# Properties

 Label 72.2.n.a Level $72$ Weight $2$ Character orbit 72.n Analytic conductor $0.575$ 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] = [72,2,Mod(13,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("72.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.n (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: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} - 2 \zeta_{12} q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10})$$ q + (z^3 - z^2 - z) * q^2 + (-z^3 - z) * q^3 + 2*z * q^4 - 2*z * q^5 + (2*z^3 + z^2 - z + 1) * q^6 - 4*z^2 * q^7 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} - 2 \zeta_{12} q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (2 \zeta_{12}^{3} + 2) q^{10} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + 2 \zeta_{12} q^{13} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{14} + (4 \zeta_{12}^{2} - 2) q^{15} + 4 \zeta_{12}^{2} q^{16} + 5 q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} - \zeta_{12}^{3} q^{19} - 4 \zeta_{12}^{2} q^{20} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{21} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{22} + (2 \zeta_{12}^{2} - 2) q^{23} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{24} - \zeta_{12}^{2} q^{25} + ( - 2 \zeta_{12}^{3} - 2) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} - 8 \zeta_{12}^{3} q^{28} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{30} + ( - 4 \zeta_{12}^{2} + 4) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 3 \zeta_{12}^{2} - 3) q^{33} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12}) q^{34} + 8 \zeta_{12}^{3} q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} - 2 \zeta_{12}^{3} q^{37} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{38} + ( - 4 \zeta_{12}^{2} + 2) q^{39} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{40} + ( - 5 \zeta_{12}^{2} + 5) q^{41} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{42} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{43} + 6 q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{45} + ( - 2 \zeta_{12}^{3} + 2) q^{46} + 6 \zeta_{12}^{2} q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + (9 \zeta_{12}^{2} - 9) q^{49} + (\zeta_{12}^{2} + \zeta_{12} - 1) q^{50} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{51} + 4 \zeta_{12}^{2} q^{52} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 6) q^{54} - 6 q^{55} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{56} + (\zeta_{12}^{2} - 2) q^{57} - \zeta_{12} q^{59} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{60} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{61} + (4 \zeta_{12}^{3} - 4) q^{62} + 12 q^{63} + 8 \zeta_{12}^{3} q^{64} - 4 \zeta_{12}^{2} q^{65} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 3) q^{66} + 3 \zeta_{12} q^{67} + 10 \zeta_{12} q^{68} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{69} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{70} - 6 q^{71} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + 9 q^{73} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{74} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{75} + ( - 2 \zeta_{12}^{2} + 2) q^{76} - 12 \zeta_{12} q^{77} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{78} + 14 \zeta_{12}^{2} q^{79} - 8 \zeta_{12}^{3} q^{80} - 9 \zeta_{12}^{2} q^{81} + (5 \zeta_{12}^{3} - 5) q^{82} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{83} + (8 \zeta_{12}^{2} - 16) q^{84} - 10 \zeta_{12} q^{85} + ( - 11 \zeta_{12}^{2} + 11 \zeta_{12} + 11) q^{86} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{88} - 14 q^{89} + (6 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{90} - 8 \zeta_{12}^{3} q^{91} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{92} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{93} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{94} + (2 \zeta_{12}^{2} - 2) q^{95} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 4) q^{96} - \zeta_{12}^{2} q^{97} + ( - 9 \zeta_{12}^{3} + 9) q^{98} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + (z^3 - z^2 - z) * q^2 + (-z^3 - z) * q^3 + 2*z * q^4 - 2*z * q^5 + (2*z^3 + z^2 - z + 1) * q^6 - 4*z^2 * q^7 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 + (2*z^3 + 2) * q^10 + (-3*z^3 + 3*z) * q^11 + (-4*z^2 + 2) * q^12 + 2*z * q^13 + (4*z^2 + 4*z - 4) * q^14 + (4*z^2 - 2) * q^15 + 4*z^2 * q^16 + 5 * q^17 + (-3*z^3 + 3) * q^18 - z^3 * q^19 - 4*z^2 * q^20 + (8*z^3 - 4*z) * q^21 + (3*z^2 - 3*z - 3) * q^22 + (2*z^2 - 2) * q^23 + (2*z^3 + 2*z^2 + 2*z - 4) * q^24 - z^2 * q^25 + (-2*z^3 - 2) * q^26 + (-3*z^3 + 6*z) * q^27 - 8*z^3 * q^28 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^30 + (-4*z^2 + 4) * q^31 + (-4*z^2 - 4*z + 4) * q^32 + (-3*z^2 - 3) * q^33 + (5*z^3 - 5*z^2 - 5*z) * q^34 + 8*z^3 * q^35 + (6*z^3 - 6*z) * q^36 - 2*z^3 * q^37 + (z^3 + z^2 - z) * q^38 + (-4*z^2 + 2) * q^39 + (4*z^2 + 4*z - 4) * q^40 + (-5*z^2 + 5) * q^41 + (-4*z^3 - 8*z^2 + 8*z + 4) * q^42 + (11*z^3 - 11*z) * q^43 + 6 * q^44 + (-6*z^3 + 6*z) * q^45 + (-2*z^3 + 2) * q^46 + 6*z^2 * q^47 + (-8*z^3 + 4*z) * q^48 + (9*z^2 - 9) * q^49 + (z^2 + z - 1) * q^50 + (-5*z^3 - 5*z) * q^51 + 4*z^2 * q^52 + (-3*z^3 + 3*z^2 - 3*z - 6) * q^54 - 6 * q^55 + (8*z^3 + 8*z^2 - 8*z) * q^56 + (z^2 - 2) * q^57 - z * q^59 + (8*z^3 - 4*z) * q^60 + (-12*z^3 + 12*z) * q^61 + (4*z^3 - 4) * q^62 + 12 * q^63 + 8*z^3 * q^64 - 4*z^2 * q^65 + (-3*z^3 + 6*z^2 + 6*z - 3) * q^66 + 3*z * q^67 + 10*z * q^68 + (-2*z^3 + 4*z) * q^69 + (-8*z^3 - 8*z^2 + 8*z) * q^70 - 6 * q^71 + (-6*z^2 + 6*z + 6) * q^72 + 9 * q^73 + (2*z^3 + 2*z^2 - 2*z) * q^74 + (2*z^3 - z) * q^75 + (-2*z^2 + 2) * q^76 - 12*z * q^77 + (2*z^3 + 2*z^2 + 2*z - 4) * q^78 + 14*z^2 * q^79 - 8*z^3 * q^80 - 9*z^2 * q^81 + (5*z^3 - 5) * q^82 + (-4*z^3 + 4*z) * q^83 + (8*z^2 - 16) * q^84 - 10*z * q^85 + (-11*z^2 + 11*z + 11) * q^86 + (6*z^3 - 6*z^2 - 6*z) * q^88 - 14 * q^89 + (6*z^2 - 6*z - 6) * q^90 - 8*z^3 * q^91 + (4*z^3 - 4*z) * q^92 + (4*z^3 - 8*z) * q^93 + (-6*z^2 - 6*z + 6) * q^94 + (2*z^2 - 2) * q^95 + (4*z^3 + 8*z^2 - 8*z - 4) * q^96 - z^2 * q^97 + (-9*z^3 + 9) * q^98 + 9*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 6 * q^6 - 8 * q^7 - 8 * q^8 - 6 * q^9 $$4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} - 6 q^{9} + 8 q^{10} - 8 q^{14} + 8 q^{16} + 20 q^{17} + 12 q^{18} - 8 q^{20} - 6 q^{22} - 4 q^{23} - 12 q^{24} - 2 q^{25} - 8 q^{26} + 12 q^{30} + 8 q^{31} + 8 q^{32} - 18 q^{33} - 10 q^{34} + 2 q^{38} - 8 q^{40} + 10 q^{41} + 24 q^{44} + 8 q^{46} + 12 q^{47} - 18 q^{49} - 2 q^{50} + 8 q^{52} - 18 q^{54} - 24 q^{55} + 16 q^{56} - 6 q^{57} - 16 q^{62} + 48 q^{63} - 8 q^{65} - 16 q^{70} - 24 q^{71} + 12 q^{72} + 36 q^{73} + 4 q^{74} + 4 q^{76} - 12 q^{78} + 28 q^{79} - 18 q^{81} - 20 q^{82} - 48 q^{84} + 22 q^{86} - 12 q^{88} - 56 q^{89} - 12 q^{90} + 12 q^{94} - 4 q^{95} - 2 q^{97} + 36 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 6 * q^6 - 8 * q^7 - 8 * q^8 - 6 * q^9 + 8 * q^10 - 8 * q^14 + 8 * q^16 + 20 * q^17 + 12 * q^18 - 8 * q^20 - 6 * q^22 - 4 * q^23 - 12 * q^24 - 2 * q^25 - 8 * q^26 + 12 * q^30 + 8 * q^31 + 8 * q^32 - 18 * q^33 - 10 * q^34 + 2 * q^38 - 8 * q^40 + 10 * q^41 + 24 * q^44 + 8 * q^46 + 12 * q^47 - 18 * q^49 - 2 * q^50 + 8 * q^52 - 18 * q^54 - 24 * q^55 + 16 * q^56 - 6 * q^57 - 16 * q^62 + 48 * q^63 - 8 * q^65 - 16 * q^70 - 24 * q^71 + 12 * q^72 + 36 * q^73 + 4 * q^74 + 4 * q^76 - 12 * q^78 + 28 * q^79 - 18 * q^81 - 20 * q^82 - 48 * q^84 + 22 * q^86 - 12 * q^88 - 56 * q^89 - 12 * q^90 + 12 * q^94 - 4 * q^95 - 2 * q^97 + 36 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i −1.73205 1.00000i 0.633975 + 2.36603i −2.00000 3.46410i −2.00000 2.00000i −1.50000 + 2.59808i 2.00000 + 2.00000i
13.2 0.366025 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.73205 + 1.00000i 2.36603 0.633975i −2.00000 3.46410i −2.00000 + 2.00000i −1.50000 + 2.59808i 2.00000 2.00000i
61.1 −1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i −1.73205 + 1.00000i 0.633975 2.36603i −2.00000 + 3.46410i −2.00000 + 2.00000i −1.50000 2.59808i 2.00000 2.00000i
61.2 0.366025 + 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.73205 1.00000i 2.36603 + 0.633975i −2.00000 + 3.46410i −2.00000 2.00000i −1.50000 2.59808i 2.00000 + 2.00000i
 $$n$$: 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
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.n.a 4
3.b odd 2 1 216.2.n.a 4
4.b odd 2 1 288.2.r.a 4
8.b even 2 1 inner 72.2.n.a 4
8.d odd 2 1 288.2.r.a 4
9.c even 3 1 inner 72.2.n.a 4
9.c even 3 1 648.2.d.d 2
9.d odd 6 1 216.2.n.a 4
9.d odd 6 1 648.2.d.a 2
12.b even 2 1 864.2.r.a 4
24.f even 2 1 864.2.r.a 4
24.h odd 2 1 216.2.n.a 4
36.f odd 6 1 288.2.r.a 4
36.f odd 6 1 2592.2.d.b 2
36.h even 6 1 864.2.r.a 4
36.h even 6 1 2592.2.d.a 2
72.j odd 6 1 216.2.n.a 4
72.j odd 6 1 648.2.d.a 2
72.l even 6 1 864.2.r.a 4
72.l even 6 1 2592.2.d.a 2
72.n even 6 1 inner 72.2.n.a 4
72.n even 6 1 648.2.d.d 2
72.p odd 6 1 288.2.r.a 4
72.p odd 6 1 2592.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.a 4 1.a even 1 1 trivial
72.2.n.a 4 8.b even 2 1 inner
72.2.n.a 4 9.c even 3 1 inner
72.2.n.a 4 72.n even 6 1 inner
216.2.n.a 4 3.b odd 2 1
216.2.n.a 4 9.d odd 6 1
216.2.n.a 4 24.h odd 2 1
216.2.n.a 4 72.j odd 6 1
288.2.r.a 4 4.b odd 2 1
288.2.r.a 4 8.d odd 2 1
288.2.r.a 4 36.f odd 6 1
288.2.r.a 4 72.p odd 6 1
648.2.d.a 2 9.d odd 6 1
648.2.d.a 2 72.j odd 6 1
648.2.d.d 2 9.c even 3 1
648.2.d.d 2 72.n even 6 1
864.2.r.a 4 12.b even 2 1
864.2.r.a 4 24.f even 2 1
864.2.r.a 4 36.h even 6 1
864.2.r.a 4 72.l even 6 1
2592.2.d.a 2 36.h even 6 1
2592.2.d.a 2 72.l even 6 1
2592.2.d.b 2 36.f odd 6 1
2592.2.d.b 2 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 4T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} - 4T^{2} + 16$$
$7$ $$(T^{2} + 4 T + 16)^{2}$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$T^{4} - 4T^{2} + 16$$
$17$ $$(T - 5)^{4}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$(T^{2} + 2 T + 4)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} - 5 T + 25)^{2}$$
$43$ $$T^{4} - 121 T^{2} + 14641$$
$47$ $$(T^{2} - 6 T + 36)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4} - T^{2} + 1$$
$61$ $$T^{4} - 144 T^{2} + 20736$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T + 6)^{4}$$
$73$ $$(T - 9)^{4}$$
$79$ $$(T^{2} - 14 T + 196)^{2}$$
$83$ $$T^{4} - 16T^{2} + 256$$
$89$ $$(T + 14)^{4}$$
$97$ $$(T^{2} + T + 1)^{2}$$