# Properties

 Label 72.2.l.a Level $72$ Weight $2$ Character orbit 72.l Analytic conductor $0.575$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,2,Mod(11,72)]

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([3, 3, 1]))

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

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

chi := DirichletCharacter("72.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.l (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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + 2 \beta_{3} q^{8} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - b1 + 1) * q^3 + 2*b2 * q^4 + (-b3 - 2*b2 + b1) * q^6 + 2*b3 * q^8 + (2*b3 + b2 - 2*b1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + 2 \beta_{3} q^{8} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{9} + ( - 3 \beta_{2} - \beta_1 - 3) q^{11} + ( - 2 \beta_{3} + 2) q^{12} + (4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{3} + 6 \beta_{2} - 3) q^{17} + (\beta_{3} - 4) q^{18} + ( - 3 \beta_{3} + 6 \beta_1 - 1) q^{19} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{22} + ( - 4 \beta_{2} + 2 \beta_1 + 4) q^{24} + ( - 5 \beta_{2} + 5) q^{25} + (\beta_{3} + 5) q^{27} + (4 \beta_{3} - 4 \beta_1) q^{32} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 6) q^{33} + (6 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 4) q^{34} + (2 \beta_{2} - 4 \beta_1 - 2) q^{36} + (6 \beta_{2} - \beta_1 + 6) q^{38} + ( - 4 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 6) q^{41} + (6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{43} + ( - 2 \beta_{3} - 12 \beta_{2} + 6) q^{44} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{48} - 7 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{50} + ( - 6 \beta_{3} + 7 \beta_{2} + \beta_1 - 1) q^{51} + (2 \beta_{2} + 5 \beta_1 - 2) q^{54} + ( - 6 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 7) q^{57} + (5 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 6) q^{59} - 8 q^{64} + (5 \beta_{3} + 12 \beta_{2} - 6 \beta_1 - 8) q^{66} + ( - 3 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{67} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 12) q^{68} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{72} + (6 \beta_{3} - 12 \beta_1 - 1) q^{73} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{75} + (6 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{76} + ( - 7 \beta_{2} - 4 \beta_1 + 7) q^{81} + ( - 3 \beta_{3} + 6 \beta_1 + 8) q^{82} + 2 \beta_1 q^{83} + (5 \beta_{3} + 6 \beta_{2} - 5 \beta_1 - 12) q^{86} + ( - 12 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 4) q^{88} + 4 \beta_{3} q^{89} + (4 \beta_{3} + 8) q^{96} + ( - 12 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 5) q^{97} - 7 \beta_{3} q^{98} + ( - 7 \beta_{3} - 6 \beta_{2} + 12 \beta_1 + 7) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - b1 + 1) * q^3 + 2*b2 * q^4 + (-b3 - 2*b2 + b1) * q^6 + 2*b3 * q^8 + (2*b3 + b2 - 2*b1) * q^9 + (-3*b2 - b1 - 3) * q^11 + (-2*b3 + 2) * q^12 + (4*b2 - 4) * q^16 + (-2*b3 + 6*b2 - 3) * q^17 + (b3 - 4) * q^18 + (-3*b3 + 6*b1 - 1) * q^19 + (-3*b3 - 2*b2 - 3*b1) * q^22 + (-4*b2 + 2*b1 + 4) * q^24 + (-5*b2 + 5) * q^25 + (b3 + 5) * q^27 + (4*b3 - 4*b1) * q^32 + (4*b3 + 5*b2 + 2*b1 - 6) * q^33 + (6*b3 - 4*b2 - 3*b1 + 4) * q^34 + (2*b2 - 4*b1 - 2) * q^36 + (6*b2 - b1 + 6) * q^38 + (-4*b3 - 3*b2 + 4*b1 + 6) * q^41 + (6*b3 + 5*b2 - 3*b1 - 5) * q^43 + (-2*b3 - 12*b2 + 6) * q^44 + (-4*b3 + 4*b2 + 4*b1) * q^48 - 7*b2 * q^49 + (-5*b3 + 5*b1) * q^50 + (-6*b3 + 7*b2 + b1 - 1) * q^51 + (2*b2 + 5*b1 - 2) * q^54 + (-6*b3 - 5*b2 + 4*b1 - 7) * q^57 + (5*b3 - 3*b2 - 5*b1 + 6) * q^59 - 8 * q^64 + (5*b3 + 12*b2 - 6*b1 - 8) * q^66 + (-3*b3 + 7*b2 - 3*b1) * q^67 + (-4*b3 + 6*b2 + 4*b1 - 12) * q^68 + (2*b3 - 8*b2 - 2*b1) * q^72 + (6*b3 - 12*b1 - 1) * q^73 + (5*b3 - 5*b2 - 5*b1) * q^75 + (6*b3 - 2*b2 + 6*b1) * q^76 + (-7*b2 - 4*b1 + 7) * q^81 + (-3*b3 + 6*b1 + 8) * q^82 + 2*b1 * q^83 + (5*b3 + 6*b2 - 5*b1 - 12) * q^86 + (-12*b3 - 4*b2 + 6*b1 + 4) * q^88 + 4*b3 * q^89 + (4*b3 + 8) * q^96 + (-12*b3 + 5*b2 + 6*b1 - 5) * q^97 - 7*b3 * q^98 + (-7*b3 - 6*b2 + 12*b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^4 - 4 * q^6 + 2 * q^9 $$4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9} - 18 q^{11} + 8 q^{12} - 8 q^{16} - 16 q^{18} - 4 q^{19} - 4 q^{22} + 8 q^{24} + 10 q^{25} + 20 q^{27} - 14 q^{33} + 8 q^{34} - 4 q^{36} + 36 q^{38} + 18 q^{41} - 10 q^{43} + 8 q^{48} - 14 q^{49} + 10 q^{51} - 4 q^{54} - 38 q^{57} + 18 q^{59} - 32 q^{64} - 8 q^{66} + 14 q^{67} - 36 q^{68} - 16 q^{72} - 4 q^{73} - 10 q^{75} - 4 q^{76} + 14 q^{81} + 32 q^{82} - 36 q^{86} + 8 q^{88} + 32 q^{96} - 10 q^{97} + 16 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^4 - 4 * q^6 + 2 * q^9 - 18 * q^11 + 8 * q^12 - 8 * q^16 - 16 * q^18 - 4 * q^19 - 4 * q^22 + 8 * q^24 + 10 * q^25 + 20 * q^27 - 14 * q^33 + 8 * q^34 - 4 * q^36 + 36 * q^38 + 18 * q^41 - 10 * q^43 + 8 * q^48 - 14 * q^49 + 10 * q^51 - 4 * q^54 - 38 * q^57 + 18 * q^59 - 32 * q^64 - 8 * q^66 + 14 * q^67 - 36 * q^68 - 16 * q^72 - 4 * q^73 - 10 * q^75 - 4 * q^76 + 14 * q^81 + 32 * q^82 - 36 * q^86 + 8 * q^88 + 32 * q^96 - 10 * q^97 + 16 * q^99

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

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

## Character values

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

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 1.72474 0.158919i 1.00000 + 1.73205i 0 −2.22474 1.02494i 0 2.82843i 2.94949 0.548188i 0
11.2 1.22474 + 0.707107i −0.724745 1.57313i 1.00000 + 1.73205i 0 0.224745 2.43916i 0 2.82843i −1.94949 + 2.28024i 0
59.1 −1.22474 + 0.707107i 1.72474 + 0.158919i 1.00000 1.73205i 0 −2.22474 + 1.02494i 0 2.82843i 2.94949 + 0.548188i 0
59.2 1.22474 0.707107i −0.724745 + 1.57313i 1.00000 1.73205i 0 0.224745 + 2.43916i 0 2.82843i −1.94949 2.28024i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.d odd 6 1 inner
72.l 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.l.a 4
3.b odd 2 1 216.2.l.a 4
4.b odd 2 1 288.2.p.a 4
8.b even 2 1 288.2.p.a 4
8.d odd 2 1 CM 72.2.l.a 4
9.c even 3 1 216.2.l.a 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 72.2.l.a 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 864.2.p.a 4
24.f even 2 1 216.2.l.a 4
24.h odd 2 1 864.2.p.a 4
36.f odd 6 1 864.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 288.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 288.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 72.2.l.a 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 864.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 216.2.l.a 4
72.p odd 6 1 648.2.f.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4} - 2 T^{3} + T^{2} - 6 T + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 18 T^{3} + 133 T^{2} + \cdots + 625$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 70T^{2} + 361$$
$19$ $$(T^{2} + 2 T - 53)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4} - 18 T^{3} + 103 T^{2} + \cdots + 25$$
$43$ $$T^{4} + 10 T^{3} + 129 T^{2} + \cdots + 841$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} - 18 T^{3} + 85 T^{2} + \cdots + 529$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 14 T^{3} + 201 T^{2} + \cdots + 25$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 2 T - 215)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} - 8T^{2} + 64$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$T^{4} + 10 T^{3} + 291 T^{2} + \cdots + 36481$$