# Properties

 Label 144.3.q.c Level $144$ Weight $3$ Character orbit 144.q Analytic conductor $3.924$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{3} + ( -3 - 3 \beta_{1} ) q^{5} + ( -\beta_{1} + 3 \beta_{2} ) q^{7} + ( 3 + 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{3} + ( -3 - 3 \beta_{1} ) q^{5} + ( -\beta_{1} + 3 \beta_{2} ) q^{7} + ( 3 + 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( -6 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{11} + ( -5 + 5 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{13} + ( -9 + 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{15} + ( -6 + 12 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 10 - 6 \beta_{3} ) q^{19} + ( -2 - 17 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{21} + ( -3 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + 2 \beta_{1} q^{25} + ( 15 - 30 \beta_{1} + 3 \beta_{3} ) q^{27} + ( 6 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -19 + 19 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{31} + ( 12 + 3 \beta_{1} + 6 \beta_{3} ) q^{33} + ( -3 + 6 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} ) q^{35} + ( 32 - 6 \beta_{3} ) q^{37} + ( 41 - 31 \beta_{1} + \beta_{2} + 11 \beta_{3} ) q^{39} + ( -21 - 21 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{41} + ( 23 \beta_{1} - 9 \beta_{2} ) q^{43} + ( -9 - 9 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{45} + ( -18 + 9 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} ) q^{47} + ( -6 + 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{49} + ( 6 + 24 \beta_{1} - 12 \beta_{2} ) q^{51} + ( -30 + 60 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} ) q^{53} + ( 27 + 9 \beta_{3} ) q^{55} + ( 46 - 20 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} ) q^{57} + ( -21 - 21 \beta_{1} - 13 \beta_{2} - 13 \beta_{3} ) q^{59} + ( 31 \beta_{1} + 18 \beta_{2} ) q^{61} + ( -72 + 33 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{63} + ( 30 - 15 \beta_{1} - 18 \beta_{2} + 36 \beta_{3} ) q^{65} + ( 53 - 53 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{67} + ( -3 + 15 \beta_{1} + 6 \beta_{2} ) q^{69} + ( -30 + 60 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{71} + ( -52 - 18 \beta_{3} ) q^{73} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( -15 - 15 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{77} + ( -7 \beta_{1} + 15 \beta_{2} ) q^{79} + ( -63 + 24 \beta_{2} - 12 \beta_{3} ) q^{81} + ( 126 - 63 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} ) q^{83} + ( 54 - 54 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} ) q^{85} + ( 24 - 21 \beta_{1} + 9 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 30 - 60 \beta_{1} + 44 \beta_{2} - 22 \beta_{3} ) q^{89} + ( -103 - 9 \beta_{3} ) q^{91} + ( 73 - 35 \beta_{1} - 10 \beta_{2} + 28 \beta_{3} ) q^{93} + ( -30 - 30 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} ) q^{95} + ( 7 \beta_{1} - 42 \beta_{2} ) q^{97} + ( -18 - 27 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 18q^{5} - 2q^{7} + 12q^{9} + O(q^{10})$$ $$4q - 18q^{5} - 2q^{7} + 12q^{9} - 18q^{11} - 10q^{13} - 18q^{15} + 40q^{19} - 42q^{21} - 18q^{23} + 4q^{25} + 18q^{29} - 38q^{31} + 54q^{33} + 128q^{37} + 102q^{39} - 126q^{41} + 46q^{43} - 54q^{45} - 54q^{47} - 12q^{49} + 72q^{51} + 108q^{55} + 144q^{57} - 126q^{59} + 62q^{61} - 222q^{63} + 90q^{65} + 106q^{67} + 18q^{69} - 208q^{73} + 12q^{75} - 90q^{77} - 14q^{79} - 252q^{81} + 378q^{83} + 108q^{85} + 54q^{87} - 412q^{91} + 222q^{93} - 180q^{95} + 14q^{97} - 126q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 −2.44949 + 1.73205i 0 −4.50000 + 2.59808i 0 3.17423 5.49794i 0 3.00000 8.48528i 0
65.2 0 2.44949 + 1.73205i 0 −4.50000 + 2.59808i 0 −4.17423 + 7.22999i 0 3.00000 + 8.48528i 0
113.1 0 −2.44949 1.73205i 0 −4.50000 2.59808i 0 3.17423 + 5.49794i 0 3.00000 + 8.48528i 0
113.2 0 2.44949 1.73205i 0 −4.50000 2.59808i 0 −4.17423 7.22999i 0 3.00000 8.48528i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.c 4
3.b odd 2 1 432.3.q.d 4
4.b odd 2 1 18.3.d.a 4
8.b even 2 1 576.3.q.e 4
8.d odd 2 1 576.3.q.f 4
9.c even 3 1 432.3.q.d 4
9.c even 3 1 1296.3.e.g 4
9.d odd 6 1 inner 144.3.q.c 4
9.d odd 6 1 1296.3.e.g 4
12.b even 2 1 54.3.d.a 4
20.d odd 2 1 450.3.i.b 4
20.e even 4 2 450.3.k.a 8
24.f even 2 1 1728.3.q.d 4
24.h odd 2 1 1728.3.q.c 4
36.f odd 6 1 54.3.d.a 4
36.f odd 6 1 162.3.b.a 4
36.h even 6 1 18.3.d.a 4
36.h even 6 1 162.3.b.a 4
60.h even 2 1 1350.3.i.b 4
60.l odd 4 2 1350.3.k.a 8
72.j odd 6 1 576.3.q.e 4
72.l even 6 1 576.3.q.f 4
72.n even 6 1 1728.3.q.c 4
72.p odd 6 1 1728.3.q.d 4
180.n even 6 1 450.3.i.b 4
180.p odd 6 1 1350.3.i.b 4
180.v odd 12 2 450.3.k.a 8
180.x even 12 2 1350.3.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 4.b odd 2 1
18.3.d.a 4 36.h even 6 1
54.3.d.a 4 12.b even 2 1
54.3.d.a 4 36.f odd 6 1
144.3.q.c 4 1.a even 1 1 trivial
144.3.q.c 4 9.d odd 6 1 inner
162.3.b.a 4 36.f odd 6 1
162.3.b.a 4 36.h even 6 1
432.3.q.d 4 3.b odd 2 1
432.3.q.d 4 9.c even 3 1
450.3.i.b 4 20.d odd 2 1
450.3.i.b 4 180.n even 6 1
450.3.k.a 8 20.e even 4 2
450.3.k.a 8 180.v odd 12 2
576.3.q.e 4 8.b even 2 1
576.3.q.e 4 72.j odd 6 1
576.3.q.f 4 8.d odd 2 1
576.3.q.f 4 72.l even 6 1
1296.3.e.g 4 9.c even 3 1
1296.3.e.g 4 9.d odd 6 1
1350.3.i.b 4 60.h even 2 1
1350.3.i.b 4 180.p odd 6 1
1350.3.k.a 8 60.l odd 4 2
1350.3.k.a 8 180.x even 12 2
1728.3.q.c 4 24.h odd 2 1
1728.3.q.c 4 72.n even 6 1
1728.3.q.d 4 24.f even 2 1
1728.3.q.d 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 9 T_{5} + 27$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 6 T^{2} + T^{4}$$
$5$ $$( 27 + 9 T + T^{2} )^{2}$$
$7$ $$2809 - 106 T + 57 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$81 + 162 T + 117 T^{2} + 18 T^{3} + T^{4}$$
$13$ $$36481 - 1910 T + 291 T^{2} + 10 T^{3} + T^{4}$$
$17$ $$1296 + 360 T^{2} + T^{4}$$
$19$ $$( -116 - 20 T + T^{2} )^{2}$$
$23$ $$81 + 162 T + 117 T^{2} + 18 T^{3} + T^{4}$$
$29$ $$2025 + 810 T + 63 T^{2} - 18 T^{3} + T^{4}$$
$31$ $$15625 - 4750 T + 1569 T^{2} + 38 T^{3} + T^{4}$$
$37$ $$( 808 - 64 T + T^{2} )^{2}$$
$41$ $$455625 + 85050 T + 5967 T^{2} + 126 T^{3} + T^{4}$$
$43$ $$1849 - 1978 T + 2073 T^{2} - 46 T^{3} + T^{4}$$
$47$ $$408321 - 34506 T + 333 T^{2} + 54 T^{3} + T^{4}$$
$53$ $$810000 + 9000 T^{2} + T^{4}$$
$59$ $$2954961 - 216594 T + 3573 T^{2} + 126 T^{3} + T^{4}$$
$61$ $$966289 + 60946 T + 4827 T^{2} - 62 T^{3} + T^{4}$$
$67$ $$5396329 - 246238 T + 8913 T^{2} - 106 T^{3} + T^{4}$$
$71$ $$2396304 + 7704 T^{2} + T^{4}$$
$73$ $$( 760 + 104 T + T^{2} )^{2}$$
$79$ $$1692601 - 18214 T + 1497 T^{2} + 14 T^{3} + T^{4}$$
$83$ $$131262849 - 4330746 T + 59085 T^{2} - 378 T^{3} + T^{4}$$
$89$ $$36144144 + 22824 T^{2} + T^{4}$$
$97$ $$110986225 + 147490 T + 10731 T^{2} - 14 T^{3} + T^{4}$$