# Properties

 Label 504.2.cj.c Level 504 Weight 2 Character orbit 504.cj Analytic conductor 4.024 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.cj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.951588245534976.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} -\beta_{3} q^{4} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} -\beta_{3} q^{4} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{10} + ( -\beta_{3} + \beta_{6} + \beta_{10} ) q^{11} + ( \beta_{2} + \beta_{9} + \beta_{10} ) q^{13} + ( 2 - \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{14} + ( 2 - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} ) q^{16} + ( -\beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - \beta_{10} ) q^{17} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{9} + 2 \beta_{10} ) q^{20} + ( \beta_{1} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{23} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{10} ) q^{25} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{9} + \beta_{11} ) q^{26} + ( 2 - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{28} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{29} + ( \beta_{4} + \beta_{5} - \beta_{8} ) q^{31} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} ) q^{32} + ( 3 - \beta_{2} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{35} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{37} + ( \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{11} ) q^{40} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{41} + ( 4 \beta_{1} + 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{11} ) q^{43} + ( 5 + \beta_{1} + \beta_{2} - 5 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{11} ) q^{44} + ( -\beta_{4} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{46} + ( -3 - 3 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{47} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} ) q^{49} + ( 2 - 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{50} + ( -6 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} ) q^{52} + ( -\beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{10} ) q^{53} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{55} + ( 5 + \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{56} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{9} + \beta_{11} ) q^{58} + ( 4 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} ) q^{59} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{9} - 3 \beta_{11} ) q^{61} + ( 2 + \beta_{1} - \beta_{9} ) q^{62} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{64} + ( -2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{9} ) q^{65} + ( 2 \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 4 \beta_{7} - \beta_{9} ) q^{68} + ( -3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + 3 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{70} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{71} + ( -4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 4 \beta_{10} ) q^{73} + ( 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{74} + ( 7 - 5 \beta_{1} + 3 \beta_{2} + 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{76} + ( -2 \beta_{1} - \beta_{2} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - 6 \beta_{11} ) q^{77} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{79} + ( 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{8} + 2 \beta_{10} ) q^{80} + ( -5 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{9} + \beta_{11} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{83} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{85} + ( -6 + 2 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{11} ) q^{86} + ( -\beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - \beta_{8} - 3 \beta_{10} ) q^{88} + ( -1 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{7} + 4 \beta_{9} ) q^{89} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{7} - 3 \beta_{8} - \beta_{10} - \beta_{11} ) q^{92} + ( 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{94} + ( -\beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} ) q^{95} + ( 6 - 4 \beta_{1} + \beta_{2} + 4 \beta_{7} + 4 \beta_{8} - \beta_{9} + \beta_{10} ) q^{97} + ( -7 + \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{2} - 4q^{7} - 4q^{8} + O(q^{10})$$ $$12q + 2q^{2} - 4q^{7} - 4q^{8} - 8q^{10} + 16q^{14} + 8q^{16} + 2q^{17} - 8q^{20} + 12q^{22} - 2q^{23} - 4q^{25} + 2q^{26} + 26q^{28} + 10q^{31} + 12q^{32} + 32q^{34} - 18q^{38} + 10q^{40} + 8q^{41} + 30q^{44} - 4q^{46} - 30q^{47} - 12q^{49} + 16q^{50} - 32q^{52} + 4q^{55} + 40q^{56} - 22q^{58} + 28q^{62} + 24q^{64} - 8q^{65} - 4q^{68} - 48q^{70} - 32q^{71} - 10q^{73} - 18q^{74} + 52q^{76} - 22q^{79} - 36q^{80} - 26q^{82} - 40q^{86} - 14q^{88} + 10q^{89} + 20q^{92} + 42q^{94} + 34q^{95} + 40q^{97} - 52q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + 32 x^{2} - 32 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{11} + 3 \nu^{10} + 8 \nu^{9} - 15 \nu^{8} - 6 \nu^{7} - 31 \nu^{6} + 51 \nu^{5} + 12 \nu^{4} + 84 \nu^{3} - 96 \nu^{2} - 48 \nu - 128$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{11} + \nu^{10} + 7 \nu^{9} - 10 \nu^{8} + \nu^{7} - 32 \nu^{6} + 39 \nu^{5} - \nu^{4} + 76 \nu^{3} - 60 \nu^{2} - 32 \nu - 112$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{11} - \nu^{10} - 3 \nu^{9} + 10 \nu^{8} + 3 \nu^{7} + 12 \nu^{6} - 27 \nu^{5} - 11 \nu^{4} - 20 \nu^{3} + 40 \nu^{2} + 40 \nu + 32$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{11} - \nu^{10} - 9 \nu^{9} + 20 \nu^{8} - 3 \nu^{7} + 42 \nu^{6} - 63 \nu^{5} + 7 \nu^{4} - 100 \nu^{3} + 104 \nu^{2} + 32 \nu + 160$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{11} - 3 \nu^{10} - 18 \nu^{9} + 45 \nu^{8} + 4 \nu^{7} + 81 \nu^{6} - 135 \nu^{5} - 30 \nu^{4} - 180 \nu^{3} + 216 \nu^{2} + 144 \nu + 288$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} - 18 \nu^{4} - 260 \nu^{3} + 248 \nu^{2} + 96 \nu + 416$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{11} - \nu^{10} - 20 \nu^{9} + 49 \nu^{8} - 14 \nu^{7} + 97 \nu^{6} - 161 \nu^{5} + 40 \nu^{4} - 232 \nu^{3} + 248 \nu^{2} + 32 \nu + 320$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$11 \nu^{11} + 7 \nu^{10} + 28 \nu^{9} - 55 \nu^{8} - 10 \nu^{7} - 135 \nu^{6} + 183 \nu^{5} + 16 \nu^{4} + 340 \nu^{3} - 312 \nu^{2} - 128 \nu - 544$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$-5 \nu^{11} - \nu^{10} - 16 \nu^{9} + 29 \nu^{8} - 10 \nu^{7} + 77 \nu^{6} - 97 \nu^{5} + 28 \nu^{4} - 200 \nu^{3} + 160 \nu^{2} + 16 \nu + 288$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$-13 \nu^{11} - 11 \nu^{10} - 38 \nu^{9} + 65 \nu^{8} + 16 \nu^{7} + 189 \nu^{6} - 231 \nu^{5} - 34 \nu^{4} - 500 \nu^{3} + 408 \nu^{2} + 208 \nu + 800$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} - 5 \nu^{7} + 68 \nu^{6} - 94 \nu^{5} + 16 \nu^{4} - 165 \nu^{3} + 160 \nu^{2} + 32 \nu + 240$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{2} + \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{10} - \beta_{8} - 3 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{11} + \beta_{9} - 5 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_{1} - 5$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{11} - \beta_{10} - 3 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - \beta_{2} + 9 \beta_{1} - 3$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{10} - 15 \beta_{8} - 9 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 3 \beta_{3}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$\beta_{11} + 9 \beta_{9} - 5 \beta_{7} - \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 3 \beta_{1} + 7$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$3 \beta_{11} + 13 \beta_{10} - 9 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + 13 \beta_{2} + 23 \beta_{1} - 7$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-11 \beta_{10} - 37 \beta_{8} - 27 \beta_{6} + 19 \beta_{5} - 13 \beta_{4} - 11 \beta_{3}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$15 \beta_{11} + 17 \beta_{9} + \beta_{7} - 15 \beta_{6} + 9 \beta_{5} - 77 \beta_{4} + 17 \beta_{3} + 9 \beta_{2} - 9 \beta_{1} + 77$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.26950 + 0.623187i −0.981777 − 1.01790i 1.41417 − 0.0105323i −0.0950561 + 1.41102i −0.716208 + 1.21944i −0.390636 − 1.35919i 1.26950 − 0.623187i −0.981777 + 1.01790i 1.41417 + 0.0105323i −0.0950561 − 1.41102i −0.716208 − 1.21944i −0.390636 + 1.35919i
−1.38687 + 0.276739i 0 1.84683 0.767603i 0.476087 + 0.274869i 0 −2.60755 0.447998i −2.34889 + 1.57566i 0 −0.736339 0.249456i
37.2 −0.804171 1.16332i 0 −0.706619 + 1.87101i 2.80486 + 1.61939i 0 1.47779 2.19457i 2.74483 0.682591i 0 −0.371724 4.56521i
37.3 0.242117 1.39333i 0 −1.88276 0.674701i −1.28690 0.742990i 0 0.129755 + 2.64257i −1.39593 + 2.45995i 0 −1.34681 + 1.61319i
37.4 0.453773 + 1.33944i 0 −1.58818 + 1.21560i −0.476087 0.274869i 0 −2.60755 0.447998i −2.34889 1.57566i 0 0.152134 0.762416i
37.5 1.08560 0.906347i 0 0.357071 1.96787i 1.28690 + 0.742990i 0 0.129755 + 2.64257i −1.39593 2.45995i 0 2.07047 0.359782i
37.6 1.40955 + 0.114773i 0 1.97365 + 0.323556i −2.80486 1.61939i 0 1.47779 2.19457i 2.74483 + 0.682591i 0 −3.76772 2.60453i
109.1 −1.38687 0.276739i 0 1.84683 + 0.767603i 0.476087 0.274869i 0 −2.60755 + 0.447998i −2.34889 1.57566i 0 −0.736339 + 0.249456i
109.2 −0.804171 + 1.16332i 0 −0.706619 1.87101i 2.80486 1.61939i 0 1.47779 + 2.19457i 2.74483 + 0.682591i 0 −0.371724 + 4.56521i
109.3 0.242117 + 1.39333i 0 −1.88276 + 0.674701i −1.28690 + 0.742990i 0 0.129755 2.64257i −1.39593 2.45995i 0 −1.34681 1.61319i
109.4 0.453773 1.33944i 0 −1.58818 1.21560i −0.476087 + 0.274869i 0 −2.60755 + 0.447998i −2.34889 + 1.57566i 0 0.152134 + 0.762416i
109.5 1.08560 + 0.906347i 0 0.357071 + 1.96787i 1.28690 0.742990i 0 0.129755 2.64257i −1.39593 + 2.45995i 0 2.07047 + 0.359782i
109.6 1.40955 0.114773i 0 1.97365 0.323556i −2.80486 + 1.61939i 0 1.47779 + 2.19457i 2.74483 0.682591i 0 −3.76772 + 2.60453i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.c 12
3.b odd 2 1 56.2.p.a 12
4.b odd 2 1 2016.2.cr.c 12
7.c even 3 1 inner 504.2.cj.c 12
8.b even 2 1 inner 504.2.cj.c 12
8.d odd 2 1 2016.2.cr.c 12
12.b even 2 1 224.2.t.a 12
21.c even 2 1 392.2.p.g 12
21.g even 6 1 392.2.b.f 6
21.g even 6 1 392.2.p.g 12
21.h odd 6 1 56.2.p.a 12
21.h odd 6 1 392.2.b.e 6
24.f even 2 1 224.2.t.a 12
24.h odd 2 1 56.2.p.a 12
28.g odd 6 1 2016.2.cr.c 12
56.k odd 6 1 2016.2.cr.c 12
56.p even 6 1 inner 504.2.cj.c 12
84.h odd 2 1 1568.2.t.g 12
84.j odd 6 1 1568.2.b.e 6
84.j odd 6 1 1568.2.t.g 12
84.n even 6 1 224.2.t.a 12
84.n even 6 1 1568.2.b.f 6
168.e odd 2 1 1568.2.t.g 12
168.i even 2 1 392.2.p.g 12
168.s odd 6 1 56.2.p.a 12
168.s odd 6 1 392.2.b.e 6
168.v even 6 1 224.2.t.a 12
168.v even 6 1 1568.2.b.f 6
168.ba even 6 1 392.2.b.f 6
168.ba even 6 1 392.2.p.g 12
168.be odd 6 1 1568.2.b.e 6
168.be odd 6 1 1568.2.t.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a 12 3.b odd 2 1
56.2.p.a 12 21.h odd 6 1
56.2.p.a 12 24.h odd 2 1
56.2.p.a 12 168.s odd 6 1
224.2.t.a 12 12.b even 2 1
224.2.t.a 12 24.f even 2 1
224.2.t.a 12 84.n even 6 1
224.2.t.a 12 168.v even 6 1
392.2.b.e 6 21.h odd 6 1
392.2.b.e 6 168.s odd 6 1
392.2.b.f 6 21.g even 6 1
392.2.b.f 6 168.ba even 6 1
392.2.p.g 12 21.c even 2 1
392.2.p.g 12 21.g even 6 1
392.2.p.g 12 168.i even 2 1
392.2.p.g 12 168.ba even 6 1
504.2.cj.c 12 1.a even 1 1 trivial
504.2.cj.c 12 7.c even 3 1 inner
504.2.cj.c 12 8.b even 2 1 inner
504.2.cj.c 12 56.p even 6 1 inner
1568.2.b.e 6 84.j odd 6 1
1568.2.b.e 6 168.be odd 6 1
1568.2.b.f 6 84.n even 6 1
1568.2.b.f 6 168.v even 6 1
1568.2.t.g 12 84.h odd 2 1
1568.2.t.g 12 84.j odd 6 1
1568.2.t.g 12 168.e odd 2 1
1568.2.t.g 12 168.be odd 6 1
2016.2.cr.c 12 4.b odd 2 1
2016.2.cr.c 12 8.d odd 2 1
2016.2.cr.c 12 28.g odd 6 1
2016.2.cr.c 12 56.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 13 T_{5}^{10} + 142 T_{5}^{8} - 337 T_{5}^{6} + 638 T_{5}^{4} - 189 T_{5}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{4} + 4 T^{5} - 4 T^{6} + 8 T^{7} - 16 T^{8} + 32 T^{10} - 64 T^{11} + 64 T^{12}$$
$3$ 
$5$ $$1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24}$$
$7$ $$( 1 + 2 T + 5 T^{2} + 32 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2}$$
$11$ $$1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24}$$
$13$ $$( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - T - 33 T^{2} - 8 T^{3} + 565 T^{4} + 425 T^{5} - 9946 T^{6} + 7225 T^{7} + 163285 T^{8} - 39304 T^{9} - 2756193 T^{10} - 1419857 T^{11} + 24137569 T^{12} )^{2}$$
$19$ $$1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24}$$
$23$ $$( 1 + T - 61 T^{2} - 12 T^{3} + 2381 T^{4} - 29 T^{5} - 63238 T^{6} - 667 T^{7} + 1259549 T^{8} - 146004 T^{9} - 17070301 T^{10} + 6436343 T^{11} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 5 T - 71 T^{2} + 126 T^{3} + 4725 T^{4} - 4133 T^{5} - 158330 T^{6} - 128123 T^{7} + 4540725 T^{8} + 3753666 T^{9} - 65569991 T^{10} - 143145755 T^{11} + 887503681 T^{12} )^{2}$$
$37$ $$1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24}$$
$41$ $$( 1 - 2 T + 83 T^{2} - 80 T^{3} + 3403 T^{4} - 3362 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 15 T + 57 T^{2} + 78 T^{3} + 2013 T^{4} + 1383 T^{5} - 125066 T^{6} + 65001 T^{7} + 4446717 T^{8} + 8098194 T^{9} + 278141817 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} )^{2}$$
$53$ $$1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} +$$$$65\!\cdots\!89$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$( 1 + 8 T + 157 T^{2} + 704 T^{3} + 11147 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 + 11 T - 137 T^{2} - 656 T^{3} + 26965 T^{4} + 70973 T^{5} - 1975630 T^{6} + 5606867 T^{7} + 168288565 T^{8} - 323433584 T^{9} - 5336161097 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 5 T - 133 T^{2} + 1452 T^{3} + 5297 T^{4} - 76103 T^{5} + 246758 T^{6} - 6773167 T^{7} + 41957537 T^{8} + 1023614988 T^{9} - 8344718053 T^{10} - 27920297245 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4}$$