# Properties

 Label 5184.2.d.r Level $5184$ Weight $2$ Character orbit 5184.d Analytic conductor $41.394$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16 x^{8} - 24 x^{7} + 96 x^{5} + 304 x^{4} + 384 x^{3} + 288 x^{2} + 144 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{5} + \beta_{4} q^{7} +O(q^{10})$$ $$q + \beta_{8} q^{5} + \beta_{4} q^{7} + ( \beta_{7} + \beta_{9} ) q^{11} + \beta_{10} q^{13} + \beta_{1} q^{17} + \beta_{7} q^{19} + ( -\beta_{3} + \beta_{4} - \beta_{6} ) q^{23} + 2 q^{25} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( \beta_{3} - \beta_{4} - \beta_{6} ) q^{31} + ( \beta_{5} + \beta_{7} + \beta_{9} ) q^{35} + ( -\beta_{8} - \beta_{10} ) q^{37} + ( 3 + \beta_{1} ) q^{41} + ( -\beta_{5} + \beta_{7} - \beta_{9} ) q^{43} + ( -\beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{47} + ( 4 + \beta_{1} - \beta_{2} ) q^{49} + ( 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{53} + ( -\beta_{4} + \beta_{6} ) q^{55} + ( \beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{59} + ( -\beta_{8} - \beta_{10} + \beta_{11} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{65} + ( 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{67} + ( -2 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{71} + ( -\beta_{1} + \beta_{2} ) q^{73} + ( 6 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{77} + ( \beta_{3} + \beta_{4} + \beta_{6} ) q^{79} + ( -\beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{83} + ( \beta_{10} - \beta_{11} ) q^{85} + ( 4 - \beta_{1} + \beta_{2} ) q^{89} + ( 3 \beta_{7} + 3 \beta_{9} ) q^{91} -\beta_{3} q^{95} + ( 1 - 3 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + O(q^{10})$$ $$12 q + 24 q^{25} + 36 q^{41} + 48 q^{49} + 12 q^{65} + 48 q^{89} + 12 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16 x^{8} - 24 x^{7} + 96 x^{5} + 304 x^{4} + 384 x^{3} + 288 x^{2} + 144 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192$$$$)/51972$$ $$\beta_{2}$$ $$=$$ $$($$$$1969 \nu^{11} - 2148 \nu^{10} + 4654 \nu^{9} - 11338 \nu^{8} - 17186 \nu^{7} - 23628 \nu^{6} - 17534 \nu^{5} + 254444 \nu^{4} + 441804 \nu^{3} + 378072 \nu^{2} + 214812 \nu - 442176$$$$)/51972$$ $$\beta_{3}$$ $$=$$ $$($$$$-34 \nu^{11} - 4 \nu^{10} + 17 \nu^{9} - 12 \nu^{8} + 546 \nu^{7} + 896 \nu^{6} - 196 \nu^{5} - 3468 \nu^{4} - 10454 \nu^{3} - 13152 \nu^{2} - 7260 \nu - 2736$$$$)/366$$ $$\beta_{4}$$ $$=$$ $$($$$$-11251 \nu^{11} - 3656 \nu^{10} + 8584 \nu^{9} - 6576 \nu^{8} + 184040 \nu^{7} + 325576 \nu^{6} - 51410 \nu^{5} - 1183104 \nu^{4} - 3657720 \nu^{3} - 4638708 \nu^{2} - 2561328 \nu - 967896$$$$)/103944$$ $$\beta_{5}$$ $$=$$ $$($$$$5765 \nu^{11} - 3533 \nu^{10} + 2602 \nu^{9} - 2490 \nu^{8} - 88123 \nu^{7} - 86504 \nu^{6} + 51058 \nu^{5} + 522216 \nu^{4} + 1425948 \nu^{3} + 1357440 \nu^{2} + 954990 \nu + 355320$$$$)/51972$$ $$\beta_{6}$$ $$=$$ $$($$$$6604 \nu^{11} + 4175 \nu^{10} - 8182 \nu^{9} + 6486 \nu^{8} - 110771 \nu^{7} - 217840 \nu^{6} + 19196 \nu^{5} + 732168 \nu^{4} + 2347908 \nu^{3} + 3011652 \nu^{2} + 1663614 \nu + 631152$$$$)/51972$$ $$\beta_{7}$$ $$=$$ $$($$$$1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120$$$$)/12993$$ $$\beta_{8}$$ $$=$$ $$($$$$-118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 4320$$$$)/852$$ $$\beta_{9}$$ $$=$$ $$($$$$26035 \nu^{11} - 16590 \nu^{10} + 14290 \nu^{9} - 9780 \nu^{8} - 416670 \nu^{7} - 347068 \nu^{6} + 155770 \nu^{5} + 2327880 \nu^{4} + 6525980 \nu^{3} + 6240620 \nu^{2} + 4402140 \nu + 1639800$$$$)/103944$$ $$\beta_{10}$$ $$=$$ $$($$$$-17920 \nu^{11} + 13314 \nu^{10} - 5985 \nu^{9} + 1146 \nu^{8} + 286651 \nu^{7} + 215040 \nu^{6} - 211740 \nu^{5} - 1622514 \nu^{4} - 4157302 \nu^{3} - 3407556 \nu^{2} - 1863546 \nu - 639072$$$$)/51972$$ $$\beta_{11}$$ $$=$$ $$($$$$-29743 \nu^{11} + 22494 \nu^{10} - 12243 \nu^{9} + 5262 \nu^{8} + 472510 \nu^{7} + 356916 \nu^{6} - 347886 \nu^{5} - 2656440 \nu^{4} - 6971578 \nu^{3} - 5717928 \nu^{2} - 3128892 \nu - 1079712$$$$)/51972$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{9} - 7 \beta_{7} + 2 \beta_{5} + 6 \beta_{4} - 9 \beta_{3}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{11} + 4 \beta_{10} + 6 \beta_{8} + 4 \beta_{6} + 8 \beta_{4} - 3 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{11} + 3 \beta_{10} + 30 \beta_{8} + 2 \beta_{2} + 7 \beta_{1} + 32$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{11} + 16 \beta_{10} + 6 \beta_{9} + 39 \beta_{8} - 57 \beta_{7} - 16 \beta_{6} + 54 \beta_{5} - 38 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 54 \beta_{1} + 120$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$16 \beta_{9} - 65 \beta_{7} + 40 \beta_{5}$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-47 \beta_{11} + 35 \beta_{10} + 24 \beta_{9} + 108 \beta_{8} - 156 \beta_{7} + 35 \beta_{6} + 129 \beta_{5} + 94 \beta_{4} - 60 \beta_{3} - 12 \beta_{2} - 129 \beta_{1} - 336$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-249 \beta_{11} + 144 \beta_{10} + 669 \beta_{8} - 35 \beta_{2} - 214 \beta_{1} - 704$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-236 \beta_{11} + 164 \beta_{10} + 570 \beta_{8} - 164 \beta_{6} - 472 \beta_{4} + 321 \beta_{3}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$164 \beta_{9} - 790 \beta_{7} - 393 \beta_{6} + 557 \beta_{5} - 1278 \beta_{4} + 954 \beta_{3}$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$1193 \beta_{11} - 800 \beta_{10} + 786 \beta_{9} - 2949 \beta_{8} - 4227 \beta_{7} - 800 \beta_{6} + 3186 \beta_{5} - 2386 \beta_{4} + 1671 \beta_{3} - 393 \beta_{2} - 3186 \beta_{1} - 9240$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2593.1
 −0.673288 + 0.180407i 0.583700 + 2.17840i −1.50511 + 0.403293i −0.403293 − 1.50511i 2.17840 − 0.583700i −0.180407 − 0.673288i −0.673288 − 0.180407i 0.583700 − 2.17840i −1.50511 − 0.403293i −0.403293 + 1.50511i 2.17840 + 0.583700i −0.180407 + 0.673288i
0 0 0 1.73205i 0 −4.35461 0 0 0
2593.2 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.3 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.4 0 0 0 1.73205i 0 0.990721 0 0 0
2593.5 0 0 0 1.73205i 0 3.61328 0 0 0
2593.6 0 0 0 1.73205i 0 4.35461 0 0 0
2593.7 0 0 0 1.73205i 0 −4.35461 0 0 0
2593.8 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.9 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.10 0 0 0 1.73205i 0 0.990721 0 0 0
2593.11 0 0 0 1.73205i 0 3.61328 0 0 0
2593.12 0 0 0 1.73205i 0 4.35461 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2593.12 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.d.r 12
3.b odd 2 1 5184.2.d.q 12
4.b odd 2 1 inner 5184.2.d.r 12
8.b even 2 1 inner 5184.2.d.r 12
8.d odd 2 1 inner 5184.2.d.r 12
9.c even 3 1 1728.2.r.e 12
9.c even 3 1 1728.2.r.f 12
9.d odd 6 1 576.2.r.e 12
9.d odd 6 1 576.2.r.f yes 12
12.b even 2 1 5184.2.d.q 12
24.f even 2 1 5184.2.d.q 12
24.h odd 2 1 5184.2.d.q 12
36.f odd 6 1 1728.2.r.e 12
36.f odd 6 1 1728.2.r.f 12
36.h even 6 1 576.2.r.e 12
36.h even 6 1 576.2.r.f yes 12
72.j odd 6 1 576.2.r.e 12
72.j odd 6 1 576.2.r.f yes 12
72.l even 6 1 576.2.r.e 12
72.l even 6 1 576.2.r.f yes 12
72.n even 6 1 1728.2.r.e 12
72.n even 6 1 1728.2.r.f 12
72.p odd 6 1 1728.2.r.e 12
72.p odd 6 1 1728.2.r.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 9.d odd 6 1
576.2.r.e 12 36.h even 6 1
576.2.r.e 12 72.j odd 6 1
576.2.r.e 12 72.l even 6 1
576.2.r.f yes 12 9.d odd 6 1
576.2.r.f yes 12 36.h even 6 1
576.2.r.f yes 12 72.j odd 6 1
576.2.r.f yes 12 72.l even 6 1
1728.2.r.e 12 9.c even 3 1
1728.2.r.e 12 36.f odd 6 1
1728.2.r.e 12 72.n even 6 1
1728.2.r.e 12 72.p odd 6 1
1728.2.r.f 12 9.c even 3 1
1728.2.r.f 12 36.f odd 6 1
1728.2.r.f 12 72.n even 6 1
1728.2.r.f 12 72.p odd 6 1
5184.2.d.q 12 3.b odd 2 1
5184.2.d.q 12 12.b even 2 1
5184.2.d.q 12 24.f even 2 1
5184.2.d.q 12 24.h odd 2 1
5184.2.d.r 12 1.a even 1 1 trivial
5184.2.d.r 12 4.b odd 2 1 inner
5184.2.d.r 12 8.b even 2 1 inner
5184.2.d.r 12 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}^{2} + 3$$ $$T_{7}^{6} - 33 T_{7}^{4} + 279 T_{7}^{2} - 243$$ $$T_{17}^{3} - 24 T_{17} - 36$$ $$T_{23}^{6} - 117 T_{23}^{4} + 4347 T_{23}^{2} - 49923$$ $$T_{41}^{3} - 9 T_{41}^{2} + 3 T_{41} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 3 + T^{2} )^{6}$$
$7$ $$( -243 + 279 T^{2} - 33 T^{4} + T^{6} )^{2}$$
$11$ $$( 81 + 171 T^{2} + 39 T^{4} + T^{6} )^{2}$$
$13$ $$( 243 + 675 T^{2} + 57 T^{4} + T^{6} )^{2}$$
$17$ $$( -36 - 24 T + T^{3} )^{4}$$
$19$ $$( 4 + T^{2} )^{6}$$
$23$ $$( -49923 + 4347 T^{2} - 117 T^{4} + T^{6} )^{2}$$
$29$ $$( 93987 + 7155 T^{2} + 153 T^{4} + T^{6} )^{2}$$
$31$ $$( -9747 + 2619 T^{2} - 117 T^{4} + T^{6} )^{2}$$
$37$ $$( 3888 + 1008 T^{2} + 60 T^{4} + T^{6} )^{2}$$
$41$ $$( 9 + 3 T - 9 T^{2} + T^{3} )^{4}$$
$43$ $$( 6889 + 1935 T^{2} + 123 T^{4} + T^{6} )^{2}$$
$47$ $$( -717363 + 27135 T^{2} - 297 T^{4} + T^{6} )^{2}$$
$53$ $$( 432 + 1728 T^{2} + 180 T^{4} + T^{6} )^{2}$$
$59$ $$( 729 + 711 T^{2} + 147 T^{4} + T^{6} )^{2}$$
$61$ $$( 4563 + 3267 T^{2} + 153 T^{4} + T^{6} )^{2}$$
$67$ $$( 18769 + 8091 T^{2} + 231 T^{4} + T^{6} )^{2}$$
$71$ $$( -124848 + 19872 T^{2} - 288 T^{4} + T^{6} )^{2}$$
$73$ $$( -164 - 84 T + T^{3} )^{4}$$
$79$ $$( -2187 + 891 T^{2} - 93 T^{4} + T^{6} )^{2}$$
$83$ $$( 6561 + 3159 T^{2} + 171 T^{4} + T^{6} )^{2}$$
$89$ $$( 108 - 36 T - 12 T^{2} + T^{3} )^{4}$$
$97$ $$( 1187 - 213 T - 3 T^{2} + T^{3} )^{4}$$