# Properties

 Label 72.2.f.a Level $72$ Weight $2$ Character orbit 72.f 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(35,72)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.f (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: $$4$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{2} - 2 \beta_1) q^{5} - 2 \beta_{3} q^{7} + 2 \beta_{2} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 1) * q^4 + (b2 - 2*b1) * q^5 - 2*b3 * q^7 + 2*b2 * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{2} - 2 \beta_1) q^{5} - 2 \beta_{3} q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 3) q^{10} - 2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + ( - 4 \beta_{2} + 2 \beta_1) q^{14} + (2 \beta_{3} - 2) q^{16} + \beta_{2} q^{17} - 4 q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{20} + ( - 2 \beta_{3} + 2) q^{22} + ( - 2 \beta_{2} + 4 \beta_1) q^{23} + q^{25} + (4 \beta_{2} - 2 \beta_1) q^{26} + ( - 2 \beta_{3} + 6) q^{28} + ( - \beta_{2} + 2 \beta_1) q^{29} + 2 \beta_{3} q^{31} + (4 \beta_{2} - 4 \beta_1) q^{32} + (\beta_{3} - 1) q^{34} + 6 \beta_{2} q^{35} - 4 \beta_1 q^{38} - 4 \beta_{3} q^{40} + \beta_{2} q^{41} + 8 q^{43} + ( - 4 \beta_{2} + 4 \beta_1) q^{44} + (2 \beta_{3} + 6) q^{46} + (2 \beta_{2} - 4 \beta_1) q^{47} - 5 q^{49} + \beta_1 q^{50} + (2 \beta_{3} - 6) q^{52} + ( - 3 \beta_{2} + 6 \beta_1) q^{53} + 4 \beta_{3} q^{55} + ( - 4 \beta_{2} + 8 \beta_1) q^{56} + (\beta_{3} + 3) q^{58} - 8 \beta_{2} q^{59} - 8 \beta_{3} q^{61} + (4 \beta_{2} - 2 \beta_1) q^{62} - 8 q^{64} - 6 \beta_{2} q^{65} - 4 q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + (6 \beta_{3} - 6) q^{70} + (6 \beta_{2} - 12 \beta_1) q^{71} - 4 q^{73} + ( - 4 \beta_{3} - 4) q^{76} + (4 \beta_{2} - 8 \beta_1) q^{77} - 2 \beta_{3} q^{79} + ( - 8 \beta_{2} + 4 \beta_1) q^{80} + (\beta_{3} - 1) q^{82} + 10 \beta_{2} q^{83} - 2 \beta_{3} q^{85} + 8 \beta_1 q^{86} + 8 q^{88} - 5 \beta_{2} q^{89} + 12 q^{91} + (4 \beta_{2} + 4 \beta_1) q^{92} + ( - 2 \beta_{3} - 6) q^{94} + ( - 4 \beta_{2} + 8 \beta_1) q^{95} + 8 q^{97} - 5 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + 1) * q^4 + (b2 - 2*b1) * q^5 - 2*b3 * q^7 + 2*b2 * q^8 + (-b3 - 3) * q^10 - 2*b2 * q^11 + 2*b3 * q^13 + (-4*b2 + 2*b1) * q^14 + (2*b3 - 2) * q^16 + b2 * q^17 - 4 * q^19 + (-2*b2 - 2*b1) * q^20 + (-2*b3 + 2) * q^22 + (-2*b2 + 4*b1) * q^23 + q^25 + (4*b2 - 2*b1) * q^26 + (-2*b3 + 6) * q^28 + (-b2 + 2*b1) * q^29 + 2*b3 * q^31 + (4*b2 - 4*b1) * q^32 + (b3 - 1) * q^34 + 6*b2 * q^35 - 4*b1 * q^38 - 4*b3 * q^40 + b2 * q^41 + 8 * q^43 + (-4*b2 + 4*b1) * q^44 + (2*b3 + 6) * q^46 + (2*b2 - 4*b1) * q^47 - 5 * q^49 + b1 * q^50 + (2*b3 - 6) * q^52 + (-3*b2 + 6*b1) * q^53 + 4*b3 * q^55 + (-4*b2 + 8*b1) * q^56 + (b3 + 3) * q^58 - 8*b2 * q^59 - 8*b3 * q^61 + (4*b2 - 2*b1) * q^62 - 8 * q^64 - 6*b2 * q^65 - 4 * q^67 + (2*b2 - 2*b1) * q^68 + (6*b3 - 6) * q^70 + (6*b2 - 12*b1) * q^71 - 4 * q^73 + (-4*b3 - 4) * q^76 + (4*b2 - 8*b1) * q^77 - 2*b3 * q^79 + (-8*b2 + 4*b1) * q^80 + (b3 - 1) * q^82 + 10*b2 * q^83 - 2*b3 * q^85 + 8*b1 * q^86 + 8 * q^88 - 5*b2 * q^89 + 12 * q^91 + (4*b2 + 4*b1) * q^92 + (-2*b3 - 6) * q^94 + (-4*b2 + 8*b1) * q^95 + 8 * q^97 - 5*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} - 12 q^{10} - 8 q^{16} - 16 q^{19} + 8 q^{22} + 4 q^{25} + 24 q^{28} - 4 q^{34} + 32 q^{43} + 24 q^{46} - 20 q^{49} - 24 q^{52} + 12 q^{58} - 32 q^{64} - 16 q^{67} - 24 q^{70} - 16 q^{73} - 16 q^{76} - 4 q^{82} + 32 q^{88} + 48 q^{91} - 24 q^{94} + 32 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 - 12 * q^10 - 8 * q^16 - 16 * q^19 + 8 * q^22 + 4 * q^25 + 24 * q^28 - 4 * q^34 + 32 * q^43 + 24 * q^46 - 20 * q^49 - 24 * q^52 + 12 * q^58 - 32 * q^64 - 16 * q^67 - 24 * q^70 - 16 * q^73 - 16 * q^76 - 4 * q^82 + 32 * q^88 + 48 * q^91 - 24 * q^94 + 32 * q^97

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 2.44949 0 3.46410i 2.82843i 0 −3.00000 1.73205i
35.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.44949 0 3.46410i 2.82843i 0 −3.00000 + 1.73205i
35.3 1.22474 0.707107i 0 1.00000 1.73205i −2.44949 0 3.46410i 2.82843i 0 −3.00000 + 1.73205i
35.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.44949 0 3.46410i 2.82843i 0 −3.00000 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f 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.f.a 4
3.b odd 2 1 inner 72.2.f.a 4
4.b odd 2 1 288.2.f.a 4
5.b even 2 1 1800.2.b.c 4
5.c odd 4 2 1800.2.m.c 8
8.b even 2 1 288.2.f.a 4
8.d odd 2 1 inner 72.2.f.a 4
9.c even 3 1 648.2.l.a 4
9.c even 3 1 648.2.l.c 4
9.d odd 6 1 648.2.l.a 4
9.d odd 6 1 648.2.l.c 4
12.b even 2 1 288.2.f.a 4
15.d odd 2 1 1800.2.b.c 4
15.e even 4 2 1800.2.m.c 8
16.e even 4 2 2304.2.c.i 8
16.f odd 4 2 2304.2.c.i 8
20.d odd 2 1 7200.2.b.c 4
20.e even 4 2 7200.2.m.c 8
24.f even 2 1 inner 72.2.f.a 4
24.h odd 2 1 288.2.f.a 4
36.f odd 6 1 2592.2.p.a 4
36.f odd 6 1 2592.2.p.c 4
36.h even 6 1 2592.2.p.a 4
36.h even 6 1 2592.2.p.c 4
40.e odd 2 1 1800.2.b.c 4
40.f even 2 1 7200.2.b.c 4
40.i odd 4 2 7200.2.m.c 8
40.k even 4 2 1800.2.m.c 8
48.i odd 4 2 2304.2.c.i 8
48.k even 4 2 2304.2.c.i 8
60.h even 2 1 7200.2.b.c 4
60.l odd 4 2 7200.2.m.c 8
72.j odd 6 1 2592.2.p.a 4
72.j odd 6 1 2592.2.p.c 4
72.l even 6 1 648.2.l.a 4
72.l even 6 1 648.2.l.c 4
72.n even 6 1 2592.2.p.a 4
72.n even 6 1 2592.2.p.c 4
72.p odd 6 1 648.2.l.a 4
72.p odd 6 1 648.2.l.c 4
120.i odd 2 1 7200.2.b.c 4
120.m even 2 1 1800.2.b.c 4
120.q odd 4 2 1800.2.m.c 8
120.w even 4 2 7200.2.m.c 8

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 6)^{2}$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$(T^{2} + 8)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T + 4)^{4}$$
$23$ $$(T^{2} - 24)^{2}$$
$29$ $$(T^{2} - 6)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 2)^{2}$$
$43$ $$(T - 8)^{4}$$
$47$ $$(T^{2} - 24)^{2}$$
$53$ $$(T^{2} - 54)^{2}$$
$59$ $$(T^{2} + 128)^{2}$$
$61$ $$(T^{2} + 192)^{2}$$
$67$ $$(T + 4)^{4}$$
$71$ $$(T^{2} - 216)^{2}$$
$73$ $$(T + 4)^{4}$$
$79$ $$(T^{2} + 12)^{2}$$
$83$ $$(T^{2} + 200)^{2}$$
$89$ $$(T^{2} + 50)^{2}$$
$97$ $$(T - 8)^{4}$$