# Properties

 Label 144.2.i.d Level 144 Weight 2 Character orbit 144.i Analytic conductor 1.150 Analytic rank 0 Dimension 4 CM no Inner twists 2

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} + \beta_{2} q^{11} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 6 - \beta_{1} - 3 \beta_{2} ) q^{15} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 3 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{21} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{23} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{25} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} + \beta_{3} ) q^{33} + ( -7 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -6 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 7 - 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{45} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{49} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{51} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{57} + ( 7 - 7 \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{61} + ( -3 + 4 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} ) q^{63} + ( 3 - 6 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{65} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{67} + ( 6 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{69} + 4 q^{71} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 3 + 3 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{75} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{77} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{79} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( 1 - 2 \beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{83} + ( 8 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{87} + 6 q^{89} + ( 5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( -6 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -8 - 4 \beta_{1} + 12 \beta_{2} + 8 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{3} + q^{5} + 3q^{7} + 5q^{9} + O(q^{10})$$ $$4q - q^{3} + q^{5} + 3q^{7} + 5q^{9} + 2q^{11} - 5q^{13} + 17q^{15} - 10q^{17} - 14q^{19} - 3q^{21} + 5q^{23} - 7q^{25} - 16q^{27} + 3q^{29} + 7q^{31} - 2q^{33} - 30q^{35} + 12q^{37} - 19q^{39} + 12q^{41} + 8q^{43} + 5q^{45} + 3q^{47} - 7q^{49} + 19q^{51} - 20q^{53} + 2q^{55} - 13q^{57} + 14q^{59} + q^{61} + 9q^{63} + 19q^{65} + 4q^{67} + 19q^{69} + 16q^{71} - 14q^{73} + 7q^{75} - 3q^{77} - 7q^{79} - 7q^{81} + 25q^{83} + 14q^{85} - 3q^{87} + 24q^{89} + 18q^{91} - 13q^{93} - 20q^{95} + 8q^{97} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
0 −1.68614 0.396143i 0 −1.18614 + 2.05446i 0 2.18614 + 3.78651i 0 2.68614 + 1.33591i 0
49.2 0 1.18614 + 1.26217i 0 1.68614 2.92048i 0 −0.686141 1.18843i 0 −0.186141 + 2.99422i 0
97.1 0 −1.68614 + 0.396143i 0 −1.18614 2.05446i 0 2.18614 3.78651i 0 2.68614 1.33591i 0
97.2 0 1.18614 1.26217i 0 1.68614 + 2.92048i 0 −0.686141 + 1.18843i 0 −0.186141 2.99422i 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 144.2.i.d 4
3.b odd 2 1 432.2.i.d 4
4.b odd 2 1 72.2.i.b 4
8.b even 2 1 576.2.i.l 4
8.d odd 2 1 576.2.i.j 4
9.c even 3 1 inner 144.2.i.d 4
9.c even 3 1 1296.2.a.n 2
9.d odd 6 1 432.2.i.d 4
9.d odd 6 1 1296.2.a.p 2
12.b even 2 1 216.2.i.b 4
24.f even 2 1 1728.2.i.i 4
24.h odd 2 1 1728.2.i.j 4
36.f odd 6 1 72.2.i.b 4
36.f odd 6 1 648.2.a.f 2
36.h even 6 1 216.2.i.b 4
36.h even 6 1 648.2.a.g 2
72.j odd 6 1 1728.2.i.j 4
72.j odd 6 1 5184.2.a.bo 2
72.l even 6 1 1728.2.i.i 4
72.l even 6 1 5184.2.a.bp 2
72.n even 6 1 576.2.i.l 4
72.n even 6 1 5184.2.a.bs 2
72.p odd 6 1 576.2.i.j 4
72.p odd 6 1 5184.2.a.bt 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8}$$
$7$ $$1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 126 T^{5} + 49 T^{6} - 1029 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8}$$
$17$ $$( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 7 T + 42 T^{2} + 133 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 230 T^{5} - 10051 T^{6} - 60835 T^{7} + 279841 T^{8}$$
$29$ $$1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8}$$
$31$ $$1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 868 T^{5} - 16337 T^{6} - 208537 T^{7} + 923521 T^{8}$$
$37$ $$( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 1476 T^{5} + 99179 T^{6} - 827052 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 5848 T^{5} - 9245 T^{6} - 636056 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 846 T^{5} - 174511 T^{6} - 311469 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 + 10 T + 98 T^{2} + 530 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 7772 T^{5} - 399521 T^{6} - 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 - 4 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 2212 T^{5} - 705233 T^{6} + 3451273 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 307100 T^{5} + 2142479 T^{6} - 14294675 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{4}$$
$97$ $$1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8}$$
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