# Properties

 Label 144.2.c.a Level $144$ Weight $2$ Character orbit 144.c Analytic conductor $1.150$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} + 4 q^{13} -\beta q^{17} -13 q^{25} -\beta q^{29} + 2 q^{37} -3 \beta q^{41} + 7 q^{49} + 3 \beta q^{53} -10 q^{61} + 4 \beta q^{65} + 16 q^{73} + 18 q^{85} + \beta q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 8q^{13} - 26q^{25} + 4q^{37} + 14q^{49} - 20q^{61} + 32q^{73} + 36q^{85} - 16q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 − 1.41421i 1.41421i
0 0 0 4.24264i 0 0 0 0 0
143.2 0 0 0 4.24264i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.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 144.2.c.a 2
3.b odd 2 1 inner 144.2.c.a 2
4.b odd 2 1 CM 144.2.c.a 2
5.b even 2 1 3600.2.h.b 2
5.c odd 4 2 3600.2.o.a 4
7.b odd 2 1 7056.2.h.b 2
8.b even 2 1 576.2.c.a 2
8.d odd 2 1 576.2.c.a 2
9.c even 3 2 1296.2.s.h 4
9.d odd 6 2 1296.2.s.h 4
12.b even 2 1 inner 144.2.c.a 2
15.d odd 2 1 3600.2.h.b 2
15.e even 4 2 3600.2.o.a 4
16.e even 4 2 2304.2.f.f 4
16.f odd 4 2 2304.2.f.f 4
20.d odd 2 1 3600.2.h.b 2
20.e even 4 2 3600.2.o.a 4
21.c even 2 1 7056.2.h.b 2
24.f even 2 1 576.2.c.a 2
24.h odd 2 1 576.2.c.a 2
28.d even 2 1 7056.2.h.b 2
36.f odd 6 2 1296.2.s.h 4
36.h even 6 2 1296.2.s.h 4
48.i odd 4 2 2304.2.f.f 4
48.k even 4 2 2304.2.f.f 4
60.h even 2 1 3600.2.h.b 2
60.l odd 4 2 3600.2.o.a 4
84.h odd 2 1 7056.2.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.c.a 2 1.a even 1 1 trivial
144.2.c.a 2 3.b odd 2 1 inner
144.2.c.a 2 4.b odd 2 1 CM
144.2.c.a 2 12.b even 2 1 inner
576.2.c.a 2 8.b even 2 1
576.2.c.a 2 8.d odd 2 1
576.2.c.a 2 24.f even 2 1
576.2.c.a 2 24.h odd 2 1
1296.2.s.h 4 9.c even 3 2
1296.2.s.h 4 9.d odd 6 2
1296.2.s.h 4 36.f odd 6 2
1296.2.s.h 4 36.h even 6 2
2304.2.f.f 4 16.e even 4 2
2304.2.f.f 4 16.f odd 4 2
2304.2.f.f 4 48.i odd 4 2
2304.2.f.f 4 48.k even 4 2
3600.2.h.b 2 5.b even 2 1
3600.2.h.b 2 15.d odd 2 1
3600.2.h.b 2 20.d odd 2 1
3600.2.h.b 2 60.h even 2 1
3600.2.o.a 4 5.c odd 4 2
3600.2.o.a 4 15.e even 4 2
3600.2.o.a 4 20.e even 4 2
3600.2.o.a 4 60.l odd 4 2
7056.2.h.b 2 7.b odd 2 1
7056.2.h.b 2 21.c even 2 1
7056.2.h.b 2 28.d even 2 1
7056.2.h.b 2 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$18 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$18 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$18 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$162 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$162 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$18 + T^{2}$$
$97$ $$( 8 + T )^{2}$$