# Properties

 Label 432.3.o.b Level $432$ Weight $3$ Character orbit 432.o Analytic conductor $11.771$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 432.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7711474204$$ 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: $$2^{2}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 144) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{5} ) q^{5} + ( -\beta_{4} - \beta_{5} - \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{5} ) q^{5} + ( -\beta_{4} - \beta_{5} - \beta_{6} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{11} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( -1 + \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -2 + 4 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 14 - 7 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{23} + ( -8 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{25} + ( -20 + 20 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{29} + ( 10 - 5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{31} + ( -18 + 36 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{35} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{37} + ( -12 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - 5 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} ) q^{41} + ( 2 + 2 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{43} + ( -18 - 18 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{47} + ( 11 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 27 + 3 \beta_{2} + 6 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} ) q^{53} + ( 9 - 18 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{55} + ( 48 - 24 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} - 2 \beta_{5} + 5 \beta_{7} ) q^{59} + ( 4 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{61} + ( -22 + 22 \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{65} + ( 2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( -33 + 66 \beta_{1} - 5 \beta_{2} - \beta_{4} + \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{71} + ( 7 + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -51 \beta_{1} + 3 \beta_{5} ) q^{77} + ( 1 + \beta_{1} + 9 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{79} + ( -44 - 44 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - \beta_{5} + 11 \beta_{6} ) q^{83} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} + 13 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} ) q^{85} + ( 25 - 3 \beta_{2} - 6 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} ) q^{89} + ( -9 + 18 \beta_{1} + 9 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 11 \beta_{6} + 11 \beta_{7} ) q^{91} + ( 124 - 62 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 28 \beta_{4} + 10 \beta_{5} - 8 \beta_{7} ) q^{95} + ( -4 + 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + \beta_{4} - 8 \beta_{5} + 8 \beta_{6} - 16 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 3q^{5} + 3q^{7} + O(q^{10})$$ $$8q - 3q^{5} + 3q^{7} - 18q^{11} + 5q^{13} - 6q^{17} + 81q^{23} - 23q^{25} - 69q^{29} + 45q^{31} - 20q^{37} - 54q^{41} - 207q^{47} + 41q^{49} + 252q^{53} + 306q^{59} + 7q^{61} - 93q^{65} + 12q^{67} + 74q^{73} - 207q^{77} + 33q^{79} - 549q^{83} - 30q^{85} + 168q^{89} + 684q^{95} - 10q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 7 \nu^{3} + 10 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{3} + 14 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 6 \nu^{4} + \nu^{3} + 24 \nu^{2} - 14 \nu - 6$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + 8 \nu^{4} - 4 \nu^{3} + 14 \nu^{2} + 8 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 8 \nu^{4} + 4 \nu^{3} + 14 \nu^{2} - 8 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + 2 \nu^{6} + 32 \nu^{5} + 22 \nu^{4} + 95 \nu^{3} + 70 \nu^{2} + 50 \nu + 42$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 2 \nu^{6} - 32 \nu^{5} + 22 \nu^{4} - 95 \nu^{3} + 70 \nu^{2} - 50 \nu + 42$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_{1} - 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 24$$$$)/9$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 2 \beta_{1} + 1$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 14 \beta_{3} + 7 \beta_{2} + 105$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$-9 \beta_{5} + 9 \beta_{4} - 16 \beta_{2} + 16 \beta_{1} - 8$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$6 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 28 \beta_{3} - 14 \beta_{2} - 168$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} + 41 \beta_{5} - 41 \beta_{4} + 84 \beta_{2} - 124 \beta_{1} + 62$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 2.06288i 2.33086i 0.385731i 1.07834i 2.06288i − 2.33086i − 0.385731i − 1.07834i
0 0 0 −4.61660 7.99619i 0 −5.33093 3.07781i 0 0 0
127.2 0 0 0 −0.355304 0.615405i 0 2.70480 + 1.56162i 0 0 0
127.3 0 0 0 0.454613 + 0.787412i 0 −6.10709 3.52593i 0 0 0
127.4 0 0 0 3.01729 + 5.22611i 0 10.2332 + 5.90815i 0 0 0
415.1 0 0 0 −4.61660 + 7.99619i 0 −5.33093 + 3.07781i 0 0 0
415.2 0 0 0 −0.355304 + 0.615405i 0 2.70480 1.56162i 0 0 0
415.3 0 0 0 0.454613 0.787412i 0 −6.10709 + 3.52593i 0 0 0
415.4 0 0 0 3.01729 5.22611i 0 10.2332 5.90815i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.4 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 432.3.o.b 8
3.b odd 2 1 144.3.o.a 8
4.b odd 2 1 432.3.o.a 8
8.b even 2 1 1728.3.o.f 8
8.d odd 2 1 1728.3.o.e 8
9.c even 3 1 432.3.o.a 8
9.c even 3 1 1296.3.g.k 8
9.d odd 6 1 144.3.o.c yes 8
9.d odd 6 1 1296.3.g.j 8
12.b even 2 1 144.3.o.c yes 8
24.f even 2 1 576.3.o.d 8
24.h odd 2 1 576.3.o.f 8
36.f odd 6 1 inner 432.3.o.b 8
36.f odd 6 1 1296.3.g.k 8
36.h even 6 1 144.3.o.a 8
36.h even 6 1 1296.3.g.j 8
72.j odd 6 1 576.3.o.d 8
72.l even 6 1 576.3.o.f 8
72.n even 6 1 1728.3.o.e 8
72.p odd 6 1 1728.3.o.f 8

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(432, [\chi])$$:

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

## Hecke characteristic polynomials

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