# Properties

 Label 144.3.m.a Level $144$ Weight $3$ Character orbit 144.m Analytic conductor $3.924$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{8} + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{10} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{11} + ( -2 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{13} + ( -2 + 10 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -6 + 10 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{16} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{17} + ( 5 - 6 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( 14 - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{20} + ( -8 + 10 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{22} + ( -11 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{23} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{25} + ( -18 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{26} + ( 14 + 2 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 8 \beta_{1} + 9 \beta_{2} + \beta_{3} - 7 \beta_{4} - \beta_{5} ) q^{29} + ( -20 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} ) q^{31} + ( 4 - 16 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{32} + ( -16 - 4 \beta_{1} + 6 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} ) q^{34} + ( 14 - 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{35} + ( 14 - 2 \beta_{1} + 5 \beta_{2} - 17 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} ) q^{37} + ( -10 - 8 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{38} + ( 6 - 2 \beta_{1} + 12 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} ) q^{40} + ( 4 \beta_{1} - 12 \beta_{3} + 12 \beta_{5} ) q^{41} + ( -24 - 17 \beta_{1} + 15 \beta_{2} + 8 \beta_{3} + \beta_{4} - 8 \beta_{5} ) q^{43} + ( 6 + 16 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{44} + ( 10 + 6 \beta_{1} - 14 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{46} + ( 8 - 36 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} - 12 \beta_{5} ) q^{47} + ( -5 - 4 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} ) q^{49} + ( -8 - 24 \beta_{1} - 7 \beta_{2} + 8 \beta_{4} + 8 \beta_{5} ) q^{50} + ( 24 - 14 \beta_{1} - 4 \beta_{2} - 20 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{52} + ( -10 + 14 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 5 \beta_{5} ) q^{53} + ( 39 - 7 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} + 9 \beta_{4} + 7 \beta_{5} ) q^{55} + ( 28 - 8 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} - 8 \beta_{5} ) q^{56} + ( -34 - 10 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 14 \beta_{5} ) q^{58} + ( -36 - 35 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{59} + ( 2 + 6 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{61} + ( 16 - 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{5} ) q^{62} + ( 4 - 20 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} ) q^{64} + ( 2 - 12 \beta_{2} - 12 \beta_{4} ) q^{65} + ( -39 + 30 \beta_{1} - 14 \beta_{2} + 23 \beta_{3} - 14 \beta_{4} - 5 \beta_{5} ) q^{67} + ( -18 - 42 \beta_{1} - 8 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{68} + ( -12 - 32 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{70} + ( 37 - 17 \beta_{1} - 15 \beta_{2} + 17 \beta_{3} + 19 \beta_{4} + 17 \beta_{5} ) q^{71} + ( -2 + 2 \beta_{2} - 20 \beta_{3} - 2 \beta_{4} + 24 \beta_{5} ) q^{73} + ( 6 + 58 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} + 10 \beta_{5} ) q^{74} + ( -32 + 6 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} ) q^{76} + ( 38 + 34 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{77} + ( 24 + 48 \beta_{1} - 24 \beta_{2} - 32 \beta_{3} + 24 \beta_{4} - 16 \beta_{5} ) q^{79} + ( -36 + 24 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 24 \beta_{5} ) q^{80} + ( 48 - 24 \beta_{1} - 16 \beta_{2} + 24 \beta_{5} ) q^{82} + ( -47 + 56 \beta_{1} - 9 \beta_{3} - 9 \beta_{5} ) q^{83} + ( -26 + 38 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 16 \beta_{5} ) q^{85} + ( -44 + 34 \beta_{1} + 25 \beta_{2} - 25 \beta_{3} + 16 \beta_{4} - 2 \beta_{5} ) q^{86} + ( 2 + 34 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 26 \beta_{4} - 6 \beta_{5} ) q^{88} + ( -18 + 16 \beta_{1} + 18 \beta_{2} + 28 \beta_{3} - 18 \beta_{4} + 8 \beta_{5} ) q^{89} + ( 32 + 38 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} ) q^{91} + ( 26 + 38 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} ) q^{92} + ( 56 \beta_{1} + 48 \beta_{2} - 16 \beta_{4} - 24 \beta_{5} ) q^{94} + ( 7 + 39 \beta_{1} - 7 \beta_{2} - 21 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} ) q^{95} + ( -2 - 26 \beta_{1} - 48 \beta_{2} + 26 \beta_{3} + 4 \beta_{4} + 26 \beta_{5} ) q^{97} + ( -56 \beta_{1} + 3 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} - 4q^{8} + O(q^{10})$$ $$6q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} - 4q^{8} + 36q^{10} + 18q^{11} - 2q^{13} - 12q^{14} - 40q^{16} + 4q^{17} + 30q^{19} + 84q^{20} - 52q^{22} - 60q^{23} - 96q^{26} + 56q^{28} + 18q^{29} - 8q^{32} - 76q^{34} + 100q^{35} + 46q^{37} - 40q^{38} + 40q^{40} - 114q^{43} - 20q^{44} + 28q^{46} - 46q^{49} - 46q^{50} + 100q^{52} - 78q^{53} + 252q^{55} + 168q^{56} - 176q^{58} - 206q^{59} + 30q^{61} + 144q^{62} + 64q^{64} - 12q^{65} - 226q^{67} - 112q^{68} - 16q^{70} + 260q^{71} + 92q^{74} - 188q^{76} + 212q^{77} - 232q^{80} + 304q^{82} - 318q^{83} - 212q^{85} - 268q^{86} - 8q^{88} + 188q^{91} + 168q^{92} + 48q^{94} - 4q^{97} - 10q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} - 2 \nu^{2} + 2 \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{3} + 2 \nu^{2} + 2 \nu + 4$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$-\nu^{5} + \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 2 \nu + 5$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{5} - 4 \nu^{4} + 11 \nu^{3} - 12 \nu^{2} + 10 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 2 \beta_{2} - 3 \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + 2 \beta_{2} - 5 \beta_{1} + 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 5 \beta_{1} + 3$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.264658 + 1.38923i −0.671462 − 1.24464i 1.40680 − 0.144584i 0.264658 − 1.38923i −0.671462 + 1.24464i 1.40680 + 0.144584i
−1.12457 + 1.65389i 0 −1.47068 3.71982i 0.0586332 0.0586332i 0 4.61555 7.80605 + 1.75086i 0 0.0310355 + 0.162910i
19.2 0.573183 1.91611i 0 −3.34292 2.19656i −3.68585 + 3.68585i 0 −9.66442 −6.12494 + 5.14637i 0 4.94981 + 9.17513i
19.3 1.55139 + 1.26222i 0 0.813607 + 3.91638i 4.62721 4.62721i 0 3.04888 −3.68111 + 7.10278i 0 13.0192 1.33804i
91.1 −1.12457 1.65389i 0 −1.47068 + 3.71982i 0.0586332 + 0.0586332i 0 4.61555 7.80605 1.75086i 0 0.0310355 0.162910i
91.2 0.573183 + 1.91611i 0 −3.34292 + 2.19656i −3.68585 3.68585i 0 −9.66442 −6.12494 5.14637i 0 4.94981 9.17513i
91.3 1.55139 1.26222i 0 0.813607 3.91638i 4.62721 + 4.62721i 0 3.04888 −3.68111 7.10278i 0 13.0192 + 1.33804i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.m.a 6
3.b odd 2 1 16.3.f.a 6
4.b odd 2 1 576.3.m.a 6
8.b even 2 1 1152.3.m.a 6
8.d odd 2 1 1152.3.m.b 6
12.b even 2 1 64.3.f.a 6
15.d odd 2 1 400.3.r.c 6
15.e even 4 1 400.3.k.c 6
15.e even 4 1 400.3.k.d 6
16.e even 4 1 576.3.m.a 6
16.e even 4 1 1152.3.m.b 6
16.f odd 4 1 inner 144.3.m.a 6
16.f odd 4 1 1152.3.m.a 6
24.f even 2 1 128.3.f.a 6
24.h odd 2 1 128.3.f.b 6
48.i odd 4 1 64.3.f.a 6
48.i odd 4 1 128.3.f.a 6
48.k even 4 1 16.3.f.a 6
48.k even 4 1 128.3.f.b 6
96.o even 8 2 1024.3.c.j 12
96.o even 8 2 1024.3.d.k 12
96.p odd 8 2 1024.3.c.j 12
96.p odd 8 2 1024.3.d.k 12
240.t even 4 1 400.3.r.c 6
240.z odd 4 1 400.3.k.c 6
240.bd odd 4 1 400.3.k.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a 6 3.b odd 2 1
16.3.f.a 6 48.k even 4 1
64.3.f.a 6 12.b even 2 1
64.3.f.a 6 48.i odd 4 1
128.3.f.a 6 24.f even 2 1
128.3.f.a 6 48.i odd 4 1
128.3.f.b 6 24.h odd 2 1
128.3.f.b 6 48.k even 4 1
144.3.m.a 6 1.a even 1 1 trivial
144.3.m.a 6 16.f odd 4 1 inner
400.3.k.c 6 15.e even 4 1
400.3.k.c 6 240.z odd 4 1
400.3.k.d 6 15.e even 4 1
400.3.k.d 6 240.bd odd 4 1
400.3.r.c 6 15.d odd 2 1
400.3.r.c 6 240.t even 4 1
576.3.m.a 6 4.b odd 2 1
576.3.m.a 6 16.e even 4 1
1024.3.c.j 12 96.o even 8 2
1024.3.c.j 12 96.p odd 8 2
1024.3.d.k 12 96.o even 8 2
1024.3.d.k 12 96.p odd 8 2
1152.3.m.a 6 8.b even 2 1
1152.3.m.a 6 16.f odd 4 1
1152.3.m.b 6 8.d odd 2 1
1152.3.m.b 6 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 2 T_{5}^{5} + 2 T_{5}^{4} + 64 T_{5}^{3} + 1156 T_{5}^{2} - 136 T_{5} + 8$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 32 T + 24 T^{2} - 8 T^{3} + 6 T^{4} - 2 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$8 - 136 T + 1156 T^{2} + 64 T^{3} + 2 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$( 136 - 60 T + 2 T^{2} + T^{3} )^{2}$$
$11$ $$587528 - 67208 T + 3844 T^{2} + 32 T^{3} + 162 T^{4} - 18 T^{5} + T^{6}$$
$13$ $$1286408 + 311176 T + 37636 T^{2} + 1216 T^{3} + 2 T^{4} + 2 T^{5} + T^{6}$$
$17$ $$( 1544 - 260 T - 2 T^{2} + T^{3} )^{2}$$
$19$ $$13448 + 5576 T + 1156 T^{2} - 1184 T^{3} + 450 T^{4} - 30 T^{5} + T^{6}$$
$23$ $$( -968 + 164 T + 30 T^{2} + T^{3} )^{2}$$
$29$ $$19046792 + 4752440 T + 592900 T^{2} + 20032 T^{3} + 162 T^{4} - 18 T^{5} + T^{6}$$
$31$ $$16777216 + 659456 T^{2} + 1920 T^{4} + T^{6}$$
$37$ $$42632 + 439752 T + 2268036 T^{2} + 69568 T^{3} + 1058 T^{4} - 46 T^{5} + T^{6}$$
$41$ $$67108864 + 6230016 T^{2} + 4992 T^{4} + T^{6}$$
$43$ $$42632 + 80008 T + 75076 T^{2} + 30944 T^{3} + 6498 T^{4} + 114 T^{5} + T^{6}$$
$47$ $$6056574976 + 15044608 T^{2} + 8576 T^{4} + T^{6}$$
$53$ $$783752 + 838840 T + 448900 T^{2} + 51008 T^{3} + 3042 T^{4} + 78 T^{5} + T^{6}$$
$59$ $$8410007432 + 853113976 T + 43270084 T^{2} + 1225376 T^{3} + 21218 T^{4} + 206 T^{5} + T^{6}$$
$61$ $$151449608 - 10059512 T + 334084 T^{2} - 64 T^{3} + 450 T^{4} - 30 T^{5} + T^{6}$$
$67$ $$87233303432 - 1203788344 T + 8305924 T^{2} + 1069024 T^{3} + 25538 T^{4} + 226 T^{5} + T^{6}$$
$71$ $$( 391864 - 3548 T - 130 T^{2} + T^{3} )^{2}$$
$73$ $$7310934016 + 79362304 T^{2} + 18848 T^{4} + T^{6}$$
$79$ $$1550483193856 + 433127424 T^{2} + 37376 T^{4} + T^{6}$$
$83$ $$105636303368 + 7200782904 T + 245423556 T^{2} + 4522144 T^{3} + 50562 T^{4} + 318 T^{5} + T^{6}$$
$89$ $$25681985536 + 41113856 T^{2} + 16288 T^{4} + T^{6}$$
$97$ $$( 519928 - 17540 T + 2 T^{2} + T^{3} )^{2}$$
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