# Properties

 Label 567.2.g.k Level $567$ Weight $2$ Character orbit 567.g Analytic conductor $4.528$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.1767277521.3 Defining polynomial: $$x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7$$ x^8 - 2*x^7 + x^6 - 10*x^5 + 38*x^4 - 40*x^3 + 64*x^2 - 38*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (b6 + b2) * q^2 + (b7 - b5) * q^4 + (b3 + b2 + b1 - 1) * q^5 + (b4 - b3 - b2) * q^7 + (-b3 - b2 + 1) * q^8 $$q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{8} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 1) q^{10} + ( - 2 \beta_{3} - \beta_1 - 1) q^{11} + (\beta_{7} - \beta_{5} + \beta_1 + 1) q^{13} + ( - \beta_{7} + 3 \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{14} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{16} + (2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_1 + 1) q^{17} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{19} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{20} + ( - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 3) q^{22} + ( - \beta_{2} + \beta_1 + 3) q^{23} + (3 \beta_{2} + \beta_1 + 1) q^{25} + (3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{26} + ( - \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{28} + ( - 2 \beta_{7} - 3 \beta_{5}) q^{29} + (2 \beta_{6} + 5 \beta_{5} + \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{32} + ( - \beta_{7} + 5 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{34} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{35} + ( - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{37} + ( - 2 \beta_{3} + 5) q^{38} + (\beta_{3} - \beta_{2} - \beta_1 - 4) q^{40} + ( - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{43} + ( - 4 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{44} + ( - \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + \beta_{4} + 5 \beta_{2} - \beta_1 - 4) q^{46} + ( - \beta_{6} + 5 \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{47} + (3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{49} + (3 \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 8) q^{50} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 6) q^{52} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{2} - \beta_1 - 2) q^{53} + (2 \beta_{3} - 5 \beta_{2} - \beta_1 - 5) q^{55} + (\beta_{7} + 4 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{56} + (2 \beta_{3} + \beta_{2} - 2) q^{58} + (3 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{59} + ( - 4 \beta_{6} - 6 \beta_{5} + \beta_{4} - 4 \beta_{2} + 6) q^{61} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{62} + ( - 3 \beta_{3} - 3 \beta_1 - 5) q^{64} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 1) q^{65} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{67} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 13) q^{68} + ( - 3 \beta_{7} - 5 \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + \cdots - 4) q^{70}+ \cdots + (\beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 5) q^{98}+O(q^{100})$$ q + (b6 + b2) * q^2 + (b7 - b5) * q^4 + (b3 + b2 + b1 - 1) * q^5 + (b4 - b3 - b2) * q^7 + (-b3 - b2 + 1) * q^8 + (2*b7 + b6 - b5 + 2*b4 + b2 + 2*b1 + 1) * q^10 + (-2*b3 - b1 - 1) * q^11 + (b7 - b5 + b1 + 1) * q^13 + (-b7 + 3*b5 - b3 - 2*b1 - 2) * q^14 + (b6 - b4 + b2) * q^16 + (2*b7 - b5 - b4 + 2*b1 + 1) * q^17 + (b7 - b6 + 2*b5 + b4 - b3) * q^19 + (b7 + 3*b6 - b5 + 2*b4 - 2*b3) * q^20 + (-2*b7 - 3*b6 - 3*b5 - 3*b4 - 3*b2 - 2*b1 + 3) * q^22 + (-b2 + b1 + 3) * q^23 + (3*b2 + b1 + 1) * q^25 + (3*b6 + b5 + b4 - b3) * q^26 + (-b7 - 4*b6 - 3*b5 - b4 + b3 - b2 - b1 + 2) * q^28 + (-2*b7 - 3*b5) * q^29 + (2*b6 + 5*b5 + b4 - b3) * q^31 + (2*b6 - 2*b5 + b4 - b3) * q^32 + (-b7 + 5*b6 + b5 + b4 - b3) * q^34 + (b6 - 2*b5 - 2*b4 - b2 - 2*b1 - 1) * q^35 + (-3*b7 - b6 - b5 - 2*b4 + 2*b3) * q^37 + (-2*b3 + 5) * q^38 + (b3 - b2 - b1 - 4) * q^40 + (-2*b7 - 3*b6 - 3*b2 - 2*b1) * q^41 + (2*b7 - b6 + 2*b5 + b4 - b3) * q^43 + (-4*b7 - b6 + 2*b5 - b4 + b3) * q^44 + (-b7 + 5*b6 + 4*b5 + b4 + 5*b2 - b1 - 4) * q^46 + (-b6 + 5*b5 - b4 - b2 - 5) * q^47 + (3*b5 - b4 + 2*b3 + 3*b1 - 2) * q^49 + (3*b7 + 3*b6 - 8*b5 + b4 + 3*b2 + 3*b1 + 8) * q^50 + (-b3 + b2 - 2*b1 - 6) * q^52 + (-b7 - 3*b6 + 2*b5 - b4 - 3*b2 - b1 - 2) * q^53 + (2*b3 - 5*b2 - b1 - 5) * q^55 + (b7 + 4*b5 + b4 - b2 + 2*b1 + 2) * q^56 + (2*b3 + b2 - 2) * q^58 + (3*b7 - b6 + 2*b5 - b4 + b3) * q^59 + (-4*b6 - 6*b5 + b4 - 4*b2 + 6) * q^61 + (-b3 + 5*b2 - 3*b1 - 5) * q^62 + (-3*b3 - 3*b1 - 5) * q^64 + (b7 + 3*b6 - b5 + 2*b4 + 3*b2 + b1 + 1) * q^65 + (-3*b7 - 2*b6 - b4 + b3) * q^67 + (-2*b3 + 3*b2 - 2*b1 - 13) * q^68 + (-3*b7 - 5*b6 - b5 - 4*b4 + 2*b3 - 7*b2 - 2*b1 - 4) * q^70 + (-3*b2 + 3) * q^71 + (-b7 + 3*b6 - b5 - 3*b4 + 3*b2 - b1 + 1) * q^73 + (5*b3 + 5*b2 + 3*b1 - 2) * q^74 + (3*b6 + 2*b5 + 3*b2 - 2) * q^76 + (3*b7 - b6 + 8*b5 + 2*b4 + b3 + 3*b2 + 2*b1 - 3) * q^77 + (3*b7 + 3*b6 - b5 + 3*b4 + 3*b2 + 3*b1 + 1) * q^79 + (2*b7 + b5 + 4*b4 + 2*b1 - 1) * q^80 + (-3*b7 - 4*b6 + 7*b5 - 2*b4 + 2*b3) * q^82 + (-2*b7 + b6 + 2*b5 - 4*b4 + 4*b3) * q^83 + (b7 + 4*b6 - b5 + 5*b4 + 4*b2 + b1 + 1) * q^85 + (-3*b3 - 2*b2 + 6) * q^86 + (-b3 + 4*b2 - 2*b1 + 4) * q^88 + (-b7 - b6 - 6*b5 - 3*b4 + 3*b3) * q^89 + (-3*b6 - b5 - b4 - 4*b2 - b1 + 3) * q^91 + (4*b7 - 4*b6 - 9*b5) * q^92 + (-2*b7 - 5*b6 + 2*b5 - b4 + b3) * q^94 + (-4*b7 - 5*b5 - 5*b4 + 5*b3) * q^95 + (-b7 + 4*b6 + 3*b5 - 4*b4 + 4*b3) * q^97 + (b7 + 4*b6 + 4*b5 + 4*b4 + b3 + 7*b2 + 2*b1 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} - 5 q^{4} - 4 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10})$$ 8 * q + q^2 - 5 * q^4 - 4 * q^5 - 2 * q^7 + 6 * q^8 $$8 q + q^{2} - 5 q^{4} - 4 q^{5} - 2 q^{7} + 6 q^{8} + 7 q^{10} - 10 q^{11} + 5 q^{13} - 7 q^{14} + q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} + 7 q^{22} + 24 q^{23} + 16 q^{25} + q^{26} + 5 q^{28} - 10 q^{29} + 18 q^{31} - 10 q^{32} - 23 q^{35} + 40 q^{38} - 36 q^{40} - 5 q^{41} + 7 q^{43} + 13 q^{44} - 12 q^{46} - 21 q^{47} + 2 q^{49} + 38 q^{50} - 50 q^{52} - 12 q^{53} - 52 q^{55} + 33 q^{56} - 14 q^{58} + 6 q^{59} + 20 q^{61} - 36 q^{62} - 46 q^{64} + 8 q^{65} + 5 q^{67} - 102 q^{68} - 46 q^{70} + 18 q^{71} + 6 q^{73} - 5 q^{76} + 16 q^{77} + 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} + 9 q^{85} + 44 q^{86} + 36 q^{88} - 22 q^{89} + 13 q^{91} - 36 q^{92} + 15 q^{94} - 16 q^{95} + 9 q^{97} - 11 q^{98}+O(q^{100})$$ 8 * q + q^2 - 5 * q^4 - 4 * q^5 - 2 * q^7 + 6 * q^8 + 7 * q^10 - 10 * q^11 + 5 * q^13 - 7 * q^14 + q^16 + 6 * q^17 + 8 * q^19 - 8 * q^20 + 7 * q^22 + 24 * q^23 + 16 * q^25 + q^26 + 5 * q^28 - 10 * q^29 + 18 * q^31 - 10 * q^32 - 23 * q^35 + 40 * q^38 - 36 * q^40 - 5 * q^41 + 7 * q^43 + 13 * q^44 - 12 * q^46 - 21 * q^47 + 2 * q^49 + 38 * q^50 - 50 * q^52 - 12 * q^53 - 52 * q^55 + 33 * q^56 - 14 * q^58 + 6 * q^59 + 20 * q^61 - 36 * q^62 - 46 * q^64 + 8 * q^65 + 5 * q^67 - 102 * q^68 - 46 * q^70 + 18 * q^71 + 6 * q^73 - 5 * q^76 + 16 * q^77 + 10 * q^79 - 2 * q^80 + 35 * q^82 + 9 * q^83 + 9 * q^85 + 44 * q^86 + 36 * q^88 - 22 * q^89 + 13 * q^91 - 36 * q^92 + 15 * q^94 - 16 * q^95 + 9 * q^97 - 11 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102$$ (-14*v^7 + 688*v^6 - 619*v^5 - 193*v^4 - 6480*v^3 + 17846*v^2 - 10595*v + 23150) / 8102 $$\beta_{2}$$ $$=$$ $$( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102$$ (74*v^7 - 743*v^6 + 957*v^5 - 716*v^4 + 8788*v^3 - 23147*v^2 + 13756*v - 21668) / 8102 $$\beta_{3}$$ $$=$$ $$( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102$$ (-139*v^7 + 465*v^6 - 648*v^5 + 688*v^4 - 5887*v^3 + 15724*v^2 - 9416*v - 2218) / 8102 $$\beta_{4}$$ $$=$$ $$( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102$$ (-1269*v^7 + 2176*v^6 - 262*v^5 + 11731*v^4 - 43953*v^3 + 36565*v^2 - 52069*v + 14432) / 8102 $$\beta_{5}$$ $$=$$ $$( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051$$ (-962*v^7 + 1557*v^6 - 288*v^5 + 9308*v^4 - 33224*v^3 + 25443*v^2 - 49196*v + 18369) / 4051 $$\beta_{6}$$ $$=$$ $$( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102$$ (-4270*v^7 + 7290*v^6 - 2449*v^5 + 42410*v^4 - 149399*v^3 + 128118*v^2 - 241837*v + 88979) / 8102 $$\beta_{7}$$ $$=$$ $$( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102$$ (4805*v^7 - 8118*v^6 + 2087*v^5 - 47477*v^4 + 167278*v^3 - 139939*v^2 + 265634*v - 98045) / 8102
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3$$ (b7 + b6 - b5 + 2*b4 - b3 - b2 - b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3$$ (-5*b7 - 5*b6 - b5 - b4 + 2*b3 - 4*b2 - 4*b1 + 2) / 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6$$ -b7 + b6 - 6*b5 + 2*b4 + b3 + 3*b2 + b1 + 6 $$\nu^{4}$$ $$=$$ $$( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3$$ (-7*b7 + 5*b6 - 38*b5 + 16*b4 - 23*b3 - 17*b2 - 17*b1 + 1) / 3 $$\nu^{5}$$ $$=$$ $$( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3$$ (-70*b7 - 70*b6 - 11*b5 - 14*b4 + 4*b3 - 14*b2 - 8*b1 - 17) / 3 $$\nu^{6}$$ $$=$$ $$17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7$$ 17*b7 + 37*b6 - 60*b5 + 35*b4 - 16*b3 + 50*b2 + 46*b1 + 7 $$\nu^{7}$$ $$=$$ $$( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3$$ (-44*b7 - 38*b6 - 22*b5 - 7*b4 - 301*b3 - 283*b2 - 97*b1 - 565) / 3

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-\beta_{5}$$ $$-1 + \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.373419 − 0.0835272i 2.11692 − 0.978886i 0.0512865 + 1.21608i −1.54162 − 1.88572i 0.373419 + 0.0835272i 2.11692 + 0.978886i 0.0512865 − 1.21608i −1.54162 + 1.88572i
−1.18584 + 2.05393i 0 −1.81242 3.13920i −1.22875 0 2.61263 + 0.417345i 3.85358 0 1.45709 2.52376i
109.2 −0.186423 + 0.322894i 0 0.930493 + 1.61166i −1.42143 0 −1.03335 2.43561i −1.43955 0 0.264988 0.458972i
109.3 0.768262 1.33067i 0 −0.180452 0.312552i −3.15761 0 −0.00900690 + 2.64574i 2.51851 0 −2.42587 + 4.20173i
109.4 1.10400 1.91218i 0 −1.43762 2.49004i 3.80779 0 −2.57027 0.627473i −1.93254 0 4.20379 7.28117i
541.1 −1.18584 2.05393i 0 −1.81242 + 3.13920i −1.22875 0 2.61263 0.417345i 3.85358 0 1.45709 + 2.52376i
541.2 −0.186423 0.322894i 0 0.930493 1.61166i −1.42143 0 −1.03335 + 2.43561i −1.43955 0 0.264988 + 0.458972i
541.3 0.768262 + 1.33067i 0 −0.180452 + 0.312552i −3.15761 0 −0.00900690 2.64574i 2.51851 0 −2.42587 4.20173i
541.4 1.10400 + 1.91218i 0 −1.43762 + 2.49004i 3.80779 0 −2.57027 + 0.627473i −1.93254 0 4.20379 + 7.28117i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.g.k 8
3.b odd 2 1 567.2.g.j 8
7.c even 3 1 567.2.h.j 8
9.c even 3 1 567.2.e.d yes 8
9.c even 3 1 567.2.h.j 8
9.d odd 6 1 567.2.e.c 8
9.d odd 6 1 567.2.h.k 8
21.h odd 6 1 567.2.h.k 8
63.g even 3 1 inner 567.2.g.k 8
63.g even 3 1 3969.2.a.s 4
63.h even 3 1 567.2.e.d yes 8
63.j odd 6 1 567.2.e.c 8
63.k odd 6 1 3969.2.a.t 4
63.n odd 6 1 567.2.g.j 8
63.n odd 6 1 3969.2.a.x 4
63.s even 6 1 3969.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c 8 9.d odd 6 1
567.2.e.c 8 63.j odd 6 1
567.2.e.d yes 8 9.c even 3 1
567.2.e.d yes 8 63.h even 3 1
567.2.g.j 8 3.b odd 2 1
567.2.g.j 8 63.n odd 6 1
567.2.g.k 8 1.a even 1 1 trivial
567.2.g.k 8 63.g even 3 1 inner
567.2.h.j 8 7.c even 3 1
567.2.h.j 8 9.c even 3 1
567.2.h.k 8 9.d odd 6 1
567.2.h.k 8 21.h odd 6 1
3969.2.a.s 4 63.g even 3 1
3969.2.a.t 4 63.k odd 6 1
3969.2.a.w 4 63.s even 6 1
3969.2.a.x 4 63.n odd 6 1

## Hecke kernels

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

 $$T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} - 6T_{2}^{5} + 39T_{2}^{4} - 30T_{2}^{3} + 54T_{2}^{2} + 18T_{2} + 9$$ T2^8 - T2^7 + 7*T2^6 - 6*T2^5 + 39*T2^4 - 30*T2^3 + 54*T2^2 + 18*T2 + 9 $$T_{13}^{8} - 5T_{13}^{7} + 25T_{13}^{6} - 40T_{13}^{5} + 107T_{13}^{4} - 70T_{13}^{3} + 400T_{13}^{2} - 140T_{13} + 49$$ T13^8 - 5*T13^7 + 25*T13^6 - 40*T13^5 + 107*T13^4 - 70*T13^3 + 400*T13^2 - 140*T13 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + 7 T^{6} - 6 T^{5} + 39 T^{4} + \cdots + 9$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 2 T^{3} - 12 T^{2} - 33 T - 21)^{2}$$
$7$ $$T^{8} + 2 T^{7} + T^{6} - 14 T^{5} + \cdots + 2401$$
$11$ $$(T^{4} + 5 T^{3} - 24 T^{2} - 174 T - 249)^{2}$$
$13$ $$T^{8} - 5 T^{7} + 25 T^{6} - 40 T^{5} + \cdots + 49$$
$17$ $$T^{8} - 6 T^{7} + 81 T^{6} + \cdots + 321489$$
$19$ $$T^{8} - 8 T^{7} + 64 T^{6} + \cdots + 2401$$
$23$ $$(T^{4} - 12 T^{3} + 36 T^{2} - 33 T + 9)^{2}$$
$29$ $$T^{8} + 10 T^{7} + 100 T^{6} + \cdots + 3969$$
$31$ $$T^{8} - 18 T^{7} + 231 T^{6} + \cdots + 441$$
$37$ $$T^{8} + 78 T^{6} + 74 T^{5} + \cdots + 904401$$
$41$ $$T^{8} + 5 T^{7} + 97 T^{6} + \cdots + 194481$$
$43$ $$T^{8} - 7 T^{7} + 79 T^{6} + \cdots + 2401$$
$47$ $$T^{8} + 21 T^{7} + 288 T^{6} + \cdots + 194481$$
$53$ $$T^{8} + 12 T^{7} + 144 T^{6} + \cdots + 6561$$
$59$ $$T^{8} - 6 T^{7} + 144 T^{6} + \cdots + 35721$$
$61$ $$T^{8} - 20 T^{7} + 373 T^{6} + \cdots + 1087849$$
$67$ $$T^{8} - 5 T^{7} + 97 T^{6} + \cdots + 124609$$
$71$ $$(T^{4} - 9 T^{3} - 27 T^{2} + 135 T + 243)^{2}$$
$73$ $$T^{8} - 6 T^{7} + 186 T^{6} + \cdots + 5239521$$
$79$ $$T^{8} - 10 T^{7} + 184 T^{6} + \cdots + 49$$
$83$ $$T^{8} - 9 T^{7} + 207 T^{6} + \cdots + 26040609$$
$89$ $$T^{8} + 22 T^{7} + 370 T^{6} + \cdots + 441$$
$97$ $$T^{8} - 9 T^{7} + 339 T^{6} + \cdots + 8277129$$