# Properties

 Label 72.4.d.a Level $72$ Weight $4$ Character orbit 72.d Analytic conductor $4.248$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 72.d (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: $$4.24813752041$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 8 q^{4} - 7 \beta q^{5} - 34 q^{7} - 8 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 8 * q^4 - 7*b * q^5 - 34 * q^7 - 8*b * q^8 $$q + \beta q^{2} - 8 q^{4} - 7 \beta q^{5} - 34 q^{7} - 8 \beta q^{8} + 56 q^{10} + 2 \beta q^{11} - 34 \beta q^{14} + 64 q^{16} + 56 \beta q^{20} - 16 q^{22} - 267 q^{25} + 272 q^{28} - 79 \beta q^{29} - 70 q^{31} + 64 \beta q^{32} + 238 \beta q^{35} - 448 q^{40} - 16 \beta q^{44} + 813 q^{49} - 267 \beta q^{50} - 205 \beta q^{53} + 112 q^{55} + 272 \beta q^{56} + 632 q^{58} - 196 \beta q^{59} - 70 \beta q^{62} - 512 q^{64} - 1904 q^{70} - 322 q^{73} - 68 \beta q^{77} + 1370 q^{79} - 448 \beta q^{80} + 434 \beta q^{83} + 128 q^{88} - 574 q^{97} + 813 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 8 * q^4 - 7*b * q^5 - 34 * q^7 - 8*b * q^8 + 56 * q^10 + 2*b * q^11 - 34*b * q^14 + 64 * q^16 + 56*b * q^20 - 16 * q^22 - 267 * q^25 + 272 * q^28 - 79*b * q^29 - 70 * q^31 + 64*b * q^32 + 238*b * q^35 - 448 * q^40 - 16*b * q^44 + 813 * q^49 - 267*b * q^50 - 205*b * q^53 + 112 * q^55 + 272*b * q^56 + 632 * q^58 - 196*b * q^59 - 70*b * q^62 - 512 * q^64 - 1904 * q^70 - 322 * q^73 - 68*b * q^77 + 1370 * q^79 - 448*b * q^80 + 434*b * q^83 + 128 * q^88 - 574 * q^97 + 813*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4} - 68 q^{7}+O(q^{10})$$ 2 * q - 16 * q^4 - 68 * q^7 $$2 q - 16 q^{4} - 68 q^{7} + 112 q^{10} + 128 q^{16} - 32 q^{22} - 534 q^{25} + 544 q^{28} - 140 q^{31} - 896 q^{40} + 1626 q^{49} + 224 q^{55} + 1264 q^{58} - 1024 q^{64} - 3808 q^{70} - 644 q^{73} + 2740 q^{79} + 256 q^{88} - 1148 q^{97}+O(q^{100})$$ 2 * q - 16 * q^4 - 68 * q^7 + 112 * q^10 + 128 * q^16 - 32 * q^22 - 534 * q^25 + 544 * q^28 - 140 * q^31 - 896 * q^40 + 1626 * q^49 + 224 * q^55 + 1264 * q^58 - 1024 * q^64 - 3808 * q^70 - 644 * q^73 + 2740 * q^79 + 256 * q^88 - 1148 * q^97

## Character values

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

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 1.41421i 1.41421i
2.82843i 0 −8.00000 19.7990i 0 −34.0000 22.6274i 0 56.0000
37.2 2.82843i 0 −8.00000 19.7990i 0 −34.0000 22.6274i 0 56.0000
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.d.a 2
3.b odd 2 1 inner 72.4.d.a 2
4.b odd 2 1 288.4.d.b 2
8.b even 2 1 inner 72.4.d.a 2
8.d odd 2 1 288.4.d.b 2
12.b even 2 1 288.4.d.b 2
16.e even 4 2 2304.4.a.bo 2
16.f odd 4 2 2304.4.a.u 2
24.f even 2 1 288.4.d.b 2
24.h odd 2 1 CM 72.4.d.a 2
48.i odd 4 2 2304.4.a.bo 2
48.k even 4 2 2304.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.d.a 2 1.a even 1 1 trivial
72.4.d.a 2 3.b odd 2 1 inner
72.4.d.a 2 8.b even 2 1 inner
72.4.d.a 2 24.h odd 2 1 CM
288.4.d.b 2 4.b odd 2 1
288.4.d.b 2 8.d odd 2 1
288.4.d.b 2 12.b even 2 1
288.4.d.b 2 24.f even 2 1
2304.4.a.u 2 16.f odd 4 2
2304.4.a.u 2 48.k even 4 2
2304.4.a.bo 2 16.e even 4 2
2304.4.a.bo 2 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 392$$ acting on $$S_{4}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 8$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 392$$
$7$ $$(T + 34)^{2}$$
$11$ $$T^{2} + 32$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 49928$$
$31$ $$(T + 70)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 336200$$
$59$ $$T^{2} + 307328$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 322)^{2}$$
$79$ $$(T - 1370)^{2}$$
$83$ $$T^{2} + 1506848$$
$89$ $$T^{2}$$
$97$ $$(T + 574)^{2}$$
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