# Properties

 Label 572.2.i.c Level $572$ Weight $2$ Character orbit 572.i Analytic conductor $4.567$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 9 x^{8} + 6 x^{7} + 59 x^{6} + 2 x^{5} + 47 x^{4} - 26 x^{3} + 38 x^{2} - 12 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{5} + ( -\beta_{3} + \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{5} + ( -\beta_{3} + \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} -\beta_{6} q^{11} + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{13} + ( 2 \beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{15} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{17} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{19} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{21} + ( -2 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{23} + ( 2 - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{25} + ( -3 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{29} + ( -2 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -\beta_{1} - \beta_{2} ) q^{33} + ( 3 + \beta_{3} - 3 \beta_{6} - \beta_{7} - \beta_{9} ) q^{35} + ( -\beta_{1} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{37} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{39} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{41} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{6} + \beta_{7} + \beta_{9} ) q^{43} + ( -6 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} + 6 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} ) q^{45} + ( -3 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{47} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{49} + ( 2 + 2 \beta_{2} + \beta_{3} ) q^{51} + ( 3 \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{53} + ( -\beta_{1} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{55} + ( 4 - 5 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{57} + ( 2 - \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{59} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{6} - \beta_{7} - \beta_{9} ) q^{61} + ( -\beta_{1} + \beta_{5} + 4 \beta_{6} + \beta_{8} + \beta_{9} ) q^{63} + ( -3 + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{65} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{67} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{69} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{71} + ( -8 - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{73} + ( 4 \beta_{1} + \beta_{5} + 10 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{75} + \beta_{3} q^{77} + ( 1 + 3 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{79} + ( -6 \beta_{1} + \beta_{5} - 4 \beta_{6} + \beta_{9} ) q^{81} + ( -4 - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{83} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} + 4 \beta_{9} ) q^{85} + ( 5 - 4 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{87} + ( \beta_{1} + 5 \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{89} + ( 3 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{93} + ( 1 - 5 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{95} + ( 4 + 2 \beta_{3} - 3 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{97} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + q^{3} + 2q^{5} + q^{7} - 2q^{9} + O(q^{10})$$ $$10q + q^{3} + 2q^{5} + q^{7} - 2q^{9} - 5q^{11} + q^{13} + 12q^{15} - 3q^{17} + 12q^{19} - 12q^{21} - 7q^{23} + 28q^{25} - 32q^{27} - 10q^{29} - 18q^{31} + q^{33} + 15q^{35} + 10q^{37} + 5q^{41} - 14q^{43} - 36q^{45} - 24q^{47} + 6q^{49} + 14q^{51} - 14q^{53} - q^{55} + 52q^{57} + 8q^{59} + 18q^{61} + 20q^{63} - 45q^{65} - q^{67} + 7q^{69} + 3q^{71} - 76q^{73} + 57q^{75} - 2q^{77} + 12q^{79} - 25q^{81} - 28q^{83} - 10q^{85} + 27q^{87} + 29q^{89} + 17q^{91} - 21q^{93} + 11q^{95} + 21q^{97} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 9 x^{8} + 6 x^{7} + 59 x^{6} + 2 x^{5} + 47 x^{4} - 26 x^{3} + 38 x^{2} - 12 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2483 \nu^{9} - 413 \nu^{8} + 767 \nu^{7} - 57824 \nu^{6} + 295 \nu^{5} - 21476 \nu^{4} + 1085227 \nu^{3} - 18408 \nu^{2} + 5900 \nu - 300404$$$$)/929970$$ $$\beta_{3}$$ $$=$$ $$($$$$19399 \nu^{9} - 143591 \nu^{8} + 266669 \nu^{7} - 984578 \nu^{6} + 102565 \nu^{5} - 7466732 \nu^{4} - 1739951 \nu^{3} - 6400056 \nu^{2} + 2051300 \nu - 1548188$$$$)/929970$$ $$\beta_{4}$$ $$=$$ $$($$$$-3302 \nu^{9} + 27853 \nu^{8} - 51727 \nu^{7} + 198199 \nu^{6} - 19895 \nu^{5} + 1448356 \nu^{4} + 95293 \nu^{3} + 1241448 \nu^{2} - 397900 \nu + 774739$$$$)/154995$$ $$\beta_{5}$$ $$=$$ $$($$$$18803 \nu^{9} - 93877 \nu^{8} + 174343 \nu^{7} - 494521 \nu^{6} + 67055 \nu^{5} - 4881604 \nu^{4} - 3035092 \nu^{3} - 4184232 \nu^{2} + 1341100 \nu - 1427971$$$$)/464985$$ $$\beta_{6}$$ $$=$$ $$($$$$75101 \nu^{9} - 77584 \nu^{8} + 675496 \nu^{7} + 451373 \nu^{6} + 4373135 \nu^{5} + 150497 \nu^{4} + 3508271 \nu^{3} - 867399 \nu^{2} + 2835430 \nu + 34658$$$$)/929970$$ $$\beta_{7}$$ $$=$$ $$($$$$-75074 \nu^{9} + 5116 \nu^{8} - 607339 \nu^{7} - 1042322 \nu^{6} - 4919210 \nu^{5} - 3918833 \nu^{4} - 3695354 \nu^{3} + 626586 \nu^{2} - 1202335 \nu - 75212$$$$)/464985$$ $$\beta_{8}$$ $$=$$ $$($$$$-8279 \nu^{9} + 4886 \nu^{8} - 71072 \nu^{7} - 79679 \nu^{6} - 509807 \nu^{5} - 210913 \nu^{4} - 396279 \nu^{3} + 83447 \nu^{2} - 255794 \nu - 5874$$$$)/20666$$ $$\beta_{9}$$ $$=$$ $$($$$$-387047 \nu^{9} + 367648 \nu^{8} - 3339832 \nu^{7} - 2588951 \nu^{6} - 21851195 \nu^{5} - 876659 \nu^{4} - 10724477 \nu^{3} + 11803173 \nu^{2} - 8307730 \nu + 2593264$$$$)/929970$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + \beta_{3} + 7 \beta_{2} - 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - 9 \beta_{8} + 9 \beta_{7} - 22 \beta_{6} + \beta_{5} - 15 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{9} - 24 \beta_{8} + 16 \beta_{7} - 45 \beta_{6} - 24 \beta_{4} - 16 \beta_{3} - 64 \beta_{2} - 64 \beta_{1} + 45$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{5} - 88 \beta_{4} - 81 \beta_{3} - 173 \beta_{2} + 200$$ $$\nu^{7}$$ $$=$$ $$-81 \beta_{9} + 261 \beta_{8} - 196 \beta_{7} + 526 \beta_{6} - 81 \beta_{5} + 630 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-196 \beta_{9} + 891 \beta_{8} - 776 \beta_{7} + 1955 \beta_{6} + 891 \beta_{4} + 776 \beta_{3} + 1874 \beta_{2} + 1874 \beta_{1} - 1955$$ $$\nu^{9}$$ $$=$$ $$776 \beta_{5} + 2765 \beta_{4} + 2185 \beta_{3} + 6387 \beta_{2} - 5737$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$353$$ $$365$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
133.1
 −1.09846 − 1.90260i −0.542661 − 0.939916i 0.194431 + 0.336764i 0.334488 + 0.579350i 1.61221 + 2.79242i −1.09846 + 1.90260i −0.542661 + 0.939916i 0.194431 − 0.336764i 0.334488 − 0.579350i 1.61221 − 2.79242i
0 −1.09846 1.90260i 0 0.151238 0 2.15500 3.73256i 0 −0.913248 + 1.58179i 0
133.2 0 −0.542661 0.939916i 0 −1.28140 0 −1.24710 + 2.16005i 0 0.911039 1.57797i 0
133.3 0 0.194431 + 0.336764i 0 2.23435 0 0.258355 0.447484i 0 1.42439 2.46712i 0
133.4 0 0.334488 + 0.579350i 0 −4.07313 0 −0.950388 + 1.64612i 0 1.27624 2.21051i 0
133.5 0 1.61221 + 2.79242i 0 3.96894 0 0.284141 0.492146i 0 −3.69842 + 6.40585i 0
529.1 0 −1.09846 + 1.90260i 0 0.151238 0 2.15500 + 3.73256i 0 −0.913248 1.58179i 0
529.2 0 −0.542661 + 0.939916i 0 −1.28140 0 −1.24710 2.16005i 0 0.911039 + 1.57797i 0
529.3 0 0.194431 0.336764i 0 2.23435 0 0.258355 + 0.447484i 0 1.42439 + 2.46712i 0
529.4 0 0.334488 0.579350i 0 −4.07313 0 −0.950388 1.64612i 0 1.27624 + 2.21051i 0
529.5 0 1.61221 2.79242i 0 3.96894 0 0.284141 + 0.492146i 0 −3.69842 6.40585i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.i.c 10
13.c even 3 1 inner 572.2.i.c 10
13.c even 3 1 7436.2.a.r 5
13.e even 6 1 7436.2.a.q 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.i.c 10 1.a even 1 1 trivial
572.2.i.c 10 13.c even 3 1 inner
7436.2.a.q 5 13.e even 6 1
7436.2.a.r 5 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$4 - 12 T + 38 T^{2} - 26 T^{3} + 47 T^{4} + 2 T^{5} + 59 T^{6} + 6 T^{7} + 9 T^{8} - T^{9} + T^{10}$$
$5$ $$( -7 + 44 T + 18 T^{2} - 19 T^{3} - T^{4} + T^{5} )^{2}$$
$7$ $$36 - 108 T + 294 T^{2} - 258 T^{3} + 271 T^{4} + 100 T^{5} + 173 T^{6} + 24 T^{7} + 15 T^{8} - T^{9} + T^{10}$$
$11$ $$( 1 + T + T^{2} )^{5}$$
$13$ $$371293 - 28561 T - 32955 T^{2} + 4732 T^{3} + 1157 T^{4} - 21 T^{5} + 89 T^{6} + 28 T^{7} - 15 T^{8} - T^{9} + T^{10}$$
$17$ $$3969 + 4158 T + 5994 T^{2} + 1434 T^{3} + 2137 T^{4} + 317 T^{5} + 637 T^{6} - 23 T^{7} + 34 T^{8} + 3 T^{9} + T^{10}$$
$19$ $$2916 - 9720 T + 27594 T^{2} - 18288 T^{3} + 12349 T^{4} - 2397 T^{5} + 1689 T^{6} - 430 T^{7} + 123 T^{8} - 12 T^{9} + T^{10}$$
$23$ $$1106704 - 100992 T + 262748 T^{2} + 77840 T^{3} + 48221 T^{4} + 8662 T^{5} + 2459 T^{6} + 300 T^{7} + 75 T^{8} + 7 T^{9} + T^{10}$$
$29$ $$463761 - 1101177 T + 3089346 T^{2} + 1186977 T^{3} + 407851 T^{4} + 63689 T^{5} + 10523 T^{6} + 954 T^{7} + 144 T^{8} + 10 T^{9} + T^{10}$$
$31$ $$( 4124 + 328 T - 393 T^{2} - 37 T^{3} + 9 T^{4} + T^{5} )^{2}$$
$37$ $$30625 + 7875 T + 15850 T^{2} - 1455 T^{3} + 4761 T^{4} - 601 T^{5} + 871 T^{6} - 218 T^{7} + 94 T^{8} - 10 T^{9} + T^{10}$$
$41$ $$80089 + 102446 T + 115762 T^{2} + 40490 T^{3} + 17725 T^{4} + 1905 T^{5} + 1277 T^{6} + 77 T^{7} + 62 T^{8} - 5 T^{9} + T^{10}$$
$43$ $$4096 + 4096 T + 8128 T^{2} + 3520 T^{3} + 8641 T^{4} + 5445 T^{5} + 2663 T^{6} + 700 T^{7} + 137 T^{8} + 14 T^{9} + T^{10}$$
$47$ $$( -866 - 1424 T - 615 T^{2} - 25 T^{3} + 12 T^{4} + T^{5} )^{2}$$
$53$ $$( 8799 - 150 T - 1300 T^{2} - 149 T^{3} + 7 T^{4} + T^{5} )^{2}$$
$59$ $$40804 - 317544 T + 2311402 T^{2} - 1213152 T^{3} + 509397 T^{4} - 84275 T^{5} + 13525 T^{6} - 982 T^{7} + 139 T^{8} - 8 T^{9} + T^{10}$$
$61$ $$1570009 - 1367023 T + 1141414 T^{2} - 268089 T^{3} + 122265 T^{4} - 34513 T^{5} + 9893 T^{6} - 1698 T^{7} + 234 T^{8} - 18 T^{9} + T^{10}$$
$67$ $$252004 - 2071252 T + 17039438 T^{2} - 28718 T^{3} + 645119 T^{4} - 13590 T^{5} + 20179 T^{6} - 218 T^{7} + 157 T^{8} + T^{9} + T^{10}$$
$71$ $$1373584 - 3295664 T + 7924924 T^{2} - 262540 T^{3} + 362269 T^{4} + 13750 T^{5} + 14133 T^{6} + 360 T^{7} + 139 T^{8} - 3 T^{9} + T^{10}$$
$73$ $$( 3672 + 5580 T + 2754 T^{2} + 505 T^{3} + 38 T^{4} + T^{5} )^{2}$$
$79$ $$( 23216 + 9736 T + 216 T^{2} - 202 T^{3} - 6 T^{4} + T^{5} )^{2}$$
$83$ $$( 626 + 392 T - 249 T^{2} - 73 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$89$ $$2663424 + 2232576 T + 2212512 T^{2} + 373416 T^{3} + 272689 T^{4} - 38758 T^{5} + 48233 T^{6} - 6276 T^{7} + 639 T^{8} - 29 T^{9} + T^{10}$$
$97$ $$35617024 - 95297024 T + 236303152 T^{2} - 49426752 T^{3} + 9197409 T^{4} - 805493 T^{5} + 83702 T^{6} - 5313 T^{7} + 486 T^{8} - 21 T^{9} + T^{10}$$