# Properties

 Label 144.3.g.c Level $144$ Weight $3$ Character orbit 144.g Analytic conductor $3.924$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 8 - 16 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 8 - 16 \zeta_{6} ) q^{7} + 22 q^{13} + ( 16 - 32 \zeta_{6} ) q^{19} -25 q^{25} + ( -24 + 48 \zeta_{6} ) q^{31} + 26 q^{37} + ( -48 + 96 \zeta_{6} ) q^{43} -143 q^{49} + 74 q^{61} + ( -32 + 64 \zeta_{6} ) q^{67} + 46 q^{73} + ( 40 - 80 \zeta_{6} ) q^{79} + ( 176 - 352 \zeta_{6} ) q^{91} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 44q^{13} - 50q^{25} + 52q^{37} - 286q^{49} + 148q^{61} + 92q^{73} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 13.8564i 0 0 0
127.2 0 0 0 0 0 13.8564i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.g.c 2
3.b odd 2 1 CM 144.3.g.c 2
4.b odd 2 1 inner 144.3.g.c 2
5.b even 2 1 3600.3.e.h 2
5.c odd 4 2 3600.3.j.e 4
8.b even 2 1 576.3.g.f 2
8.d odd 2 1 576.3.g.f 2
9.c even 3 1 1296.3.o.f 2
9.c even 3 1 1296.3.o.k 2
9.d odd 6 1 1296.3.o.f 2
9.d odd 6 1 1296.3.o.k 2
12.b even 2 1 inner 144.3.g.c 2
15.d odd 2 1 3600.3.e.h 2
15.e even 4 2 3600.3.j.e 4
16.e even 4 2 2304.3.b.m 4
16.f odd 4 2 2304.3.b.m 4
20.d odd 2 1 3600.3.e.h 2
20.e even 4 2 3600.3.j.e 4
24.f even 2 1 576.3.g.f 2
24.h odd 2 1 576.3.g.f 2
36.f odd 6 1 1296.3.o.f 2
36.f odd 6 1 1296.3.o.k 2
36.h even 6 1 1296.3.o.f 2
36.h even 6 1 1296.3.o.k 2
48.i odd 4 2 2304.3.b.m 4
48.k even 4 2 2304.3.b.m 4
60.h even 2 1 3600.3.e.h 2
60.l odd 4 2 3600.3.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.g.c 2 1.a even 1 1 trivial
144.3.g.c 2 3.b odd 2 1 CM
144.3.g.c 2 4.b odd 2 1 inner
144.3.g.c 2 12.b even 2 1 inner
576.3.g.f 2 8.b even 2 1
576.3.g.f 2 8.d odd 2 1
576.3.g.f 2 24.f even 2 1
576.3.g.f 2 24.h odd 2 1
1296.3.o.f 2 9.c even 3 1
1296.3.o.f 2 9.d odd 6 1
1296.3.o.f 2 36.f odd 6 1
1296.3.o.f 2 36.h even 6 1
1296.3.o.k 2 9.c even 3 1
1296.3.o.k 2 9.d odd 6 1
1296.3.o.k 2 36.f odd 6 1
1296.3.o.k 2 36.h even 6 1
2304.3.b.m 4 16.e even 4 2
2304.3.b.m 4 16.f odd 4 2
2304.3.b.m 4 48.i odd 4 2
2304.3.b.m 4 48.k even 4 2
3600.3.e.h 2 5.b even 2 1
3600.3.e.h 2 15.d odd 2 1
3600.3.e.h 2 20.d odd 2 1
3600.3.e.h 2 60.h even 2 1
3600.3.j.e 4 5.c odd 4 2
3600.3.j.e 4 15.e even 4 2
3600.3.j.e 4 20.e even 4 2
3600.3.j.e 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$192 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -22 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$768 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$1728 + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$6912 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -74 + T )^{2}$$
$67$ $$3072 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -46 + T )^{2}$$
$79$ $$4800 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 2 + T )^{2}$$