# Properties

 Label 72.4.d.b Level $72$ Weight $4$ Character orbit 72.d Analytic conductor $4.248$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

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: $$\Q(\sqrt{-7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 2$$ x^2 - x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) 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 $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta - 6) q^{4} + 4 \beta q^{5} - 8 q^{7} + ( - 4 \beta - 20) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b - 6) * q^4 + 4*b * q^5 - 8 * q^7 + (-4*b - 20) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta - 6) q^{4} + 4 \beta q^{5} - 8 q^{7} + ( - 4 \beta - 20) q^{8} + (4 \beta - 28) q^{10} + 6 \beta q^{11} + 20 \beta q^{13} + ( - 8 \beta - 8) q^{14} + ( - 24 \beta + 8) q^{16} + 14 q^{17} - 14 \beta q^{19} + ( - 24 \beta - 56) q^{20} + (6 \beta - 42) q^{22} + 152 q^{23} + 13 q^{25} + (20 \beta - 140) q^{26} + ( - 16 \beta + 48) q^{28} + 60 \beta q^{29} + 224 q^{31} + ( - 16 \beta + 176) q^{32} + (14 \beta + 14) q^{34} - 32 \beta q^{35} + 92 \beta q^{37} + ( - 14 \beta + 98) q^{38} + ( - 80 \beta + 112) q^{40} + 70 q^{41} - 166 \beta q^{43} + ( - 36 \beta - 84) q^{44} + (152 \beta + 152) q^{46} - 336 q^{47} - 279 q^{49} + (13 \beta + 13) q^{50} + ( - 120 \beta - 280) q^{52} - 12 \beta q^{53} - 168 q^{55} + (32 \beta + 160) q^{56} + (60 \beta - 420) q^{58} - 202 \beta q^{59} + 36 \beta q^{61} + (224 \beta + 224) q^{62} + (160 \beta + 288) q^{64} - 560 q^{65} + 66 \beta q^{67} + (28 \beta - 84) q^{68} + ( - 32 \beta + 224) q^{70} + 72 q^{71} - 294 q^{73} + (92 \beta - 644) q^{74} + (84 \beta + 196) q^{76} - 48 \beta q^{77} - 464 q^{79} + (32 \beta + 672) q^{80} + (70 \beta + 70) q^{82} + 206 \beta q^{83} + 56 \beta q^{85} + ( - 166 \beta + 1162) q^{86} + ( - 120 \beta + 168) q^{88} - 266 q^{89} - 160 \beta q^{91} + (304 \beta - 912) q^{92} + ( - 336 \beta - 336) q^{94} + 392 q^{95} + 994 q^{97} + ( - 279 \beta - 279) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b - 6) * q^4 + 4*b * q^5 - 8 * q^7 + (-4*b - 20) * q^8 + (4*b - 28) * q^10 + 6*b * q^11 + 20*b * q^13 + (-8*b - 8) * q^14 + (-24*b + 8) * q^16 + 14 * q^17 - 14*b * q^19 + (-24*b - 56) * q^20 + (6*b - 42) * q^22 + 152 * q^23 + 13 * q^25 + (20*b - 140) * q^26 + (-16*b + 48) * q^28 + 60*b * q^29 + 224 * q^31 + (-16*b + 176) * q^32 + (14*b + 14) * q^34 - 32*b * q^35 + 92*b * q^37 + (-14*b + 98) * q^38 + (-80*b + 112) * q^40 + 70 * q^41 - 166*b * q^43 + (-36*b - 84) * q^44 + (152*b + 152) * q^46 - 336 * q^47 - 279 * q^49 + (13*b + 13) * q^50 + (-120*b - 280) * q^52 - 12*b * q^53 - 168 * q^55 + (32*b + 160) * q^56 + (60*b - 420) * q^58 - 202*b * q^59 + 36*b * q^61 + (224*b + 224) * q^62 + (160*b + 288) * q^64 - 560 * q^65 + 66*b * q^67 + (28*b - 84) * q^68 + (-32*b + 224) * q^70 + 72 * q^71 - 294 * q^73 + (92*b - 644) * q^74 + (84*b + 196) * q^76 - 48*b * q^77 - 464 * q^79 + (32*b + 672) * q^80 + (70*b + 70) * q^82 + 206*b * q^83 + 56*b * q^85 + (-166*b + 1162) * q^86 + (-120*b + 168) * q^88 - 266 * q^89 - 160*b * q^91 + (304*b - 912) * q^92 + (-336*b - 336) * q^94 + 392 * q^95 + 994 * q^97 + (-279*b - 279) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 12 q^{4} - 16 q^{7} - 40 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 12 * q^4 - 16 * q^7 - 40 * q^8 $$2 q + 2 q^{2} - 12 q^{4} - 16 q^{7} - 40 q^{8} - 56 q^{10} - 16 q^{14} + 16 q^{16} + 28 q^{17} - 112 q^{20} - 84 q^{22} + 304 q^{23} + 26 q^{25} - 280 q^{26} + 96 q^{28} + 448 q^{31} + 352 q^{32} + 28 q^{34} + 196 q^{38} + 224 q^{40} + 140 q^{41} - 168 q^{44} + 304 q^{46} - 672 q^{47} - 558 q^{49} + 26 q^{50} - 560 q^{52} - 336 q^{55} + 320 q^{56} - 840 q^{58} + 448 q^{62} + 576 q^{64} - 1120 q^{65} - 168 q^{68} + 448 q^{70} + 144 q^{71} - 588 q^{73} - 1288 q^{74} + 392 q^{76} - 928 q^{79} + 1344 q^{80} + 140 q^{82} + 2324 q^{86} + 336 q^{88} - 532 q^{89} - 1824 q^{92} - 672 q^{94} + 784 q^{95} + 1988 q^{97} - 558 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 12 * q^4 - 16 * q^7 - 40 * q^8 - 56 * q^10 - 16 * q^14 + 16 * q^16 + 28 * q^17 - 112 * q^20 - 84 * q^22 + 304 * q^23 + 26 * q^25 - 280 * q^26 + 96 * q^28 + 448 * q^31 + 352 * q^32 + 28 * q^34 + 196 * q^38 + 224 * q^40 + 140 * q^41 - 168 * q^44 + 304 * q^46 - 672 * q^47 - 558 * q^49 + 26 * q^50 - 560 * q^52 - 336 * q^55 + 320 * q^56 - 840 * q^58 + 448 * q^62 + 576 * q^64 - 1120 * q^65 - 168 * q^68 + 448 * q^70 + 144 * q^71 - 588 * q^73 - 1288 * q^74 + 392 * q^76 - 928 * q^79 + 1344 * q^80 + 140 * q^82 + 2324 * q^86 + 336 * q^88 - 532 * q^89 - 1824 * q^92 - 672 * q^94 + 784 * q^95 + 1988 * q^97 - 558 * q^98

## Character values

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

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

## 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
 0.5 − 1.32288i 0.5 + 1.32288i
1.00000 2.64575i 0 −6.00000 5.29150i 10.5830i 0 −8.00000 −20.0000 + 10.5830i 0 −28.0000 10.5830i
37.2 1.00000 + 2.64575i 0 −6.00000 + 5.29150i 10.5830i 0 −8.00000 −20.0000 10.5830i 0 −28.0000 + 10.5830i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.d.b 2
3.b odd 2 1 8.4.b.a 2
4.b odd 2 1 288.4.d.a 2
8.b even 2 1 inner 72.4.d.b 2
8.d odd 2 1 288.4.d.a 2
12.b even 2 1 32.4.b.a 2
15.d odd 2 1 200.4.d.a 2
15.e even 4 2 200.4.f.a 4
16.e even 4 2 2304.4.a.bn 2
16.f odd 4 2 2304.4.a.v 2
24.f even 2 1 32.4.b.a 2
24.h odd 2 1 8.4.b.a 2
48.i odd 4 2 256.4.a.l 2
48.k even 4 2 256.4.a.j 2
60.h even 2 1 800.4.d.a 2
60.l odd 4 2 800.4.f.a 4
120.i odd 2 1 200.4.d.a 2
120.m even 2 1 800.4.d.a 2
120.q odd 4 2 800.4.f.a 4
120.w even 4 2 200.4.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 3.b odd 2 1
8.4.b.a 2 24.h odd 2 1
32.4.b.a 2 12.b even 2 1
32.4.b.a 2 24.f even 2 1
72.4.d.b 2 1.a even 1 1 trivial
72.4.d.b 2 8.b even 2 1 inner
200.4.d.a 2 15.d odd 2 1
200.4.d.a 2 120.i odd 2 1
200.4.f.a 4 15.e even 4 2
200.4.f.a 4 120.w even 4 2
256.4.a.j 2 48.k even 4 2
256.4.a.l 2 48.i odd 4 2
288.4.d.a 2 4.b odd 2 1
288.4.d.a 2 8.d odd 2 1
800.4.d.a 2 60.h even 2 1
800.4.d.a 2 120.m even 2 1
800.4.f.a 4 60.l odd 4 2
800.4.f.a 4 120.q odd 4 2
2304.4.a.v 2 16.f odd 4 2
2304.4.a.bn 2 16.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 112$$ acting on $$S_{4}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 8$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 112$$
$7$ $$(T + 8)^{2}$$
$11$ $$T^{2} + 252$$
$13$ $$T^{2} + 2800$$
$17$ $$(T - 14)^{2}$$
$19$ $$T^{2} + 1372$$
$23$ $$(T - 152)^{2}$$
$29$ $$T^{2} + 25200$$
$31$ $$(T - 224)^{2}$$
$37$ $$T^{2} + 59248$$
$41$ $$(T - 70)^{2}$$
$43$ $$T^{2} + 192892$$
$47$ $$(T + 336)^{2}$$
$53$ $$T^{2} + 1008$$
$59$ $$T^{2} + 285628$$
$61$ $$T^{2} + 9072$$
$67$ $$T^{2} + 30492$$
$71$ $$(T - 72)^{2}$$
$73$ $$(T + 294)^{2}$$
$79$ $$(T + 464)^{2}$$
$83$ $$T^{2} + 297052$$
$89$ $$(T + 266)^{2}$$
$97$ $$(T - 994)^{2}$$