# Properties

 Label 72.2.d.b Level $72$ Weight $2$ Character orbit 72.d Analytic conductor $0.575$ 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] = [72,2,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, 2, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} - 2 i q^{5} - 2 q^{7} + (2 i - 2) q^{8} +O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 - 2*i * q^5 - 2 * q^7 + (2*i - 2) * q^8 $$q + (i + 1) q^{2} + 2 i q^{4} - 2 i q^{5} - 2 q^{7} + (2 i - 2) q^{8} + ( - 2 i + 2) q^{10} - 4 i q^{13} + ( - 2 i - 2) q^{14} - 4 q^{16} + 2 q^{17} + 4 i q^{19} + 4 q^{20} - 4 q^{23} + q^{25} + ( - 4 i + 4) q^{26} - 4 i q^{28} + 6 i q^{29} + 2 q^{31} + ( - 4 i - 4) q^{32} + (2 i + 2) q^{34} + 4 i q^{35} + 8 i q^{37} + (4 i - 4) q^{38} + (4 i + 4) q^{40} - 2 q^{41} - 4 i q^{43} + ( - 4 i - 4) q^{46} + 12 q^{47} - 3 q^{49} + (i + 1) q^{50} + 8 q^{52} - 6 i q^{53} + ( - 4 i + 4) q^{56} + (6 i - 6) q^{58} - 4 i q^{59} + (2 i + 2) q^{62} - 8 i q^{64} - 8 q^{65} - 12 i q^{67} + 4 i q^{68} + (4 i - 4) q^{70} - 12 q^{71} - 6 q^{73} + (8 i - 8) q^{74} - 8 q^{76} + 10 q^{79} + 8 i q^{80} + ( - 2 i - 2) q^{82} - 16 i q^{83} - 4 i q^{85} + ( - 4 i + 4) q^{86} + 10 q^{89} + 8 i q^{91} - 8 i q^{92} + (12 i + 12) q^{94} + 8 q^{95} - 2 q^{97} + ( - 3 i - 3) q^{98} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 - 2*i * q^5 - 2 * q^7 + (2*i - 2) * q^8 + (-2*i + 2) * q^10 - 4*i * q^13 + (-2*i - 2) * q^14 - 4 * q^16 + 2 * q^17 + 4*i * q^19 + 4 * q^20 - 4 * q^23 + q^25 + (-4*i + 4) * q^26 - 4*i * q^28 + 6*i * q^29 + 2 * q^31 + (-4*i - 4) * q^32 + (2*i + 2) * q^34 + 4*i * q^35 + 8*i * q^37 + (4*i - 4) * q^38 + (4*i + 4) * q^40 - 2 * q^41 - 4*i * q^43 + (-4*i - 4) * q^46 + 12 * q^47 - 3 * q^49 + (i + 1) * q^50 + 8 * q^52 - 6*i * q^53 + (-4*i + 4) * q^56 + (6*i - 6) * q^58 - 4*i * q^59 + (2*i + 2) * q^62 - 8*i * q^64 - 8 * q^65 - 12*i * q^67 + 4*i * q^68 + (4*i - 4) * q^70 - 12 * q^71 - 6 * q^73 + (8*i - 8) * q^74 - 8 * q^76 + 10 * q^79 + 8*i * q^80 + (-2*i - 2) * q^82 - 16*i * q^83 - 4*i * q^85 + (-4*i + 4) * q^86 + 10 * q^89 + 8*i * q^91 - 8*i * q^92 + (12*i + 12) * q^94 + 8 * q^95 - 2 * q^97 + (-3*i - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{7} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^7 - 4 * q^8 $$2 q + 2 q^{2} - 4 q^{7} - 4 q^{8} + 4 q^{10} - 4 q^{14} - 8 q^{16} + 4 q^{17} + 8 q^{20} - 8 q^{23} + 2 q^{25} + 8 q^{26} + 4 q^{31} - 8 q^{32} + 4 q^{34} - 8 q^{38} + 8 q^{40} - 4 q^{41} - 8 q^{46} + 24 q^{47} - 6 q^{49} + 2 q^{50} + 16 q^{52} + 8 q^{56} - 12 q^{58} + 4 q^{62} - 16 q^{65} - 8 q^{70} - 24 q^{71} - 12 q^{73} - 16 q^{74} - 16 q^{76} + 20 q^{79} - 4 q^{82} + 8 q^{86} + 20 q^{89} + 24 q^{94} + 16 q^{95} - 4 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^7 - 4 * q^8 + 4 * q^10 - 4 * q^14 - 8 * q^16 + 4 * q^17 + 8 * q^20 - 8 * q^23 + 2 * q^25 + 8 * q^26 + 4 * q^31 - 8 * q^32 + 4 * q^34 - 8 * q^38 + 8 * q^40 - 4 * q^41 - 8 * q^46 + 24 * q^47 - 6 * q^49 + 2 * q^50 + 16 * q^52 + 8 * q^56 - 12 * q^58 + 4 * q^62 - 16 * q^65 - 8 * q^70 - 24 * q^71 - 12 * q^73 - 16 * q^74 - 16 * q^76 + 20 * q^79 - 4 * q^82 + 8 * q^86 + 20 * q^89 + 24 * q^94 + 16 * q^95 - 4 * q^97 - 6 * 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
 − 1.00000i 1.00000i
1.00000 1.00000i 0 2.00000i 2.00000i 0 −2.00000 −2.00000 2.00000i 0 2.00000 + 2.00000i
37.2 1.00000 + 1.00000i 0 2.00000i 2.00000i 0 −2.00000 −2.00000 + 2.00000i 0 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.d.b 2
3.b odd 2 1 24.2.d.a 2
4.b odd 2 1 288.2.d.b 2
5.b even 2 1 1800.2.k.a 2
5.c odd 4 1 1800.2.d.b 2
5.c odd 4 1 1800.2.d.i 2
8.b even 2 1 inner 72.2.d.b 2
8.d odd 2 1 288.2.d.b 2
9.c even 3 2 648.2.n.c 4
9.d odd 6 2 648.2.n.k 4
12.b even 2 1 96.2.d.a 2
15.d odd 2 1 600.2.k.b 2
15.e even 4 1 600.2.d.b 2
15.e even 4 1 600.2.d.c 2
16.e even 4 1 2304.2.a.e 1
16.e even 4 1 2304.2.a.o 1
16.f odd 4 1 2304.2.a.b 1
16.f odd 4 1 2304.2.a.l 1
20.d odd 2 1 7200.2.k.d 2
20.e even 4 1 7200.2.d.d 2
20.e even 4 1 7200.2.d.g 2
21.c even 2 1 1176.2.c.a 2
24.f even 2 1 96.2.d.a 2
24.h odd 2 1 24.2.d.a 2
36.f odd 6 2 2592.2.r.g 4
36.h even 6 2 2592.2.r.f 4
40.e odd 2 1 7200.2.k.d 2
40.f even 2 1 1800.2.k.a 2
40.i odd 4 1 1800.2.d.b 2
40.i odd 4 1 1800.2.d.i 2
40.k even 4 1 7200.2.d.d 2
40.k even 4 1 7200.2.d.g 2
48.i odd 4 1 768.2.a.a 1
48.i odd 4 1 768.2.a.h 1
48.k even 4 1 768.2.a.d 1
48.k even 4 1 768.2.a.e 1
60.h even 2 1 2400.2.k.a 2
60.l odd 4 1 2400.2.d.b 2
60.l odd 4 1 2400.2.d.c 2
72.j odd 6 2 648.2.n.k 4
72.l even 6 2 2592.2.r.f 4
72.n even 6 2 648.2.n.c 4
72.p odd 6 2 2592.2.r.g 4
84.h odd 2 1 4704.2.c.a 2
120.i odd 2 1 600.2.k.b 2
120.m even 2 1 2400.2.k.a 2
120.q odd 4 1 2400.2.d.b 2
120.q odd 4 1 2400.2.d.c 2
120.w even 4 1 600.2.d.b 2
120.w even 4 1 600.2.d.c 2
168.e odd 2 1 4704.2.c.a 2
168.i even 2 1 1176.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 3.b odd 2 1
24.2.d.a 2 24.h odd 2 1
72.2.d.b 2 1.a even 1 1 trivial
72.2.d.b 2 8.b even 2 1 inner
96.2.d.a 2 12.b even 2 1
96.2.d.a 2 24.f even 2 1
288.2.d.b 2 4.b odd 2 1
288.2.d.b 2 8.d odd 2 1
600.2.d.b 2 15.e even 4 1
600.2.d.b 2 120.w even 4 1
600.2.d.c 2 15.e even 4 1
600.2.d.c 2 120.w even 4 1
600.2.k.b 2 15.d odd 2 1
600.2.k.b 2 120.i odd 2 1
648.2.n.c 4 9.c even 3 2
648.2.n.c 4 72.n even 6 2
648.2.n.k 4 9.d odd 6 2
648.2.n.k 4 72.j odd 6 2
768.2.a.a 1 48.i odd 4 1
768.2.a.d 1 48.k even 4 1
768.2.a.e 1 48.k even 4 1
768.2.a.h 1 48.i odd 4 1
1176.2.c.a 2 21.c even 2 1
1176.2.c.a 2 168.i even 2 1
1800.2.d.b 2 5.c odd 4 1
1800.2.d.b 2 40.i odd 4 1
1800.2.d.i 2 5.c odd 4 1
1800.2.d.i 2 40.i odd 4 1
1800.2.k.a 2 5.b even 2 1
1800.2.k.a 2 40.f even 2 1
2304.2.a.b 1 16.f odd 4 1
2304.2.a.e 1 16.e even 4 1
2304.2.a.l 1 16.f odd 4 1
2304.2.a.o 1 16.e even 4 1
2400.2.d.b 2 60.l odd 4 1
2400.2.d.b 2 120.q odd 4 1
2400.2.d.c 2 60.l odd 4 1
2400.2.d.c 2 120.q odd 4 1
2400.2.k.a 2 60.h even 2 1
2400.2.k.a 2 120.m even 2 1
2592.2.r.f 4 36.h even 6 2
2592.2.r.f 4 72.l even 6 2
2592.2.r.g 4 36.f odd 6 2
2592.2.r.g 4 72.p odd 6 2
4704.2.c.a 2 84.h odd 2 1
4704.2.c.a 2 168.e odd 2 1
7200.2.d.d 2 20.e even 4 1
7200.2.d.d 2 40.k even 4 1
7200.2.d.g 2 20.e even 4 1
7200.2.d.g 2 40.k even 4 1
7200.2.k.d 2 20.d odd 2 1
7200.2.k.d 2 40.e odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 12)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 10)^{2}$$
$97$ $$(T + 2)^{2}$$