# Properties

 Label 144.3.e.a Level $144$ Weight $3$ Character orbit 144.e Analytic conductor $3.924$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} -12 q^{7} +O(q^{10})$$ $$q + 5 \beta q^{5} -12 q^{7} + 4 \beta q^{11} -8 q^{13} + 7 \beta q^{17} + 16 q^{19} + 28 \beta q^{23} -25 q^{25} -21 \beta q^{29} + 4 q^{31} -60 \beta q^{35} + 30 q^{37} -15 \beta q^{41} + 8 q^{43} + 12 \beta q^{47} + 95 q^{49} + 35 \beta q^{53} -40 q^{55} -56 \beta q^{59} -14 q^{61} -40 \beta q^{65} + 88 q^{67} + 20 \beta q^{71} -80 q^{73} -48 \beta q^{77} -100 q^{79} + 92 \beta q^{83} -70 q^{85} + 105 \beta q^{89} + 96 q^{91} + 80 \beta q^{95} -112 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 24q^{7} + O(q^{10})$$ $$2q - 24q^{7} - 16q^{13} + 32q^{19} - 50q^{25} + 8q^{31} + 60q^{37} + 16q^{43} + 190q^{49} - 80q^{55} - 28q^{61} + 176q^{67} - 160q^{73} - 200q^{79} - 140q^{85} + 192q^{91} - 224q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 0 0 7.07107i 0 −12.0000 0 0 0
17.2 0 0 0 7.07107i 0 −12.0000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.e.a 2
3.b odd 2 1 inner 144.3.e.a 2
4.b odd 2 1 72.3.e.a 2
5.b even 2 1 3600.3.l.l 2
5.c odd 4 2 3600.3.c.c 4
8.b even 2 1 576.3.e.a 2
8.d odd 2 1 576.3.e.h 2
9.c even 3 2 1296.3.q.k 4
9.d odd 6 2 1296.3.q.k 4
12.b even 2 1 72.3.e.a 2
15.d odd 2 1 3600.3.l.l 2
15.e even 4 2 3600.3.c.c 4
16.e even 4 2 2304.3.h.h 4
16.f odd 4 2 2304.3.h.a 4
20.d odd 2 1 1800.3.l.a 2
20.e even 4 2 1800.3.c.a 4
24.f even 2 1 576.3.e.h 2
24.h odd 2 1 576.3.e.a 2
28.d even 2 1 3528.3.d.a 2
36.f odd 6 2 648.3.m.a 4
36.h even 6 2 648.3.m.a 4
48.i odd 4 2 2304.3.h.h 4
48.k even 4 2 2304.3.h.a 4
60.h even 2 1 1800.3.l.a 2
60.l odd 4 2 1800.3.c.a 4
84.h odd 2 1 3528.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.e.a 2 4.b odd 2 1
72.3.e.a 2 12.b even 2 1
144.3.e.a 2 1.a even 1 1 trivial
144.3.e.a 2 3.b odd 2 1 inner
576.3.e.a 2 8.b even 2 1
576.3.e.a 2 24.h odd 2 1
576.3.e.h 2 8.d odd 2 1
576.3.e.h 2 24.f even 2 1
648.3.m.a 4 36.f odd 6 2
648.3.m.a 4 36.h even 6 2
1296.3.q.k 4 9.c even 3 2
1296.3.q.k 4 9.d odd 6 2
1800.3.c.a 4 20.e even 4 2
1800.3.c.a 4 60.l odd 4 2
1800.3.l.a 2 20.d odd 2 1
1800.3.l.a 2 60.h even 2 1
2304.3.h.a 4 16.f odd 4 2
2304.3.h.a 4 48.k even 4 2
2304.3.h.h 4 16.e even 4 2
2304.3.h.h 4 48.i odd 4 2
3528.3.d.a 2 28.d even 2 1
3528.3.d.a 2 84.h odd 2 1
3600.3.c.c 4 5.c odd 4 2
3600.3.c.c 4 15.e even 4 2
3600.3.l.l 2 5.b even 2 1
3600.3.l.l 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 50$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$50 + T^{2}$$
$7$ $$( 12 + T )^{2}$$
$11$ $$32 + T^{2}$$
$13$ $$( 8 + T )^{2}$$
$17$ $$98 + T^{2}$$
$19$ $$( -16 + T )^{2}$$
$23$ $$1568 + T^{2}$$
$29$ $$882 + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$( -30 + T )^{2}$$
$41$ $$450 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$288 + T^{2}$$
$53$ $$2450 + T^{2}$$
$59$ $$6272 + T^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$( -88 + T )^{2}$$
$71$ $$800 + T^{2}$$
$73$ $$( 80 + T )^{2}$$
$79$ $$( 100 + T )^{2}$$
$83$ $$16928 + T^{2}$$
$89$ $$22050 + T^{2}$$
$97$ $$( 112 + T )^{2}$$