# Properties

 Label 567.2.e.e Level $567$ Weight $2$ Character orbit 567.e Analytic conductor $4.528$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Defining polynomial: $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} + ( 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{10} + ( \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{9} ) q^{13} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{14} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{16} + ( -\beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( 2 + \beta_{3} - 2 \beta_{4} - 2 \beta_{8} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} ) q^{22} + ( -\beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{23} + ( \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{9} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{26} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{28} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{29} + ( \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + ( -\beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{32} + ( -2 + 4 \beta_{1} - \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{34} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{37} + ( -3 \beta_{3} - 2 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{38} + ( -1 - \beta_{1} + \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{40} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{41} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 3 \beta_{9} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{44} + ( 2 \beta_{3} + 5 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{46} + ( -4 - \beta_{1} - \beta_{2} - \beta_{4} - 4 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{49} + ( 6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{50} + ( -\beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{52} + ( \beta_{5} - 5 \beta_{6} + 2 \beta_{9} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55} + ( -1 + \beta_{2} + \beta_{5} + 4 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{56} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{58} + ( -\beta_{3} + \beta_{5} + 6 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{61} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{62} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{65} + ( -2 \beta_{3} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( -7 + 2 \beta_{1} + \beta_{2} - 7 \beta_{6} - 3 \beta_{7} ) q^{68} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{70} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -\beta_{3} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{73} + ( 4 \beta_{3} + 2 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{5} + \beta_{9} ) q^{76} + ( -3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} ) q^{77} + ( -3 + 4 \beta_{1} + \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 2 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{80} + ( 2 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( 1 - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{85} + ( -1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{7} ) q^{86} + ( \beta_{3} + 4 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{88} + ( -7 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 7 \beta_{6} ) q^{89} + ( -\beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{91} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{92} + ( 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} + 2 \beta_{9} ) q^{94} + ( -2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{95} + ( 2 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{97} + ( -5 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 2q^{2} - 4q^{4} - 4q^{5} + 5q^{7} + 6q^{8} + O(q^{10})$$ $$10q - 2q^{2} - 4q^{4} - 4q^{5} + 5q^{7} + 6q^{8} - 7q^{10} - 4q^{11} + 16q^{13} - 4q^{14} + 2q^{16} - 12q^{17} + q^{19} + 10q^{20} + 2q^{22} - 3q^{23} - q^{25} - 11q^{26} - 2q^{28} + 14q^{29} - 3q^{31} + 2q^{32} - 6q^{34} - 5q^{35} - 20q^{38} - 3q^{40} + 10q^{41} + 14q^{43} + 10q^{44} + 3q^{46} - 27q^{47} - 17q^{49} + 38q^{50} - 10q^{52} + 21q^{53} + 4q^{55} - 27q^{56} - 10q^{58} - 30q^{59} - 14q^{61} + 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 27q^{68} - 11q^{70} + 6q^{71} + 15q^{73} + 36q^{74} - 10q^{76} - 20q^{77} - 4q^{79} - 20q^{80} - 5q^{82} + 18q^{83} + 12q^{85} + 8q^{86} - 18q^{88} - 28q^{89} - 4q^{91} + 54q^{92} - 3q^{94} + 14q^{95} + 24q^{97} - 59q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557$$$$)/567$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$-53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368$$$$)/567$$ $$\beta_{8}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 4 \beta_{5} + \beta_{4} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{7} - 14 \beta_{6} + \beta_{4} + \beta_{2} - 14$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 9 \beta_{6} - 17 \beta_{5} + \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{9} - 10 \beta_{8} - \beta_{5} - 10 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + \beta_{1} + 70$$ $$\nu^{7}$$ $$=$$ $$11 \beta_{7} + 65 \beta_{6} - 43 \beta_{4} - 19 \beta_{2} + 75 \beta_{1} + 65$$ $$\nu^{8}$$ $$=$$ $$-62 \beta_{9} + 73 \beta_{8} + 118 \beta_{7} + 360 \beta_{6} + 14 \beta_{5} - 118 \beta_{3}$$ $$\nu^{9}$$ $$=$$ $$-135 \beta_{9} + 253 \beta_{8} + 343 \beta_{5} + 253 \beta_{4} - 87 \beta_{3} + 135 \beta_{2} - 343 \beta_{1} - 430$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1 - \beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 1.19343 − 2.06709i 0.920620 − 1.59456i 0.247934 − 0.429435i −0.335166 + 0.580525i −1.02682 + 1.77851i 1.19343 + 2.06709i 0.920620 + 1.59456i 0.247934 + 0.429435i −0.335166 − 0.580525i −1.02682 − 1.77851i
−1.19343 + 2.06709i 0 −1.84857 3.20182i −1.46043 + 2.52954i 0 −2.21886 1.44106i 4.05086 0 −3.48586 6.03769i
163.2 −0.920620 + 1.59456i 0 −0.695084 1.20392i 0.667377 1.15593i 0 0.640804 + 2.56698i −1.12285 0 1.22880 + 2.12835i
163.3 −0.247934 + 0.429435i 0 0.877057 + 1.51911i −1.84629 + 3.19787i 0 1.68284 + 2.04158i −1.86155 0 −0.915516 1.58572i
163.4 0.335166 0.580525i 0 0.775327 + 1.34291i 0.712469 1.23403i 0 0.145107 2.64177i 2.38012 0 −0.477591 0.827212i
163.5 1.02682 1.77851i 0 −1.10873 1.92038i −0.0731228 + 0.126652i 0 2.25011 1.39176i −0.446582 0 0.150168 + 0.260099i
487.1 −1.19343 2.06709i 0 −1.84857 + 3.20182i −1.46043 2.52954i 0 −2.21886 + 1.44106i 4.05086 0 −3.48586 + 6.03769i
487.2 −0.920620 1.59456i 0 −0.695084 + 1.20392i 0.667377 + 1.15593i 0 0.640804 2.56698i −1.12285 0 1.22880 2.12835i
487.3 −0.247934 0.429435i 0 0.877057 1.51911i −1.84629 3.19787i 0 1.68284 2.04158i −1.86155 0 −0.915516 + 1.58572i
487.4 0.335166 + 0.580525i 0 0.775327 1.34291i 0.712469 + 1.23403i 0 0.145107 + 2.64177i 2.38012 0 −0.477591 + 0.827212i
487.5 1.02682 + 1.77851i 0 −1.10873 + 1.92038i −0.0731228 0.126652i 0 2.25011 + 1.39176i −0.446582 0 0.150168 0.260099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.e.e 10
3.b odd 2 1 567.2.e.f 10
7.c even 3 1 inner 567.2.e.e 10
7.c even 3 1 3969.2.a.bc 5
7.d odd 6 1 3969.2.a.bb 5
9.c even 3 1 189.2.g.b 10
9.c even 3 1 189.2.h.b 10
9.d odd 6 1 63.2.g.b 10
9.d odd 6 1 63.2.h.b yes 10
21.g even 6 1 3969.2.a.ba 5
21.h odd 6 1 567.2.e.f 10
21.h odd 6 1 3969.2.a.z 5
36.f odd 6 1 3024.2.q.i 10
36.f odd 6 1 3024.2.t.i 10
36.h even 6 1 1008.2.q.i 10
36.h even 6 1 1008.2.t.i 10
63.g even 3 1 189.2.h.b 10
63.g even 3 1 1323.2.f.e 10
63.h even 3 1 189.2.g.b 10
63.h even 3 1 1323.2.f.e 10
63.i even 6 1 441.2.f.f 10
63.i even 6 1 441.2.g.f 10
63.j odd 6 1 63.2.g.b 10
63.j odd 6 1 441.2.f.e 10
63.k odd 6 1 1323.2.f.f 10
63.k odd 6 1 1323.2.h.f 10
63.l odd 6 1 1323.2.g.f 10
63.l odd 6 1 1323.2.h.f 10
63.n odd 6 1 63.2.h.b yes 10
63.n odd 6 1 441.2.f.e 10
63.o even 6 1 441.2.g.f 10
63.o even 6 1 441.2.h.f 10
63.s even 6 1 441.2.f.f 10
63.s even 6 1 441.2.h.f 10
63.t odd 6 1 1323.2.f.f 10
63.t odd 6 1 1323.2.g.f 10
252.o even 6 1 1008.2.q.i 10
252.u odd 6 1 3024.2.t.i 10
252.bb even 6 1 1008.2.t.i 10
252.bl odd 6 1 3024.2.q.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 9.d odd 6 1
63.2.g.b 10 63.j odd 6 1
63.2.h.b yes 10 9.d odd 6 1
63.2.h.b yes 10 63.n odd 6 1
189.2.g.b 10 9.c even 3 1
189.2.g.b 10 63.h even 3 1
189.2.h.b 10 9.c even 3 1
189.2.h.b 10 63.g even 3 1
441.2.f.e 10 63.j odd 6 1
441.2.f.e 10 63.n odd 6 1
441.2.f.f 10 63.i even 6 1
441.2.f.f 10 63.s even 6 1
441.2.g.f 10 63.i even 6 1
441.2.g.f 10 63.o even 6 1
441.2.h.f 10 63.o even 6 1
441.2.h.f 10 63.s even 6 1
567.2.e.e 10 1.a even 1 1 trivial
567.2.e.e 10 7.c even 3 1 inner
567.2.e.f 10 3.b odd 2 1
567.2.e.f 10 21.h odd 6 1
1008.2.q.i 10 36.h even 6 1
1008.2.q.i 10 252.o even 6 1
1008.2.t.i 10 36.h even 6 1
1008.2.t.i 10 252.bb even 6 1
1323.2.f.e 10 63.g even 3 1
1323.2.f.e 10 63.h even 3 1
1323.2.f.f 10 63.k odd 6 1
1323.2.f.f 10 63.t odd 6 1
1323.2.g.f 10 63.l odd 6 1
1323.2.g.f 10 63.t odd 6 1
1323.2.h.f 10 63.k odd 6 1
1323.2.h.f 10 63.l odd 6 1
3024.2.q.i 10 36.f odd 6 1
3024.2.q.i 10 252.bl odd 6 1
3024.2.t.i 10 36.f odd 6 1
3024.2.t.i 10 252.u odd 6 1
3969.2.a.z 5 21.h odd 6 1
3969.2.a.ba 5 21.g even 6 1
3969.2.a.bb 5 7.d odd 6 1
3969.2.a.bc 5 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$.