# Properties

 Label 72.2.i.a Level $72$ Weight $2$ Character orbit 72.i Analytic conductor $0.575$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} -3 q^{9} -5 \zeta_{6} q^{11} + ( 5 - 5 \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{15} -2 q^{17} -4 q^{19} + ( -6 + 3 \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + 9 \zeta_{6} q^{29} + ( 1 - \zeta_{6} ) q^{31} + ( 10 - 5 \zeta_{6} ) q^{33} + 3 q^{35} -6 q^{37} + ( 5 + 5 \zeta_{6} ) q^{39} + ( -3 + 3 \zeta_{6} ) q^{41} -\zeta_{6} q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} + ( 2 - 4 \zeta_{6} ) q^{51} + 2 q^{53} -5 q^{55} + ( 4 - 8 \zeta_{6} ) q^{57} + ( -11 + 11 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} -9 \zeta_{6} q^{63} -5 \zeta_{6} q^{65} + ( 1 - \zeta_{6} ) q^{67} + ( 1 + \zeta_{6} ) q^{69} + 4 q^{71} -2 q^{73} + ( -8 + 4 \zeta_{6} ) q^{75} + ( 15 - 15 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 9 q^{81} -\zeta_{6} q^{83} + ( -2 + 2 \zeta_{6} ) q^{85} + ( -18 + 9 \zeta_{6} ) q^{87} -18 q^{89} + 15 q^{91} + ( 1 + \zeta_{6} ) q^{93} + ( -4 + 4 \zeta_{6} ) q^{95} + 13 \zeta_{6} q^{97} + 15 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} + 3q^{7} - 6q^{9} + O(q^{10})$$ $$2q + q^{5} + 3q^{7} - 6q^{9} - 5q^{11} + 5q^{13} + 3q^{15} - 4q^{17} - 8q^{19} - 9q^{21} + q^{23} + 4q^{25} + 9q^{29} + q^{31} + 15q^{33} + 6q^{35} - 12q^{37} + 15q^{39} - 3q^{41} - q^{43} - 3q^{45} + 3q^{47} - 2q^{49} + 4q^{53} - 10q^{55} - 11q^{59} - 7q^{61} - 9q^{63} - 5q^{65} + q^{67} + 3q^{69} + 8q^{71} - 4q^{73} - 12q^{75} + 15q^{77} - q^{79} + 18q^{81} - q^{83} - 2q^{85} - 27q^{87} - 36q^{89} + 30q^{91} + 3q^{93} - 4q^{95} + 13q^{97} + 15q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \zeta_{6}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 −3.00000 0
49.1 0 1.73205i 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 −3.00000 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.i.a 2
3.b odd 2 1 216.2.i.a 2
4.b odd 2 1 144.2.i.b 2
8.b even 2 1 576.2.i.d 2
8.d odd 2 1 576.2.i.c 2
9.c even 3 1 inner 72.2.i.a 2
9.c even 3 1 648.2.a.a 1
9.d odd 6 1 216.2.i.a 2
9.d odd 6 1 648.2.a.c 1
12.b even 2 1 432.2.i.a 2
24.f even 2 1 1728.2.i.g 2
24.h odd 2 1 1728.2.i.h 2
36.f odd 6 1 144.2.i.b 2
36.f odd 6 1 1296.2.a.e 1
36.h even 6 1 432.2.i.a 2
36.h even 6 1 1296.2.a.i 1
72.j odd 6 1 1728.2.i.h 2
72.j odd 6 1 5184.2.a.i 1
72.l even 6 1 1728.2.i.g 2
72.l even 6 1 5184.2.a.n 1
72.n even 6 1 576.2.i.d 2
72.n even 6 1 5184.2.a.s 1
72.p odd 6 1 576.2.i.c 2
72.p odd 6 1 5184.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 1.a even 1 1 trivial
72.2.i.a 2 9.c even 3 1 inner
144.2.i.b 2 4.b odd 2 1
144.2.i.b 2 36.f odd 6 1
216.2.i.a 2 3.b odd 2 1
216.2.i.a 2 9.d odd 6 1
432.2.i.a 2 12.b even 2 1
432.2.i.a 2 36.h even 6 1
576.2.i.c 2 8.d odd 2 1
576.2.i.c 2 72.p odd 6 1
576.2.i.d 2 8.b even 2 1
576.2.i.d 2 72.n even 6 1
648.2.a.a 1 9.c even 3 1
648.2.a.c 1 9.d odd 6 1
1296.2.a.e 1 36.f odd 6 1
1296.2.a.i 1 36.h even 6 1
1728.2.i.g 2 24.f even 2 1
1728.2.i.g 2 72.l even 6 1
1728.2.i.h 2 24.h odd 2 1
1728.2.i.h 2 72.j odd 6 1
5184.2.a.i 1 72.j odd 6 1
5184.2.a.n 1 72.l even 6 1
5184.2.a.s 1 72.n even 6 1
5184.2.a.x 1 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$1 - T + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$121 + 11 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$1 - T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$1 + T + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$169 - 13 T + T^{2}$$