*b3 - 13*b2 + 6*b1 - 6) * q^47 + (-10*b3 - 10*b1 - 1) * q^49 + 5*b1 * q^50 + (-10*b3 - 2*b2 - 2*b1 + 10) * q^52 + (-10*b3 - 10*b1 + 44) * q^53 + (-8*b3 - 6*b2 + 8*b1 - 8) * q^55 + (-24*b3 - 8*b2 - 16*b1 - 24) * q^56 + (-4*b3 - 4*b2 + 2*b1 - 12) * q^58 + (-12*b3 + 12*b2 + 12*b1 - 12) * q^59 + (-26*b3 - 26*b1 + 32) * q^61 + (26*b3 - 14*b2 + 6*b1 + 30) * q^62 - 32*b1 * q^64 + (4*b3 + 4*b1 - 10) * q^65 + (20*b3 - 11*b2 - 20*b1 + 20) * q^67 + (-18*b3 + 14*b2 + 14*b1 + 18) * q^68 + (15*b3 - 5*b2 + 5*b1 + 5) * q^70 + (20*b3 + 2*b2 - 20*b1 + 20) * q^71 + (32*b3 + 32*b1 + 66) * q^73 + (10*b3 + 10*b2 + 4*b1 + 30) * q^74 + (-8*b3 + 8*b2 - 24*b1 + 8) * q^76 + (20*b3 + 20*b1 - 60) * q^77 + (-16*b3 + 20*b2 + 16*b1 - 16) * q^79 + (-8*b2 + 8*b1 + 16) * q^80 + (6*b3 + 6*b2 + 28*b1 + 18) * q^82 + (12*b3 + 13*b2 - 12*b1 + 12) * q^83 + (-6*b3 - 6*b1 - 40) * q^85 + (-13*b3 - b2 - 7*b1 + 17) * q^86 + (-48*b3 - 16*b2 + 16) * q^88 + (-40*b3 - 40*b1 + 22) * q^89 + (8*b3 + 6*b2 - 8*b1 + 8) * q^91 + (16*b3 - 4*b2 + 12*b1 + 32) * q^92 + (45*b3 - 7*b2 + 19*b1 - 17) * q^94 + (-4*b3 - 8*b2 + 4*b1 - 4) * q^95 + (12*b3 + 12*b1 - 66) * q^97 + (-10*b3 - 10*b2 - b1 - 30) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^4 + 8 * q^8 $$4 q + 2 q^{2} - 4 q^{4} + 8 q^{8} - 10 q^{10} - 16 q^{13} - 40 q^{14} - 16 q^{16} + 24 q^{17} - 20 q^{20} - 80 q^{22} + 20 q^{25} + 12 q^{26} - 40 q^{28} + 8 q^{29} - 128 q^{32} + 92 q^{34} + 16 q^{37} - 80 q^{38} + 40 q^{40} + 112 q^{41} + 80 q^{44} + 40 q^{46} - 4 q^{49} + 10 q^{50} + 56 q^{52} + 176 q^{53} - 80 q^{56} - 36 q^{58} + 128 q^{61} + 80 q^{62} - 64 q^{64} - 40 q^{65} + 136 q^{68} + 264 q^{73} + 108 q^{74} - 240 q^{77} + 80 q^{80} + 116 q^{82} - 160 q^{85} + 80 q^{86} + 160 q^{88} + 88 q^{89} + 120 q^{92} - 120 q^{94} - 264 q^{97} - 102 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^4 + 8 * q^8 - 10 * q^10 - 16 * q^13 - 40 * q^14 - 16 * q^16 + 24 * q^17 - 20 * q^20 - 80 * q^22 + 20 * q^25 + 12 * q^26 - 40 * q^28 + 8 * q^29 - 128 * q^32 + 92 * q^34 + 16 * q^37 - 80 * q^38 + 40 * q^40 + 112 * q^41 + 80 * q^44 + 40 * q^46 - 4 * q^49 + 10 * q^50 + 56 * q^52 + 176 * q^53 - 80 * q^56 - 36 * q^58 + 128 * q^61 + 80 * q^62 - 64 * q^64 - 40 * q^65 + 136 * q^68 + 264 * q^73 + 108 * q^74 - 240 * q^77 + 80 * q^80 + 116 * q^82 - 160 * q^85 + 80 * q^86 + 160 * q^88 + 88 * q^89 + 120 * q^92 - 120 * q^94 - 264 * q^97 - 102 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{10}$$ 2*v $$\beta_{2}$$ $$=$$ $$2\zeta_{10}^{3} + 2\zeta_{10}^{2}$$ 2*v^3 + 2*v^2 $$\beta_{3}$$ $$=$$ $$-2\zeta_{10}^{3} + 2\zeta_{10}^{2} - 2\zeta_{10} + 1$$ -2*v^3 + 2*v^2 - 2*v + 1
 $$\zeta_{10}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{10}^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta _1 - 1 ) / 4$$ (b3 + b2 + b1 - 1) / 4 $$\zeta_{10}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4$$ (-b3 + b2 - b1 + 1) / 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
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i
−0.618034 1.90211i 0 −3.23607 + 2.35114i 2.23607 0 5.25731i 6.47214 + 4.70228i 0 −1.38197 4.25325i
91.2 −0.618034 + 1.90211i 0 −3.23607 2.35114i 2.23607 0 5.25731i 6.47214 4.70228i 0 −1.38197 + 4.25325i
91.3 1.61803 1.17557i 0 1.23607 3.80423i −2.23607 0 8.50651i −2.47214 7.60845i 0 −3.61803 + 2.62866i
91.4 1.61803 + 1.17557i 0 1.23607 + 3.80423i −2.23607 0 8.50651i −2.47214 + 7.60845i 0 −3.61803 2.62866i
 $$n$$: 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
4.b 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.c.a 4
3.b odd 2 1 20.3.b.a 4
4.b odd 2 1 inner 180.3.c.a 4
5.b even 2 1 900.3.c.k 4
5.c odd 4 2 900.3.f.e 8
8.b even 2 1 2880.3.e.e 4
8.d odd 2 1 2880.3.e.e 4
12.b even 2 1 20.3.b.a 4
15.d odd 2 1 100.3.b.f 4
15.e even 4 2 100.3.d.b 8
20.d odd 2 1 900.3.c.k 4
20.e even 4 2 900.3.f.e 8
24.f even 2 1 320.3.b.c 4
24.h odd 2 1 320.3.b.c 4
48.i odd 4 2 1280.3.g.e 8
48.k even 4 2 1280.3.g.e 8
60.h even 2 1 100.3.b.f 4
60.l odd 4 2 100.3.d.b 8
120.i odd 2 1 1600.3.b.s 4
120.m even 2 1 1600.3.b.s 4
120.q odd 4 2 1600.3.h.n 8
120.w even 4 2 1600.3.h.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 3.b odd 2 1
20.3.b.a 4 12.b even 2 1
100.3.b.f 4 15.d odd 2 1
100.3.b.f 4 60.h even 2 1
100.3.d.b 8 15.e even 4 2
100.3.d.b 8 60.l odd 4 2
180.3.c.a 4 1.a even 1 1 trivial
180.3.c.a 4 4.b odd 2 1 inner
320.3.b.c 4 24.f even 2 1
320.3.b.c 4 24.h odd 2 1
900.3.c.k 4 5.b even 2 1
900.3.c.k 4 20.d odd 2 1
900.3.f.e 8 5.c odd 4 2
900.3.f.e 8 20.e even 4 2
1280.3.g.e 8 48.i odd 4 2
1280.3.g.e 8 48.k even 4 2
1600.3.b.s 4 120.i odd 2 1
1600.3.b.s 4 120.m even 2 1
1600.3.h.n 8 120.q odd 4 2
1600.3.h.n 8 120.w even 4 2
2880.3.e.e 4 8.b even 2 1
2880.3.e.e 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 100T_{7}^{2} + 2000$$ acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$T^{4} + 100T^{2} + 2000$$
$11$ $$T^{4} + 400T^{2} + 1280$$
$13$ $$(T^{2} + 8 T - 4)^{2}$$
$17$ $$(T^{2} - 12 T - 284)^{2}$$
$19$ $$T^{4} + 320 T^{2} + 20480$$
$23$ $$T^{4} + 260T^{2} + 80$$
$29$ $$(T^{2} - 4 T - 76)^{2}$$
$31$ $$T^{4} + 2320 T^{2} + 154880$$
$37$ $$(T^{2} - 8 T - 484)^{2}$$
$41$ $$(T^{2} - 56 T + 604)^{2}$$
$43$ $$T^{4} + 500T^{2} + 2000$$
$47$ $$T^{4} + 4100 T^{2} + \cdots + 3561680$$
$53$ $$(T^{2} - 88 T + 1436)^{2}$$
$59$ $$T^{4} + 5760 T^{2} + \cdots + 1658880$$
$61$ $$(T^{2} - 64 T - 2356)^{2}$$
$67$ $$T^{4} + 10420 T^{2} + \cdots + 19920080$$
$71$ $$T^{4} + 8080 T^{2} + \cdots + 10138880$$
$73$ $$(T^{2} - 132 T - 764)^{2}$$
$79$ $$T^{4} + 13120 T^{2} + \cdots + 2478080$$
$83$ $$T^{4} + 6260 T^{2} + \cdots + 2620880$$
$89$ $$(T^{2} - 44 T - 7516)^{2}$$
$97$ $$(T^{2} + 132 T + 3636)^{2}$$