# Properties

 Label 144.3.o.c Level $144$ Weight $3$ Character orbit 144.o Analytic conductor $3.924$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.856615824.2 Defining polynomial: $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{4} ) q^{3} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{4} ) q^{3} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{9} + ( -2 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{13} + ( 6 - 9 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{15} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -2 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{19} + ( -6 + 3 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{21} + ( 13 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{6} ) q^{23} + ( -1 + 6 \beta_{1} - \beta_{2} + 6 \beta_{3} - 9 \beta_{4} - 5 \beta_{6} - 3 \beta_{7} ) q^{25} + ( -12 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{27} + ( 15 - 14 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -5 + 5 \beta_{1} + \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{31} + ( 6 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} ) q^{33} + ( -15 + 33 \beta_{1} + \beta_{2} + 8 \beta_{3} - 4 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{35} + ( -7 + 6 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{37} + ( 6 + 24 \beta_{1} - \beta_{4} - 6 \beta_{5} - 3 \beta_{7} ) q^{39} + ( 6 + 12 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} - 12 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{41} + ( 1 + 3 \beta_{1} + 3 \beta_{2} - 9 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{43} + ( -9 - 21 \beta_{1} + 9 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 9 \beta_{6} + 6 \beta_{7} ) q^{45} + ( -18 - 15 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 7 \beta_{6} - 5 \beta_{7} ) q^{47} + ( -5 + 12 \beta_{1} + 5 \beta_{2} + 9 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} ) q^{49} + ( 42 - 42 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{51} + ( -33 - 6 \beta_{1} - 3 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} + 9 \beta_{7} ) q^{53} + ( -6 + 15 \beta_{1} + 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} ) q^{55} + ( -45 + 24 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{4} - 12 \beta_{5} - 3 \beta_{6} - 9 \beta_{7} ) q^{57} + ( 54 - 21 \beta_{1} + 2 \beta_{2} + \beta_{3} - 11 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} ) q^{59} + ( -3 - 2 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 5 \beta_{6} + 12 \beta_{7} ) q^{61} + ( -63 + 6 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 9 \beta_{6} + 6 \beta_{7} ) q^{63} + ( 27 - 22 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} - 8 \beta_{7} ) q^{65} + ( -2 - 8 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} ) q^{67} + ( 24 - 3 \beta_{1} - 12 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{69} + ( -27 + 66 \beta_{1} - 11 \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{71} + ( 11 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 21 + 57 \beta_{1} - 9 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 15 \beta_{5} + 12 \beta_{7} ) q^{75} + ( -3 + 54 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{77} + ( -13 + 4 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} + 21 \beta_{4} - 4 \beta_{5} - 11 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 54 - 63 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} - 15 \beta_{4} + 9 \beta_{5} ) q^{81} + ( -52 - 43 \beta_{1} + 6 \beta_{2} - 14 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} ) q^{83} + ( 14 - 19 \beta_{1} - 14 \beta_{2} + 9 \beta_{3} - 12 \beta_{4} + 21 \beta_{5} + 5 \beta_{6} ) q^{85} + ( 54 - 84 \beta_{1} + 6 \beta_{2} - 13 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 9 \beta_{7} ) q^{87} + ( -19 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} - 7 \beta_{5} - \beta_{6} - 7 \beta_{7} ) q^{89} + ( 14 - 17 \beta_{1} - 14 \beta_{2} - 12 \beta_{3} - 3 \beta_{4} + 8 \beta_{5} + 3 \beta_{6} - 8 \beta_{7} ) q^{91} + ( -24 + 51 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{93} + ( 104 - 70 \beta_{1} - 2 \beta_{3} + 28 \beta_{4} + 10 \beta_{5} + 18 \beta_{6} + 20 \beta_{7} ) q^{95} + ( 2 \beta_{1} - 13 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - 2 \beta_{6} - 15 \beta_{7} ) q^{97} + ( -84 + 6 \beta_{1} - 9 \beta_{3} + 18 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 3q^{3} + 3q^{5} - 3q^{7} - 3q^{9} + O(q^{10})$$ $$8q + 3q^{3} + 3q^{5} - 3q^{7} - 3q^{9} - 18q^{11} + 5q^{13} + 21q^{15} + 6q^{17} - 33q^{21} + 81q^{23} - 23q^{25} - 108q^{27} + 69q^{29} - 45q^{31} + 72q^{33} - 20q^{37} + 141q^{39} + 54q^{41} - 117q^{45} - 207q^{47} + 41q^{49} + 141q^{51} - 252q^{53} - 273q^{57} + 306q^{59} + 7q^{61} - 441q^{63} + 93q^{65} - 12q^{67} + 189q^{69} + 74q^{73} + 387q^{75} + 207q^{77} - 33q^{79} + 117q^{81} - 549q^{83} - 30q^{85} + 87q^{87} - 168q^{89} - 27q^{93} + 684q^{95} - 10q^{97} - 585q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 7 \nu^{3} + 10 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 6 \nu^{4} - \nu^{3} + 36 \nu^{2} + 14 \nu + 26$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 10 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} - 14 \nu^{2} - 6 \nu - 14$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 10 \nu^{5} - 4 \nu^{4} + 31 \nu^{3} - 22 \nu^{2} + 30 \nu - 10$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + 11 \nu^{5} + 10 \nu^{4} + 35 \nu^{3} + 28 \nu^{2} + 22 \nu + 14$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + 31 \nu^{5} + 88 \nu^{3} - 6 \nu^{2} + 52 \nu - 16$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} - 21 \nu^{5} + 7 \nu^{4} - 63 \nu^{3} + 10 \nu^{2} - 40 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 4 \beta_{1} - 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{1} - 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 5 \beta_{3} - \beta_{2} - 7 \beta_{1} + 5$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$6 \beta_{6} - 7 \beta_{4} + 11 \beta_{3} + \beta_{2} - 5 \beta_{1} + 41$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} + 2 \beta_{4} - 25 \beta_{3} + 2 \beta_{2} + 41 \beta_{1} - 26$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{7} - 31 \beta_{6} + 3 \beta_{5} + 42 \beta_{4} - 57 \beta_{3} - 8 \beta_{2} + 23 \beta_{1} - 202$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-43 \beta_{7} - 41 \beta_{6} + 43 \beta_{5} + 2 \beta_{4} + 125 \beta_{3} - 251 \beta_{1} + 146$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 − 2.33086i 2.06288i − 0.385731i − 1.07834i 2.33086i − 2.06288i 0.385731i 1.07834i
0 −2.89248 0.795973i 0 0.355304 0.615405i 0 −2.70480 + 1.56162i 0 7.73285 + 4.60467i 0
31.2 0 0.456412 + 2.96508i 0 4.61660 7.99619i 0 5.33093 3.07781i 0 −8.58338 + 2.70659i 0
31.3 0 1.28651 2.71015i 0 −0.454613 + 0.787412i 0 6.10709 3.52593i 0 −5.68980 6.97325i 0
31.4 0 2.64956 + 1.40707i 0 −3.01729 + 5.22611i 0 −10.2332 + 5.90815i 0 5.04032 + 7.45622i 0
79.1 0 −2.89248 + 0.795973i 0 0.355304 + 0.615405i 0 −2.70480 1.56162i 0 7.73285 4.60467i 0
79.2 0 0.456412 2.96508i 0 4.61660 + 7.99619i 0 5.33093 + 3.07781i 0 −8.58338 2.70659i 0
79.3 0 1.28651 + 2.71015i 0 −0.454613 0.787412i 0 6.10709 + 3.52593i 0 −5.68980 + 6.97325i 0
79.4 0 2.64956 1.40707i 0 −3.01729 5.22611i 0 −10.2332 5.90815i 0 5.04032 7.45622i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f 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.o.c yes 8
3.b odd 2 1 432.3.o.a 8
4.b odd 2 1 144.3.o.a 8
8.b even 2 1 576.3.o.d 8
8.d odd 2 1 576.3.o.f 8
9.c even 3 1 144.3.o.a 8
9.c even 3 1 1296.3.g.j 8
9.d odd 6 1 432.3.o.b 8
9.d odd 6 1 1296.3.g.k 8
12.b even 2 1 432.3.o.b 8
24.f even 2 1 1728.3.o.f 8
24.h odd 2 1 1728.3.o.e 8
36.f odd 6 1 inner 144.3.o.c yes 8
36.f odd 6 1 1296.3.g.j 8
36.h even 6 1 432.3.o.a 8
36.h even 6 1 1296.3.g.k 8
72.j odd 6 1 1728.3.o.f 8
72.l even 6 1 1728.3.o.e 8
72.n even 6 1 576.3.o.f 8
72.p odd 6 1 576.3.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 4.b odd 2 1
144.3.o.a 8 9.c even 3 1
144.3.o.c yes 8 1.a even 1 1 trivial
144.3.o.c yes 8 36.f odd 6 1 inner
432.3.o.a 8 3.b odd 2 1
432.3.o.a 8 36.h even 6 1
432.3.o.b 8 9.d odd 6 1
432.3.o.b 8 12.b even 2 1
576.3.o.d 8 8.b even 2 1
576.3.o.d 8 72.p odd 6 1
576.3.o.f 8 8.d odd 2 1
576.3.o.f 8 72.n even 6 1
1296.3.g.j 8 9.c even 3 1
1296.3.g.j 8 36.f odd 6 1
1296.3.g.k 8 9.d odd 6 1
1296.3.g.k 8 36.h even 6 1
1728.3.o.e 8 24.h odd 2 1
1728.3.o.e 8 72.l even 6 1
1728.3.o.f 8 24.f even 2 1
1728.3.o.f 8 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$:

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 - 2187 T + 486 T^{2} + 243 T^{3} - 126 T^{4} + 27 T^{5} + 6 T^{6} - 3 T^{7} + T^{8}$$
$5$ $$1296 - 324 T + 2133 T^{2} + 729 T^{3} + 3186 T^{4} + 189 T^{5} + 66 T^{6} - 3 T^{7} + T^{8}$$
$7$ $$2566404 + 446958 T - 161487 T^{2} - 32643 T^{3} + 12366 T^{4} - 351 T^{5} - 114 T^{6} + 3 T^{7} + T^{8}$$
$11$ $$12131289 - 2821230 T - 282852 T^{2} + 116640 T^{3} + 12393 T^{4} - 2592 T^{5} - 36 T^{6} + 18 T^{7} + T^{8}$$
$13$ $$10201636 - 7515482 T + 4607155 T^{2} - 716663 T^{3} + 99640 T^{4} - 3251 T^{5} + 316 T^{6} - 5 T^{7} + T^{8}$$
$17$ $$( 84168 + 1908 T - 822 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$19$ $$2931572736 + 85791744 T^{2} + 739584 T^{4} + 1731 T^{6} + T^{8}$$
$23$ $$19131876 + 3188646 T - 2302911 T^{2} - 413343 T^{3} + 345546 T^{4} - 45927 T^{5} + 2754 T^{6} - 81 T^{7} + T^{8}$$
$29$ $$4046639163876 - 240302807082 T + 13105243395 T^{2} - 346769991 T^{3} + 10589400 T^{4} - 198963 T^{5} + 5340 T^{6} - 69 T^{7} + T^{8}$$
$31$ $$944784 - 11573604 T + 46287855 T^{2} + 11895093 T^{3} + 818424 T^{4} - 44955 T^{5} - 324 T^{6} + 45 T^{7} + T^{8}$$
$37$ $$( -613568 - 117320 T - 3756 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$41$ $$5431756955769 - 445142421774 T + 30187580904 T^{2} - 767400804 T^{3} + 19934505 T^{4} - 236196 T^{5} + 5616 T^{6} - 54 T^{7} + T^{8}$$
$43$ $$29016737649 + 20365527708 T + 5307258510 T^{2} + 380905416 T^{3} + 10320939 T^{4} - 3186 T^{6} + T^{8}$$
$47$ $$28643839776036 + 1972043877186 T + 38368451709 T^{2} - 474219603 T^{3} - 18415998 T^{4} + 266409 T^{5} + 15570 T^{6} + 207 T^{7} + T^{8}$$
$53$ $$( -6508512 - 274104 T + 972 T^{2} + 126 T^{3} + T^{4} )^{2}$$
$59$ $$48359409452649 - 1123183376802 T - 44378047044 T^{2} + 1232674848 T^{3} + 48727089 T^{4} - 2335392 T^{5} + 38844 T^{6} - 306 T^{7} + T^{8}$$
$61$ $$309954973696 - 123735132736 T + 45856336249 T^{2} - 1420643911 T^{3} + 42523942 T^{4} - 400003 T^{5} + 6406 T^{6} - 7 T^{7} + T^{8}$$
$67$ $$68036119056801 - 520837032744 T - 49299630426 T^{2} + 387577872 T^{3} + 29174067 T^{4} - 73656 T^{5} - 6090 T^{6} + 12 T^{7} + T^{8}$$
$71$ $$726110197530624 + 765915906816 T^{2} + 231242688 T^{4} + 26208 T^{6} + T^{8}$$
$73$ $$( 416536 + 25628 T - 1002 T^{2} - 37 T^{3} + T^{4} )^{2}$$
$79$ $$240627852449856 - 798427622664 T - 176002346205 T^{2} + 586923813 T^{3} + 113950044 T^{4} - 376299 T^{5} - 11040 T^{6} + 33 T^{7} + T^{8}$$
$83$ $$1517530356962064 - 28477438539300 T - 984805785801 T^{2} + 21823289325 T^{3} + 1063934676 T^{4} + 16389297 T^{5} + 130320 T^{6} + 549 T^{7} + T^{8}$$
$89$ $$( -1161936 - 109152 T - 984 T^{2} + 84 T^{3} + T^{4} )^{2}$$
$97$ $$30429664983481 + 3216990050002 T + 257418140392 T^{2} + 8850998044 T^{3} + 235988233 T^{4} + 1016476 T^{5} + 15088 T^{6} + 10 T^{7} + T^{8}$$