# Properties

 Label 567.2.s.f Level $567$ Weight $2$ Character orbit 567.s Analytic conductor $4.528$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} + \beta_{5} q^{5} + ( 1 - \beta_{3} - \beta_{4} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} + \beta_{5} q^{5} + ( 1 - \beta_{3} - \beta_{4} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} + ( -2 \beta_{4} + \beta_{9} ) q^{10} + ( -\beta_{8} + \beta_{10} - \beta_{11} ) q^{11} + ( -2 + \beta_{3} - \beta_{4} + \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{6} + \beta_{8} ) q^{14} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{16} + ( \beta_{1} + 2 \beta_{11} ) q^{17} + ( \beta_{7} + \beta_{9} ) q^{19} + ( 2 \beta_{8} - \beta_{10} ) q^{20} + ( -2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{7} - 2 \beta_{9} ) q^{22} + ( \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{23} + ( 4 + \beta_{2} + 2 \beta_{4} + \beta_{7} - \beta_{9} ) q^{25} + ( 2 \beta_{1} - \beta_{6} - 2 \beta_{11} ) q^{26} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} + 3 \beta_{9} ) q^{28} + ( \beta_{1} + \beta_{10} + \beta_{11} ) q^{29} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{31} + ( -2 \beta_{5} - \beta_{6} ) q^{32} + ( -3 - 3 \beta_{3} + \beta_{7} + \beta_{9} ) q^{34} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{10} ) q^{35} + ( 2 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{37} + ( 4 \beta_{1} - \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{38} + ( -\beta_{2} - \beta_{7} + 3 \beta_{9} ) q^{40} + ( \beta_{1} - \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{41} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{43} + ( -6 \beta_{1} + 2 \beta_{5} + \beta_{6} + 2 \beta_{10} - 6 \beta_{11} ) q^{44} + ( -2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{46} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - 4 \beta_{11} ) q^{47} + ( 3 - 4 \beta_{3} + \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{49} + ( -\beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{50} + ( -4 + 8 \beta_{3} + \beta_{7} - 2 \beta_{9} ) q^{52} + ( 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{53} + ( \beta_{2} + \beta_{7} - 3 \beta_{9} ) q^{55} + ( \beta_{5} + \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{56} + ( -3 - \beta_{2} - 2 \beta_{4} + \beta_{9} ) q^{58} + ( -\beta_{1} + \beta_{6} + \beta_{11} ) q^{59} + ( -2 + \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{61} + ( 4 \beta_{1} - \beta_{5} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{62} + ( 2 - \beta_{7} ) q^{64} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{65} + ( 7 + 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{67} + ( 6 \beta_{1} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{68} + ( 9 - \beta_{2} - 9 \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{70} + ( \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{71} + ( -6 + 3 \beta_{3} + 3 \beta_{4} - \beta_{7} - \beta_{9} ) q^{73} + ( -\beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{74} + ( -12 + 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} - 3 \beta_{9} ) q^{76} + ( -\beta_{5} - \beta_{8} + 2 \beta_{10} - 4 \beta_{11} ) q^{77} + ( -4 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{79} + ( 3 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{10} + 6 \beta_{11} ) q^{80} + ( -3 - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{82} + ( 2 \beta_{1} - \beta_{6} - 2 \beta_{11} ) q^{83} + ( 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{85} + ( -\beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{86} + ( 12 - 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{7} + 2 \beta_{9} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{89} + ( 2 - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} - 3 \beta_{9} ) q^{91} + ( -2 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{92} + ( 6 - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{94} + ( 2 \beta_{5} + \beta_{6} - 3 \beta_{10} ) q^{95} + ( -4 + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{97} + ( -4 \beta_{1} + \beta_{6} + \beta_{8} + \beta_{10} + 3 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} + 4q^{7} + O(q^{10})$$ $$12q + 8q^{4} + 4q^{7} - 6q^{10} - 24q^{13} - 4q^{16} - 6q^{19} + 20q^{22} + 48q^{25} + 28q^{28} + 12q^{31} - 60q^{34} + 8q^{37} - 10q^{43} + 14q^{46} + 24q^{49} - 40q^{58} - 18q^{61} + 28q^{64} + 36q^{67} + 66q^{70} - 42q^{73} - 108q^{76} - 36q^{79} - 54q^{82} + 6q^{85} + 148q^{88} + 6q^{91} + 114q^{94} - 60q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 76298 \nu$$$$)/21995$$ $$\beta_{2}$$ $$=$$ $$($$$$117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864$$$$)/21995$$ $$\beta_{3}$$ $$=$$ $$($$$$-1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966$$$$)/197955$$ $$\beta_{4}$$ $$=$$ $$($$$$461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237$$$$)/65985$$ $$\beta_{5}$$ $$=$$ $$($$$$1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu$$$$)/197955$$ $$\beta_{6}$$ $$=$$ $$($$$$1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} + 154944 \nu$$$$)/197955$$ $$\beta_{7}$$ $$=$$ $$($$$$-288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439$$$$)/21995$$ $$\beta_{8}$$ $$=$$ $$($$$$-288 \nu^{11} + 1888 \nu^{9} - 9933 \nu^{7} + 12896 \nu^{5} - 6336 \nu^{3} - 90434 \nu$$$$)/21995$$ $$\beta_{9}$$ $$=$$ $$($$$$1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656$$$$)/65985$$ $$\beta_{10}$$ $$=$$ $$($$$$4712 \nu^{11} - 47997 \nu^{9} + 314647 \nu^{7} - 1022364 \nu^{5} + 2149199 \nu^{3} - 1055934 \nu$$$$)/197955$$ $$\beta_{11}$$ $$=$$ $$($$$$-5192 \nu^{11} + 43812 \nu^{9} - 287212 \nu^{7} + 809244 \nu^{5} - 1763849 \nu^{3} + 172044 \nu$$$$)/197955$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{9} + 2 \beta_{7} + \beta_{4} - 9 \beta_{3} - \beta_{2} + 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{11} + 8 \beta_{6} + 4 \beta_{5}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 2 \beta_{7} - 2 \beta_{4} - 12 \beta_{3} - 4 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{11} + 3 \beta_{10} + 17 \beta_{6} + 34 \beta_{5} + 18 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$32 \beta_{9} - 5 \beta_{7} - 64 \beta_{4} - 32 \beta_{2} - 153$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$27 \beta_{8} - 74 \beta_{6} + 74 \beta_{5} + 96 \beta_{1}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$51 \beta_{9} - 51 \beta_{7} - 55 \beta_{4} + 222 \beta_{3} + 55 \beta_{2} - 222$$ $$\nu^{9}$$ $$=$$ $$($$$$-495 \beta_{11} - 177 \beta_{10} + 177 \beta_{8} - 656 \beta_{6} - 328 \beta_{5}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-191 \beta_{9} - 835 \beta_{7} + 835 \beta_{4} + 2952 \beta_{3} + 1670 \beta_{2}$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-2505 \beta_{11} - 1026 \beta_{10} - 1477 \beta_{6} - 2954 \beta_{5} - 2505 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1 - \beta_{3}$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.617942 − 0.356769i −1.90412 + 1.09935i 1.65604 − 0.956115i −1.65604 + 0.956115i 1.90412 − 1.09935i −0.617942 + 0.356769i 0.617942 + 0.356769i −1.90412 − 1.09935i 1.65604 + 0.956115i −1.65604 − 0.956115i 1.90412 + 1.09935i −0.617942 − 0.356769i
−2.15715 + 1.24543i 0 2.10220 3.64112i −1.23588 0 2.63409 0.248083i 5.49086i 0 2.66599 1.53921i
26.2 −1.58850 + 0.917122i 0 0.682224 1.18165i 3.80824 0 −2.43359 + 1.03810i 1.16576i 0 −6.04940 + 3.49262i
26.3 −0.568650 + 0.328310i 0 −0.784425 + 1.35866i −3.31208 0 0.799494 2.52206i 2.34338i 0 1.88341 1.08739i
26.4 0.568650 0.328310i 0 −0.784425 + 1.35866i 3.31208 0 0.799494 2.52206i 2.34338i 0 1.88341 1.08739i
26.5 1.58850 0.917122i 0 0.682224 1.18165i −3.80824 0 −2.43359 + 1.03810i 1.16576i 0 −6.04940 + 3.49262i
26.6 2.15715 1.24543i 0 2.10220 3.64112i 1.23588 0 2.63409 0.248083i 5.49086i 0 2.66599 1.53921i
458.1 −2.15715 1.24543i 0 2.10220 + 3.64112i −1.23588 0 2.63409 + 0.248083i 5.49086i 0 2.66599 + 1.53921i
458.2 −1.58850 0.917122i 0 0.682224 + 1.18165i 3.80824 0 −2.43359 1.03810i 1.16576i 0 −6.04940 3.49262i
458.3 −0.568650 0.328310i 0 −0.784425 1.35866i −3.31208 0 0.799494 + 2.52206i 2.34338i 0 1.88341 + 1.08739i
458.4 0.568650 + 0.328310i 0 −0.784425 1.35866i 3.31208 0 0.799494 + 2.52206i 2.34338i 0 1.88341 + 1.08739i
458.5 1.58850 + 0.917122i 0 0.682224 + 1.18165i −3.80824 0 −2.43359 1.03810i 1.16576i 0 −6.04940 3.49262i
458.6 2.15715 + 1.24543i 0 2.10220 + 3.64112i 1.23588 0 2.63409 + 0.248083i 5.49086i 0 2.66599 + 1.53921i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 458.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.k odd 6 1 inner
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.s.f 12
3.b odd 2 1 inner 567.2.s.f 12
7.d odd 6 1 567.2.i.f 12
9.c even 3 1 189.2.p.d 12
9.c even 3 1 567.2.i.f 12
9.d odd 6 1 189.2.p.d 12
9.d odd 6 1 567.2.i.f 12
21.g even 6 1 567.2.i.f 12
63.g even 3 1 1323.2.c.d 12
63.i even 6 1 189.2.p.d 12
63.k odd 6 1 inner 567.2.s.f 12
63.k odd 6 1 1323.2.c.d 12
63.n odd 6 1 1323.2.c.d 12
63.s even 6 1 inner 567.2.s.f 12
63.s even 6 1 1323.2.c.d 12
63.t odd 6 1 189.2.p.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 9.c even 3 1
189.2.p.d 12 9.d odd 6 1
189.2.p.d 12 63.i even 6 1
189.2.p.d 12 63.t odd 6 1
567.2.i.f 12 7.d odd 6 1
567.2.i.f 12 9.c even 3 1
567.2.i.f 12 9.d odd 6 1
567.2.i.f 12 21.g even 6 1
567.2.s.f 12 1.a even 1 1 trivial
567.2.s.f 12 3.b odd 2 1 inner
567.2.s.f 12 63.k odd 6 1 inner
567.2.s.f 12 63.s even 6 1 inner
1323.2.c.d 12 63.g even 3 1
1323.2.c.d 12 63.k odd 6 1
1323.2.c.d 12 63.n odd 6 1
1323.2.c.d 12 63.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2}^{12} - 10 T_{2}^{10} + 75 T_{2}^{8} - 232 T_{2}^{6} + 535 T_{2}^{4} - 225 T_{2}^{2} + 81$$ $$T_{11}^{6} + 37 T_{11}^{4} + 214 T_{11}^{2} + 225$$ $$T_{13}^{6} + 12 T_{13}^{5} + 51 T_{13}^{4} + 36 T_{13}^{3} - 171 T_{13}^{2} - 135 T_{13} + 675$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 - 225 T^{2} + 535 T^{4} - 232 T^{6} + 75 T^{8} - 10 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$( -243 + 198 T^{2} - 27 T^{4} + T^{6} )^{2}$$
$7$ $$( 343 - 98 T - 28 T^{2} + 13 T^{3} - 4 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$11$ $$( 225 + 214 T^{2} + 37 T^{4} + T^{6} )^{2}$$
$13$ $$( 675 - 135 T - 171 T^{2} + 36 T^{3} + 51 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$17$ $$59049 + 54675 T^{2} + 43335 T^{4} + 6264 T^{6} + 675 T^{8} + 30 T^{10} + T^{12}$$
$19$ $$( 243 + 648 T + 549 T^{2} - 72 T^{3} - 21 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$23$ $$( 729 + 517 T^{2} + 94 T^{4} + T^{6} )^{2}$$
$29$ $$6561 - 26082 T^{2} + 100687 T^{4} - 11752 T^{6} + 1047 T^{8} - 37 T^{10} + T^{12}$$
$31$ $$( 64827 - 27783 T + 3087 T^{2} + 378 T^{3} - 51 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$37$ $$( 4489 - 1273 T + 629 T^{2} - 58 T^{3} + 35 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$41$ $$59049 + 474579 T^{2} + 3786507 T^{4} + 222156 T^{6} + 11043 T^{8} + 114 T^{10} + T^{12}$$
$43$ $$( 1 - 16 T + 251 T^{2} - 82 T^{3} + 41 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$47$ $$387420489 + 76173210 T^{2} + 12319695 T^{4} + 483084 T^{6} + 14355 T^{8} + 135 T^{10} + T^{12}$$
$53$ $$81 - 15453 T^{2} + 2947027 T^{4} - 202588 T^{6} + 12207 T^{8} - 118 T^{10} + T^{12}$$
$59$ $$4782969 + 1614006 T^{2} + 406863 T^{4} + 42120 T^{6} + 3231 T^{8} + 63 T^{10} + T^{12}$$
$61$ $$( 49923 + 19737 T + 1440 T^{2} - 459 T^{3} - 24 T^{4} + 9 T^{5} + T^{6} )^{2}$$
$67$ $$( 458329 + 10155 T + 12411 T^{2} - 1624 T^{3} + 309 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$71$ $$( 19881 + 2290 T^{2} + 85 T^{4} + T^{6} )^{2}$$
$73$ $$( 177147 + 17496 T - 4527 T^{2} - 504 T^{3} + 123 T^{4} + 21 T^{5} + T^{6} )^{2}$$
$79$ $$( 49 - 105 T + 351 T^{2} + 284 T^{3} + 309 T^{4} + 18 T^{5} + T^{6} )^{2}$$
$83$ $$36905625 + 35320050 T^{2} + 32836671 T^{4} + 912276 T^{6} + 19467 T^{8} + 159 T^{10} + T^{12}$$
$89$ $$7695324729 + 848720025 T^{2} + 75973302 T^{4} + 1769229 T^{6} + 30726 T^{8} + 201 T^{10} + T^{12}$$
$97$ $$( 1728 - 3456 T + 1584 T^{2} + 1440 T^{3} + 348 T^{4} + 30 T^{5} + T^{6} )^{2}$$