# Properties

 Label 144.3.o.b Level $144$ Weight $3$ Character orbit 144.o Analytic conductor $3.924$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: 8.0.121550625.1 Defining polynomial: $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{5} ) q^{3} + ( -1 - \beta_{1} - \beta_{7} ) q^{5} + ( -\beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{1} - \beta_{4} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{5} ) q^{3} + ( -1 - \beta_{1} - \beta_{7} ) q^{5} + ( -\beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{1} - \beta_{4} - \beta_{7} ) q^{9} + ( -\beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{11} + ( -1 - \beta_{1} - 3 \beta_{7} ) q^{13} + ( -\beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{15} + ( 8 - \beta_{4} ) q^{17} + ( -\beta_{2} - \beta_{3} - 5 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 6 - 9 \beta_{1} + 3 \beta_{4} - 3 \beta_{7} ) q^{21} + ( -3 \beta_{2} - 5 \beta_{3} + 6 \beta_{5} + 4 \beta_{6} ) q^{23} + ( 2 \beta_{1} - 3 \beta_{4} + 3 \beta_{7} ) q^{25} + ( -3 \beta_{2} + 6 \beta_{3} - 3 \beta_{6} ) q^{27} + ( 17 \beta_{1} + \beta_{4} - \beta_{7} ) q^{29} + ( 13 \beta_{2} - \beta_{3} + 8 \beta_{5} - 6 \beta_{6} ) q^{31} + ( -24 - 3 \beta_{1} - 3 \beta_{4} + 6 \beta_{7} ) q^{33} + ( 12 \beta_{2} - 5 \beta_{3} + 9 \beta_{5} + 7 \beta_{6} ) q^{35} + ( 2 + 6 \beta_{4} ) q^{37} + ( -3 \beta_{2} + 9 \beta_{3} - 4 \beta_{5} ) q^{39} + ( -39 - 39 \beta_{1} + 6 \beta_{7} ) q^{41} + ( -10 \beta_{2} - 3 \beta_{3} + \beta_{5} - 4 \beta_{6} ) q^{43} + ( 27 + 51 \beta_{1} + 3 \beta_{7} ) q^{45} + ( -9 \beta_{2} - 8 \beta_{3} - 15 \beta_{5} + 7 \beta_{6} ) q^{47} + ( -4 - 4 \beta_{1} + 9 \beta_{7} ) q^{49} + ( 7 \beta_{2} + 3 \beta_{3} + 8 \beta_{5} - 3 \beta_{6} ) q^{51} + ( 42 + 6 \beta_{4} ) q^{53} + ( -12 \beta_{2} + 3 \beta_{3} - 15 \beta_{5} - 9 \beta_{6} ) q^{55} + ( 12 - 42 \beta_{1} - 3 \beta_{4} + 9 \beta_{7} ) q^{57} + ( -15 \beta_{2} + 5 \beta_{3} - 15 \beta_{5} + 5 \beta_{6} ) q^{59} + ( 5 \beta_{1} + 3 \beta_{4} - 3 \beta_{7} ) q^{61} + ( 15 \beta_{2} + 3 \beta_{5} + 9 \beta_{6} ) q^{63} + ( 79 \beta_{1} - 7 \beta_{4} + 7 \beta_{7} ) q^{65} + ( -3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{67} + ( -93 - 51 \beta_{1} + 12 \beta_{4} - 15 \beta_{7} ) q^{69} + ( 12 \beta_{2} - 8 \beta_{3} + 4 \beta_{6} ) q^{71} + ( -20 - 15 \beta_{4} ) q^{73} + ( -2 \beta_{2} + 3 \beta_{5} - 9 \beta_{6} ) q^{75} + ( -63 - 63 \beta_{1} - 9 \beta_{7} ) q^{77} + ( 5 \beta_{2} + 6 \beta_{3} + 13 \beta_{5} - 7 \beta_{6} ) q^{79} + ( 75 + 78 \beta_{1} + 6 \beta_{4} - 3 \beta_{7} ) q^{81} + ( 9 \beta_{2} + 10 \beta_{3} + 21 \beta_{5} - 11 \beta_{6} ) q^{83} + ( 18 + 18 \beta_{1} - 6 \beta_{7} ) q^{85} + ( -17 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{87} + ( 118 - 8 \beta_{4} ) q^{89} + ( 32 \beta_{2} - 13 \beta_{3} + 25 \beta_{5} + 19 \beta_{6} ) q^{91} + ( 63 - 51 \beta_{1} - 18 \beta_{4} - 3 \beta_{7} ) q^{93} + ( 24 \beta_{2} - 8 \beta_{3} + 24 \beta_{5} - 8 \beta_{6} ) q^{95} + ( -41 \beta_{1} + 18 \beta_{4} - 18 \beta_{7} ) q^{97} + ( -18 \beta_{2} - 9 \beta_{3} - 18 \beta_{5} - 9 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{5} - 6q^{9} + O(q^{10})$$ $$8q - 6q^{5} - 6q^{9} - 10q^{13} + 60q^{17} + 90q^{21} - 14q^{25} - 66q^{29} - 180q^{33} + 40q^{37} - 144q^{41} + 18q^{45} + 2q^{49} + 360q^{53} + 270q^{57} - 14q^{61} - 330q^{65} - 522q^{69} - 220q^{73} - 270q^{77} + 306q^{81} + 60q^{85} + 912q^{89} + 630q^{93} + 200q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-41 \nu^{7} + 111 \nu^{6} + 74 \nu^{5} + 173 \nu^{4} - 1517 \nu^{3} + 1036 \nu^{2} + 768 \nu - 1480$$$$)/816$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 5 \nu^{4} - 33 \nu^{3} + 64 \nu^{2} - 8 \nu - 40$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$41 \nu^{7} - 111 \nu^{6} - 74 \nu^{5} + 31 \nu^{4} + 1517 \nu^{3} - 1036 \nu^{2} - 2604 \nu + 1480$$$$)/408$$ $$\beta_{4}$$ $$=$$ $$($$$$65 \nu^{7} - 69 \nu^{6} - 284 \nu^{5} - 533 \nu^{4} + 1691 \nu^{3} + 1226 \nu^{2} - 2556 \nu + 376$$$$)/408$$ $$\beta_{5}$$ $$=$$ $$($$$$-49 \nu^{7} + 63 \nu^{6} + 178 \nu^{5} + 429 \nu^{4} - 1133 \nu^{3} - 636 \nu^{2} + 344 \nu - 840$$$$)/272$$ $$\beta_{6}$$ $$=$$ $$($$$$-295 \nu^{7} + 525 \nu^{6} + 826 \nu^{5} + 2419 \nu^{4} - 8671 \nu^{3} - 472 \nu^{2} + 1416 \nu - 1832$$$$)/816$$ $$\beta_{7}$$ $$=$$ $$($$$$-169 \nu^{7} + 261 \nu^{6} + 412 \nu^{5} + 1345 \nu^{4} - 4315 \nu^{3} + 566 \nu^{2} - 168 \nu - 2528$$$$)/408$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{6} - 2 \beta_{4} + 3 \beta_{3} + 13 \beta_{1} + 14$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{7} - 3 \beta_{6} + 14 \beta_{5} + 2 \beta_{4} + 7 \beta_{2} + 2 \beta_{1} + 31$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{7} + 3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 5 \beta_{1} + 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 33 \beta_{6} + 22 \beta_{5} - 10 \beta_{4} + 33 \beta_{3} - 22 \beta_{2} + 179 \beta_{1} + 184$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$-80 \beta_{7} + 9 \beta_{6} + 162 \beta_{5} + 40 \beta_{4} + 81 \beta_{2} + 40 \beta_{1} + 263$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-91 \beta_{7} + 17 \beta_{5} - 91 \beta_{4} + 186 \beta_{3} + 34 \beta_{2} + 611 \beta_{1} + 91$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.553538 − 0.676408i −0.862555 + 0.141174i −1.44918 + 1.77086i 2.25820 − 0.369600i 0.553538 + 0.676408i −0.862555 − 0.141174i −1.44918 − 1.77086i 2.25820 + 0.369600i
0 −2.82269 + 1.01607i 0 1.81174 3.13802i 0 −1.59422 + 0.920424i 0 6.93521 5.73610i 0
31.2 0 −0.531407 2.95256i 0 −3.31174 + 5.73610i 0 −8.46808 + 4.88905i 0 −8.43521 + 3.13802i 0
31.3 0 0.531407 + 2.95256i 0 −3.31174 + 5.73610i 0 8.46808 4.88905i 0 −8.43521 + 3.13802i 0
31.4 0 2.82269 1.01607i 0 1.81174 3.13802i 0 1.59422 0.920424i 0 6.93521 5.73610i 0
79.1 0 −2.82269 1.01607i 0 1.81174 + 3.13802i 0 −1.59422 0.920424i 0 6.93521 + 5.73610i 0
79.2 0 −0.531407 + 2.95256i 0 −3.31174 5.73610i 0 −8.46808 4.88905i 0 −8.43521 3.13802i 0
79.3 0 0.531407 2.95256i 0 −3.31174 5.73610i 0 8.46808 + 4.88905i 0 −8.43521 3.13802i 0
79.4 0 2.82269 + 1.01607i 0 1.81174 + 3.13802i 0 1.59422 + 0.920424i 0 6.93521 + 5.73610i 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
4.b odd 2 1 inner
9.c even 3 1 inner
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.b 8
3.b odd 2 1 432.3.o.c 8
4.b odd 2 1 inner 144.3.o.b 8
8.b even 2 1 576.3.o.e 8
8.d odd 2 1 576.3.o.e 8
9.c even 3 1 inner 144.3.o.b 8
9.c even 3 1 1296.3.g.h 4
9.d odd 6 1 432.3.o.c 8
9.d odd 6 1 1296.3.g.d 4
12.b even 2 1 432.3.o.c 8
24.f even 2 1 1728.3.o.d 8
24.h odd 2 1 1728.3.o.d 8
36.f odd 6 1 inner 144.3.o.b 8
36.f odd 6 1 1296.3.g.h 4
36.h even 6 1 432.3.o.c 8
36.h even 6 1 1296.3.g.d 4
72.j odd 6 1 1728.3.o.d 8
72.l even 6 1 1728.3.o.d 8
72.n even 6 1 576.3.o.e 8
72.p odd 6 1 576.3.o.e 8

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

## Hecke kernels

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

 $$T_{5}^{4} + 3 T_{5}^{3} + 33 T_{5}^{2} - 72 T_{5} + 576$$ $$T_{7}^{8} - 99 T_{7}^{6} + 9477 T_{7}^{4} - 32076 T_{7}^{2} + 104976$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 + 243 T^{2} - 72 T^{4} + 3 T^{6} + T^{8}$$
$5$ $$( 576 - 72 T + 33 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$7$ $$104976 - 32076 T^{2} + 9477 T^{4} - 99 T^{6} + T^{8}$$
$11$ $$( 18225 - 135 T^{2} + T^{4} )^{2}$$
$13$ $$( 52900 - 1150 T + 255 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$17$ $$( 30 - 15 T + T^{2} )^{4}$$
$19$ $$( 129600 + 855 T^{2} + T^{4} )^{2}$$
$23$ $$2379503694096 - 4039975116 T^{2} + 5316597 T^{4} - 2619 T^{6} + T^{8}$$
$29$ $$( 60516 + 8118 T + 843 T^{2} + 33 T^{3} + T^{4} )^{2}$$
$31$ $$2520473760000 - 6501222000 T^{2} + 15181425 T^{4} - 4095 T^{6} + T^{8}$$
$37$ $$( -920 - 10 T + T^{2} )^{4}$$
$41$ $$( 123201 + 25272 T + 4833 T^{2} + 72 T^{3} + T^{4} )^{2}$$
$43$ $$1147523000625 - 4531281750 T^{2} + 16821675 T^{4} - 4230 T^{6} + T^{8}$$
$47$ $$20415837456 - 837157356 T^{2} + 34184997 T^{4} - 5859 T^{6} + T^{8}$$
$53$ $$( 1080 - 90 T + T^{2} )^{4}$$
$59$ $$( 11390625 - 3375 T^{2} + T^{4} )^{2}$$
$61$ $$( 50176 - 1568 T + 273 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$67$ $$( 321489 - 567 T^{2} + T^{4} )^{2}$$
$71$ $$( 2160 + T^{2} )^{4}$$
$73$ $$( -5150 + 55 T + T^{2} )^{4}$$
$79$ $$2520473760000 - 6501222000 T^{2} + 15181425 T^{4} - 4095 T^{6} + T^{8}$$
$83$ $$62171080298496 - 84517857216 T^{2} + 107012097 T^{4} - 10719 T^{6} + T^{8}$$
$89$ $$( 11316 - 228 T + T^{2} )^{4}$$
$97$ $$( 36060025 + 600500 T + 16005 T^{2} - 100 T^{3} + T^{4} )^{2}$$