# Properties

 Label 567.3.r.c Level $567$ Weight $3$ Character orbit 567.r Analytic conductor $15.450$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$15.4496309892$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.39033114624.8 Defining polynomial: $$x^{8} - 2 x^{7} + 6 x^{6} - 30 x^{5} + 34 x^{4} - 102 x^{3} + 486 x^{2} - 730 x + 373$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( 3 - 2 \beta_{1} + 3 \beta_{3} ) q^{4} + ( -\beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} + ( -2 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( 3 - 2 \beta_{1} + 3 \beta_{3} ) q^{4} + ( -\beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} + ( -2 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{8} + ( 7 - \beta_{6} ) q^{10} + 2 \beta_{4} q^{11} + ( 9 - \beta_{1} + 9 \beta_{3} ) q^{13} + ( -\beta_{5} + 2 \beta_{7} ) q^{14} + ( -4 \beta_{1} + 9 \beta_{3} - 4 \beta_{6} ) q^{16} -2 \beta_{2} q^{17} + ( 3 + 5 \beta_{6} ) q^{19} + ( -3 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{20} + ( 14 - 4 \beta_{1} + 14 \beta_{3} ) q^{22} + ( -2 \beta_{5} + 6 \beta_{7} ) q^{23} + ( -10 \beta_{1} - 3 \beta_{3} - 10 \beta_{6} ) q^{25} + ( -\beta_{2} + 11 \beta_{4} + 11 \beta_{7} ) q^{26} + ( -14 - 3 \beta_{6} ) q^{28} + ( -2 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{29} + ( -34 - 2 \beta_{1} - 34 \beta_{3} ) q^{31} + ( 4 \beta_{5} + 5 \beta_{7} ) q^{32} + ( -6 \beta_{1} - 6 \beta_{6} ) q^{34} + ( -3 \beta_{2} - \beta_{4} - \beta_{7} ) q^{35} + ( 4 - 14 \beta_{6} ) q^{37} + ( -5 \beta_{2} + 13 \beta_{4} + 5 \beta_{5} ) q^{38} + ( -21 - 15 \beta_{1} - 21 \beta_{3} ) q^{40} + ( 8 \beta_{5} - 14 \beta_{7} ) q^{41} + ( -6 \beta_{1} - 40 \beta_{3} - 6 \beta_{6} ) q^{43} + ( -4 \beta_{2} + 14 \beta_{4} + 14 \beta_{7} ) q^{44} + ( -42 - 18 \beta_{6} ) q^{46} + ( 4 \beta_{2} + 12 \beta_{4} - 4 \beta_{5} ) q^{47} + ( -7 - 7 \beta_{3} ) q^{49} + ( -10 \beta_{5} + 17 \beta_{7} ) q^{50} + ( -21 \beta_{1} + 41 \beta_{3} - 21 \beta_{6} ) q^{52} + ( 16 \beta_{2} + 6 \beta_{4} + 6 \beta_{7} ) q^{53} + ( 14 - 2 \beta_{6} ) q^{55} + ( -\beta_{2} - 12 \beta_{4} + \beta_{5} ) q^{56} + ( -28 + 2 \beta_{1} - 28 \beta_{3} ) q^{58} + ( -\beta_{5} - 27 \beta_{7} ) q^{59} + ( 7 \beta_{1} - 39 \beta_{3} + 7 \beta_{6} ) q^{61} + ( -2 \beta_{2} - 30 \beta_{4} - 30 \beta_{7} ) q^{62} + ( 1 + 18 \beta_{6} ) q^{64} + ( 6 \beta_{2} + 8 \beta_{4} - 6 \beta_{5} ) q^{65} + ( 6 + 8 \beta_{1} + 6 \beta_{3} ) q^{67} + ( 2 \beta_{5} + 12 \beta_{7} ) q^{68} + ( -7 \beta_{1} - 7 \beta_{3} - 7 \beta_{6} ) q^{70} + ( -6 \beta_{2} - 24 \beta_{4} - 24 \beta_{7} ) q^{71} + ( -8 - 26 \beta_{6} ) q^{73} + ( 14 \beta_{2} - 24 \beta_{4} - 14 \beta_{5} ) q^{74} + ( 79 - 21 \beta_{1} + 79 \beta_{3} ) q^{76} + ( -2 \beta_{5} + 4 \beta_{7} ) q^{77} + ( -36 \beta_{1} + 32 \beta_{3} - 36 \beta_{6} ) q^{79} + ( -3 \beta_{2} + 5 \beta_{4} + 5 \beta_{7} ) q^{80} + ( 98 + 52 \beta_{6} ) q^{82} + ( -9 \beta_{2} - 27 \beta_{4} + 9 \beta_{5} ) q^{83} + ( -42 - 18 \beta_{1} - 42 \beta_{3} ) q^{85} + ( -6 \beta_{5} - 28 \beta_{7} ) q^{86} + ( -24 \beta_{1} + 42 \beta_{3} - 24 \beta_{6} ) q^{88} + ( 16 \beta_{2} - 26 \beta_{4} - 26 \beta_{7} ) q^{89} + ( -7 - 9 \beta_{6} ) q^{91} + ( 10 \beta_{2} - 54 \beta_{4} - 10 \beta_{5} ) q^{92} + ( 84 - 12 \beta_{1} + 84 \beta_{3} ) q^{94} + ( 12 \beta_{5} + 2 \beta_{7} ) q^{95} + ( 8 \beta_{1} - 2 \beta_{3} + 8 \beta_{6} ) q^{97} + ( -7 \beta_{4} - 7 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{4} + O(q^{10})$$ $$8q + 12q^{4} + 56q^{10} + 36q^{13} - 36q^{16} + 24q^{19} + 56q^{22} + 12q^{25} - 112q^{28} - 136q^{31} + 32q^{37} - 84q^{40} + 160q^{43} - 336q^{46} - 28q^{49} - 164q^{52} + 112q^{55} - 112q^{58} + 156q^{61} + 8q^{64} + 24q^{67} + 28q^{70} - 64q^{73} + 316q^{76} - 128q^{79} + 784q^{82} - 168q^{85} - 168q^{88} - 56q^{91} + 336q^{94} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 6 x^{6} - 30 x^{5} + 34 x^{4} - 102 x^{3} + 486 x^{2} - 730 x + 373$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-62 \nu^{7} - 1809 \nu^{6} - 3078 \nu^{5} - 13500 \nu^{4} + 11602 \nu^{3} + 37037 \nu^{2} + 174898 \nu - 164120$$$$)/46998$$ $$\beta_{2}$$ $$=$$ $$($$$$56 \nu^{7} - 207 \nu^{6} + 121 \nu^{5} - 2558 \nu^{4} + 2287 \nu^{3} - 4070 \nu^{2} + 43592 \nu - 43268$$$$)/2611$$ $$\beta_{3}$$ $$=$$ $$($$$$163 \nu^{7} - 63 \nu^{6} + 945 \nu^{5} - 3276 \nu^{4} + 469 \nu^{3} - 15883 \nu^{2} + 51751 \nu - 43268$$$$)/6714$$ $$\beta_{4}$$ $$=$$ $$($$$$1223 \nu^{7} + 255 \nu^{6} + 7365 \nu^{5} - 20310 \nu^{4} - 655 \nu^{3} - 114503 \nu^{2} + 346799 \nu - 202912$$$$)/46998$$ $$\beta_{5}$$ $$=$$ $$($$$$-448 \nu^{7} - 582 \nu^{6} - 2460 \nu^{5} + 4425 \nu^{4} + 14528 \nu^{3} + 50464 \nu^{2} - 53320 \nu - 78703$$$$)/15666$$ $$\beta_{6}$$ $$=$$ $$($$$$-152 \nu^{7} + 29 \nu^{6} - 808 \nu^{5} + 3373 \nu^{4} + 986 \nu^{3} + 16429 \nu^{2} - 45746 \nu + 19023$$$$)/5222$$ $$\beta_{7}$$ $$=$$ $$($$$$2323 \nu^{7} - 534 \nu^{6} + 13605 \nu^{5} - 44553 \nu^{4} + 4597 \nu^{3} - 247522 \nu^{2} + 669787 \nu - 532271$$$$)/46998$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} - \beta_{5} + 3 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{7} + \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 3 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$27 \beta_{7} + 6 \beta_{6} + 7 \beta_{5} + 24 \beta_{4} - 63 \beta_{3} - 2 \beta_{2} + 9 \beta_{1} + 15$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} + 10 \beta_{3} + 4 \beta_{2} - 4 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$($$$$-243 \beta_{7} - 138 \beta_{6} + 73 \beta_{5} - 138 \beta_{4} + 597 \beta_{3} - 44 \beta_{2} - 93 \beta_{1} + 315$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$57 \beta_{7} - 168 \beta_{6} + 77 \beta_{5} + 375 \beta_{4} - 630 \beta_{3} + 2 \beta_{2} + 9 \beta_{1} - 732$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$1743 \beta_{7} + 2100 \beta_{6} - 595 \beta_{5} + 882 \beta_{4} - 2097 \beta_{3} - 280 \beta_{2} + 1575 \beta_{1} + 255$$$$)/6$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 −0.279898 − 3.02113i −1.85391 + 1.90397i 1.03103 − 0.478705i 2.10277 − 0.136187i −0.279898 + 3.02113i −1.85391 − 1.90397i 1.03103 + 0.478705i 2.10277 + 0.136187i
−3.03622 + 1.75296i 0 4.14575 7.18065i −1.07558 0.620984i 0 −1.32288 2.29129i 15.0457i 0 4.35425
134.2 −1.13198 + 0.653548i 0 −1.14575 + 1.98450i −6.39086 3.68977i 0 1.32288 + 2.29129i 8.22359i 0 9.64575
134.3 1.13198 0.653548i 0 −1.14575 + 1.98450i 6.39086 + 3.68977i 0 1.32288 + 2.29129i 8.22359i 0 9.64575
134.4 3.03622 1.75296i 0 4.14575 7.18065i 1.07558 + 0.620984i 0 −1.32288 2.29129i 15.0457i 0 4.35425
512.1 −3.03622 1.75296i 0 4.14575 + 7.18065i −1.07558 + 0.620984i 0 −1.32288 + 2.29129i 15.0457i 0 4.35425
512.2 −1.13198 0.653548i 0 −1.14575 1.98450i −6.39086 + 3.68977i 0 1.32288 2.29129i 8.22359i 0 9.64575
512.3 1.13198 + 0.653548i 0 −1.14575 1.98450i 6.39086 3.68977i 0 1.32288 2.29129i 8.22359i 0 9.64575
512.4 3.03622 + 1.75296i 0 4.14575 + 7.18065i 1.07558 0.620984i 0 −1.32288 + 2.29129i 15.0457i 0 4.35425
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 512.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.3.r.c 8
3.b odd 2 1 inner 567.3.r.c 8
9.c even 3 1 21.3.b.a 4
9.c even 3 1 inner 567.3.r.c 8
9.d odd 6 1 21.3.b.a 4
9.d odd 6 1 inner 567.3.r.c 8
36.f odd 6 1 336.3.d.c 4
36.h even 6 1 336.3.d.c 4
45.h odd 6 1 525.3.c.a 4
45.j even 6 1 525.3.c.a 4
45.k odd 12 2 525.3.f.a 8
45.l even 12 2 525.3.f.a 8
63.g even 3 1 147.3.h.e 8
63.h even 3 1 147.3.h.e 8
63.i even 6 1 147.3.h.c 8
63.j odd 6 1 147.3.h.e 8
63.k odd 6 1 147.3.h.c 8
63.l odd 6 1 147.3.b.f 4
63.n odd 6 1 147.3.h.e 8
63.o even 6 1 147.3.b.f 4
63.s even 6 1 147.3.h.c 8
63.t odd 6 1 147.3.h.c 8
72.j odd 6 1 1344.3.d.f 4
72.l even 6 1 1344.3.d.b 4
72.n even 6 1 1344.3.d.f 4
72.p odd 6 1 1344.3.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 9.c even 3 1
21.3.b.a 4 9.d odd 6 1
147.3.b.f 4 63.l odd 6 1
147.3.b.f 4 63.o even 6 1
147.3.h.c 8 63.i even 6 1
147.3.h.c 8 63.k odd 6 1
147.3.h.c 8 63.s even 6 1
147.3.h.c 8 63.t odd 6 1
147.3.h.e 8 63.g even 3 1
147.3.h.e 8 63.h even 3 1
147.3.h.e 8 63.j odd 6 1
147.3.h.e 8 63.n odd 6 1
336.3.d.c 4 36.f odd 6 1
336.3.d.c 4 36.h even 6 1
525.3.c.a 4 45.h odd 6 1
525.3.c.a 4 45.j even 6 1
525.3.f.a 8 45.k odd 12 2
525.3.f.a 8 45.l even 12 2
567.3.r.c 8 1.a even 1 1 trivial
567.3.r.c 8 3.b odd 2 1 inner
567.3.r.c 8 9.c even 3 1 inner
567.3.r.c 8 9.d odd 6 1 inner
1344.3.d.b 4 72.l even 6 1
1344.3.d.b 4 72.p odd 6 1
1344.3.d.f 4 72.j odd 6 1
1344.3.d.f 4 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2}^{8} - 14 T_{2}^{6} + 175 T_{2}^{4} - 294 T_{2}^{2} + 441$$ $$T_{5}^{8} - 56 T_{5}^{6} + 3052 T_{5}^{4} - 4704 T_{5}^{2} + 7056$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} - T^{4} - 54 T^{6} - 295 T^{8} - 864 T^{10} - 256 T^{12} + 8192 T^{14} + 65536 T^{16}$$
$3$ 1
$5$ $$1 + 44 T^{2} + 902 T^{4} - 9504 T^{6} - 531469 T^{8} - 5940000 T^{10} + 352343750 T^{12} + 10742187500 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 + 428 T^{2} + 108554 T^{4} + 19408944 T^{6} + 2673281075 T^{8} + 284166349104 T^{10} + 23269513968074 T^{12} + 1343247345236588 T^{14} + 45949729863572161 T^{16}$$
$13$ $$( 1 - 18 T - 88 T^{2} - 1332 T^{3} + 86427 T^{4} - 225108 T^{5} - 2513368 T^{6} - 86882562 T^{7} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 - 988 T^{2} + 407046 T^{4} - 82518748 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 6 T + 556 T^{2} - 2166 T^{3} + 130321 T^{4} )^{4}$$
$23$ $$1 + 1444 T^{2} + 1104970 T^{4} + 607178896 T^{6} + 298907858899 T^{8} + 169913549435536 T^{10} + 86531289405946570 T^{12} + 31644717679837343524 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 + 2972 T^{2} + 5221226 T^{4} + 6529472112 T^{6} + 6295844474099 T^{8} + 4618171564847472 T^{10} + 2611899577758710186 T^{12} +$$$$10\!\cdots\!52$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$( 1 + 68 T + 1574 T^{2} + 76704 T^{3} + 3935315 T^{4} + 73712544 T^{5} + 1453622054 T^{6} + 60350250308 T^{7} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 1382 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$1 + 1292 T^{2} - 1162774 T^{4} - 3642773328 T^{6} - 4681427105581 T^{8} - 10293606802102608 T^{10} - 9284663448365941654 T^{12} +$$$$29\!\cdots\!52$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 - 80 T + 1354 T^{2} - 107840 T^{3} + 10209715 T^{4} - 199396160 T^{5} + 4629056554 T^{6} - 505709043920 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 + 6148 T^{2} + 18653578 T^{4} + 57698758672 T^{6} + 158252217343891 T^{8} + 281551536415343632 T^{10} +$$$$44\!\cdots\!58$$$$T^{12} +$$$$71\!\cdots\!68$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 + 20 T^{2} - 13350138 T^{4} + 157809620 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 + 3676 T^{2} - 2451290 T^{4} - 30402196256 T^{6} - 55714257061901 T^{8} - 368394387226800416 T^{10} -$$$$35\!\cdots\!90$$$$T^{12} +$$$$65\!\cdots\!56$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 - 78 T - 2536 T^{2} - 91884 T^{3} + 37819995 T^{4} - 341900364 T^{5} - 35113052776 T^{6} - 4018589200158 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 - 12 T - 8422 T^{2} + 4944 T^{3} + 52578819 T^{4} + 22193616 T^{5} - 169712741062 T^{6} - 1085500586028 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 + 16 T + 5990 T^{2} + 85264 T^{3} + 28398241 T^{4} )^{4}$$
$79$ $$( 1 + 64 T - 338 T^{2} - 515072 T^{3} - 44852861 T^{4} - 3214564352 T^{5} - 13165127378 T^{6} + 15557597153344 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 + 13948 T^{2} + 52646758 T^{4} + 655323124192 T^{6} + 8650442514739891 T^{8} + 31100535186626801632 T^{10} +$$$$11\!\cdots\!78$$$$T^{12} +$$$$14\!\cdots\!28$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 4 T - 18358 T^{2} + 1776 T^{3} + 248924051 T^{4} + 16710384 T^{5} - 1625220540598 T^{6} - 3331888019716 T^{7} + 7837433594376961 T^{8} )^{2}$$