# Properties

 Label 273.2.p.f Level $273$ Weight $2$ Character orbit 273.p Analytic conductor $2.180$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 60 x^{8} - 8 x^{7} + 80 x^{5} + 320 x^{4} + 160 x^{3} + 32 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} + \beta_{4} q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{11} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{5} + \beta_{2} q^{6} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{7} + ( -\beta_{1} + 3 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{8} q^{2} + \beta_{4} q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{11} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{5} + \beta_{2} q^{6} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{7} + ( -\beta_{1} + 3 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{8} - q^{9} + ( -\beta_{3} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{10} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{11} + ( -3 + \beta_{7} - \beta_{9} ) q^{12} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{13} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{14} + ( 1 + \beta_{4} + \beta_{5} ) q^{15} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{16} + ( -2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{17} -\beta_{8} q^{18} + ( -2 \beta_{2} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{19} + ( 1 + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{10} ) q^{20} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} ) q^{21} + ( 2 - 2 \beta_{3} + \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{22} + ( -\beta_{2} - 2 \beta_{4} - \beta_{8} + 2 \beta_{11} ) q^{23} + ( -\beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{11} ) q^{24} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{25} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{26} -\beta_{4} q^{27} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{28} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{29} + ( \beta_{3} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{30} + ( 2 + 2 \beta_{1} - 2 \beta_{4} ) q^{31} + ( 4 + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} ) q^{32} + ( \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 1 + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{34} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{35} + ( -3 \beta_{4} + \beta_{6} + \beta_{11} ) q^{36} + ( 1 + 2 \beta_{1} + 4 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 3 \beta_{11} ) q^{37} + ( 6 + \beta_{1} + 2 \beta_{3} + \beta_{5} - 4 \beta_{7} + 4 \beta_{9} + 2 \beta_{10} ) q^{38} + ( -1 + \beta_{1} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{39} + ( -4 \beta_{1} + \beta_{2} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{8} - 3 \beta_{11} ) q^{40} + ( -3 - \beta_{1} + 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{10} ) q^{43} + ( 2 + 2 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{11} ) q^{44} + ( -1 - \beta_{1} + \beta_{4} ) q^{45} + ( 3 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 3 \beta_{11} ) q^{46} + ( 3 + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{47} + ( \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{48} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{11} ) q^{49} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{50} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{10} + 2 \beta_{11} ) q^{51} + ( -6 + 3 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{52} + ( -2 - 2 \beta_{9} ) q^{53} -\beta_{2} q^{54} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{10} ) q^{55} + ( -3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 5 \beta_{7} + 4 \beta_{8} + \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{56} + ( \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{57} + ( 2 + 2 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} - 4 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{58} + ( 1 + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{60} + ( 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{8} - 3 \beta_{10} + 4 \beta_{11} ) q^{61} + ( -2 \beta_{3} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{62} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{63} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + 7 \beta_{6} + 5 \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{64} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{65} + ( 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{10} + \beta_{11} ) q^{66} + ( -2 - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 8 \beta_{4} - 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{68} + ( 2 - \beta_{2} + \beta_{8} + 2 \beta_{9} ) q^{69} + ( -2 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{10} - 4 \beta_{11} ) q^{70} + ( 5 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{71} + ( \beta_{1} - 3 \beta_{2} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{72} + ( -3 - 3 \beta_{4} + 4 \beta_{5} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{73} + ( -8 - 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{5} + 8 \beta_{7} + 3 \beta_{8} ) q^{74} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{75} + ( -2 - 2 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 8 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{76} + ( 2 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{77} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{78} + ( 4 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} - 2 \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{10} ) q^{79} + ( -1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{80} + q^{81} + ( -2 \beta_{1} + 6 \beta_{2} + \beta_{3} - 2 \beta_{5} - 3 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{82} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{83} + ( 4 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{84} + ( -4 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{11} ) q^{85} + ( 2 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} + 4 \beta_{11} ) q^{86} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{87} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{88} + ( -1 - \beta_{4} - 5 \beta_{5} - 4 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} ) q^{89} + ( \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{90} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} + 5 \beta_{11} ) q^{91} + ( 10 - \beta_{1} - 3 \beta_{2} - \beta_{5} + 3 \beta_{8} + 2 \beta_{9} ) q^{92} + ( 2 + 2 \beta_{4} + 2 \beta_{5} ) q^{93} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 14 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 3 \beta_{8} + 2 \beta_{10} - 7 \beta_{11} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{11} ) q^{95} + ( -4 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{9} - 3 \beta_{11} ) q^{96} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{97} + ( 4 + 2 \beta_{1} + \beta_{2} - 12 \beta_{4} + 4 \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{9} + \beta_{11} ) q^{98} + ( -\beta_{1} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{5} - 4q^{7} - 12q^{9} + O(q^{10})$$ $$12q + 12q^{5} - 4q^{7} - 12q^{9} - 4q^{11} - 28q^{12} + 12q^{15} - 36q^{16} - 8q^{17} + 8q^{20} + 12q^{21} + 32q^{22} + 4q^{26} + 12q^{28} - 8q^{29} + 24q^{31} + 20q^{32} - 4q^{33} - 20q^{35} - 4q^{37} + 40q^{38} - 16q^{39} - 20q^{41} + 8q^{44} - 12q^{45} + 20q^{46} + 32q^{47} + 20q^{50} - 56q^{52} - 16q^{53} - 20q^{56} + 8q^{59} - 8q^{60} + 4q^{63} - 16q^{65} - 32q^{67} + 16q^{69} - 20q^{70} - 12q^{71} - 32q^{73} - 64q^{74} + 4q^{75} - 12q^{77} + 16q^{78} + 24q^{79} - 4q^{80} + 12q^{81} + 28q^{84} - 32q^{85} + 4q^{89} + 32q^{91} + 112q^{92} + 24q^{93} - 20q^{96} + 32q^{98} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 60 x^{8} - 8 x^{7} + 80 x^{5} + 320 x^{4} + 160 x^{3} + 32 x^{2} - 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-12309 \nu^{11} + 5381 \nu^{10} - 140579 \nu^{9} + 28834 \nu^{8} - 745616 \nu^{7} + 479716 \nu^{6} - 11267844 \nu^{5} + 1884316 \nu^{4} - 4189816 \nu^{3} - 6794816 \nu^{2} - 156757872 \nu - 956416$$$$)/54445816$$ $$\beta_{3}$$ $$=$$ $$($$$$170557 \nu^{11} - 60738 \nu^{10} + 1270592 \nu^{9} - 400085 \nu^{8} + 10325938 \nu^{7} - 5541268 \nu^{6} + 74667404 \nu^{5} - 23444608 \nu^{4} + 58720980 \nu^{3} + 93750596 \nu^{2} + 262576728 \nu + 13274488$$$$)/54445816$$ $$\beta_{4}$$ $$=$$ $$($$$$-556438 \nu^{11} - 583343 \nu^{10} + 119552 \nu^{9} - 24618 \nu^{8} - 33375518 \nu^{7} - 30830234 \nu^{6} + 11897532 \nu^{5} - 46962688 \nu^{4} - 223768168 \nu^{3} - 288671368 \nu^{2} - 69115624 \nu + 9887728$$$$)/54445816$$ $$\beta_{5}$$ $$=$$ $$($$$$-583343 \nu^{11} + 119552 \nu^{10} - 24618 \nu^{9} + 10762 \nu^{8} - 35281738 \nu^{7} + 11897532 \nu^{6} - 2447648 \nu^{5} - 45708008 \nu^{4} - 199641288 \nu^{3} - 51309608 \nu^{2} - 7918288 \nu + 8903008$$$$)/54445816$$ $$\beta_{6}$$ $$=$$ $$($$$$2412741 \nu^{11} - 3033606 \nu^{10} + 2016198 \nu^{9} - 400372 \nu^{8} + 144220708 \nu^{7} - 197893628 \nu^{6} + 146207356 \nu^{5} + 152924728 \nu^{4} + 487489968 \nu^{3} - 256163344 \nu^{2} + 159724320 \nu - 22588976$$$$)/ 108891632$$ $$\beta_{7}$$ $$=$$ $$($$$$3446659 \nu^{11} - 676828 \nu^{10} - 1302018 \nu^{9} + 2509892 \nu^{8} + 205726644 \nu^{7} - 67932876 \nu^{6} - 74231220 \nu^{5} + 427286336 \nu^{4} + 952913392 \nu^{3} + 255016704 \nu^{2} - 227943584 \nu + 62241040$$$$)/ 108891632$$ $$\beta_{8}$$ $$=$$ $$($$$$-1750029 \nu^{11} + 358656 \nu^{10} - 73854 \nu^{9} + 32286 \nu^{8} - 105845214 \nu^{7} + 35692596 \nu^{6} - 7342944 \nu^{5} - 137124024 \nu^{4} - 571700956 \nu^{3} - 153928824 \nu^{2} - 23754864 \nu + 26709024$$$$)/54445816$$ $$\beta_{9}$$ $$=$$ $$($$$$5569761 \nu^{11} - 1112876 \nu^{10} - 1166686 \nu^{9} + 239104 \nu^{8} + 334136424 \nu^{7} - 111309124 \nu^{6} - 61660468 \nu^{5} + 469375944 \nu^{4} + 1688398144 \nu^{3} + 443625424 \nu^{2} - 399110384 \nu - 316463600$$$$)/ 108891632$$ $$\beta_{10}$$ $$=$$ $$($$$$-3299703 \nu^{11} + 676169 \nu^{10} - 135251 \nu^{9} - 134391 \nu^{8} - 196850238 \nu^{7} + 67292332 \nu^{6} - 13524756 \nu^{5} - 274186516 \nu^{4} - 937941424 \nu^{3} - 328183716 \nu^{2} - 52760848 \nu + 58170888$$$$)/54445816$$ $$\beta_{11}$$ $$=$$ $$($$$$-8041693 \nu^{11} - 1112876 \nu^{10} - 1166686 \nu^{9} + 239104 \nu^{8} - 482550816 \nu^{7} - 2417492 \nu^{6} - 61660468 \nu^{5} - 619540376 \nu^{4} - 2667267136 \nu^{3} - 1734207216 \nu^{2} - 834676912 \nu + 119102928$$$$)/ 108891632$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{11} - \beta_{10} - \beta_{8} - \beta_{5} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{8} - 6 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 2 \beta_{7} - 6 \beta_{5} - 8 \beta_{3} + 8 \beta_{2} - 6 \beta_{1} - 24$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{9} - 4 \beta_{4} - 20 \beta_{2} - 40 \beta_{1} + 4$$ $$\nu^{6}$$ $$=$$ $$-100 \beta_{11} + 60 \beta_{10} + 60 \beta_{8} + 20 \beta_{6} + 36 \beta_{5} + 164 \beta_{4} - 60 \beta_{3} + 60 \beta_{2} - 36 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$28 \beta_{11} - 8 \beta_{10} - 28 \beta_{9} - 168 \beta_{8} + 276 \beta_{5} - 56 \beta_{4} - 56$$ $$\nu^{8}$$ $$=$$ $$444 \beta_{10} + 728 \beta_{9} + 452 \beta_{8} + 160 \beta_{7} + 220 \beta_{5} + 444 \beta_{3} - 452 \beta_{2} + 220 \beta_{1} + 1176$$ $$\nu^{9}$$ $$=$$ $$-296 \beta_{11} - 296 \beta_{9} - 8 \beta_{7} + 8 \beta_{6} + 576 \beta_{4} - 144 \beta_{3} + 1352 \beta_{2} + 1936 \beta_{1} - 576$$ $$\nu^{10}$$ $$=$$ $$5352 \beta_{11} - 3272 \beta_{10} - 3432 \beta_{8} - 1208 \beta_{6} - 1344 \beta_{5} - 8608 \beta_{4} + 3272 \beta_{3} - 3432 \beta_{2} + 1344 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-2808 \beta_{11} + 1760 \beta_{10} + 2808 \beta_{9} + 10720 \beta_{8} + 160 \beta_{7} + 160 \beta_{6} - 13712 \beta_{5} + 5312 \beta_{4} + 5312$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −0.863233 + 0.863233i −0.528642 + 0.528642i 1.85068 − 1.85068i 0.236276 − 0.236276i −1.96818 + 1.96818i 1.27310 − 1.27310i −0.863233 − 0.863233i −0.528642 − 0.528642i 1.85068 + 1.85068i 0.236276 + 0.236276i −1.96818 − 1.96818i 1.27310 + 1.27310i
−1.94644 1.94644i 1.00000i 5.57728i 0.136767 0.136767i 1.94644 1.94644i −2.24723 1.39641i 6.96297 6.96297i −1.00000 −0.532416
34.2 −1.43819 1.43819i 1.00000i 2.13679i 0.471358 0.471358i 1.43819 1.43819i 2.56584 + 0.645342i 0.196726 0.196726i −1.00000 −1.35580
34.3 −0.786556 0.786556i 1.00000i 0.762660i 2.85068 2.85068i 0.786556 0.786556i −1.83993 1.90123i −2.17299 + 2.17299i −1.00000 −4.48443
34.4 0.695639 + 0.695639i 1.00000i 1.03217i 1.23628 1.23628i −0.695639 + 0.695639i 1.20338 2.35624i 2.10930 2.10930i −1.00000 1.72000
34.5 1.71968 + 1.71968i 1.00000i 3.91457i −0.968182 + 0.968182i −1.71968 + 1.71968i 0.407631 2.61416i −3.29244 + 3.29244i −1.00000 −3.32992
34.6 1.75588 + 1.75588i 1.00000i 4.16620i 2.27310 2.27310i −1.75588 + 1.75588i −2.08970 + 1.62270i −3.80357 + 3.80357i −1.00000 7.98257
265.1 −1.94644 + 1.94644i 1.00000i 5.57728i 0.136767 + 0.136767i 1.94644 + 1.94644i −2.24723 + 1.39641i 6.96297 + 6.96297i −1.00000 −0.532416
265.2 −1.43819 + 1.43819i 1.00000i 2.13679i 0.471358 + 0.471358i 1.43819 + 1.43819i 2.56584 0.645342i 0.196726 + 0.196726i −1.00000 −1.35580
265.3 −0.786556 + 0.786556i 1.00000i 0.762660i 2.85068 + 2.85068i 0.786556 + 0.786556i −1.83993 + 1.90123i −2.17299 2.17299i −1.00000 −4.48443
265.4 0.695639 0.695639i 1.00000i 1.03217i 1.23628 + 1.23628i −0.695639 0.695639i 1.20338 + 2.35624i 2.10930 + 2.10930i −1.00000 1.72000
265.5 1.71968 1.71968i 1.00000i 3.91457i −0.968182 0.968182i −1.71968 1.71968i 0.407631 + 2.61416i −3.29244 3.29244i −1.00000 −3.32992
265.6 1.75588 1.75588i 1.00000i 4.16620i 2.27310 + 2.27310i −1.75588 1.75588i −2.08970 1.62270i −3.80357 3.80357i −1.00000 7.98257
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 265.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.p.f yes 12
3.b odd 2 1 819.2.y.f 12
7.b odd 2 1 273.2.p.e 12
13.d odd 4 1 273.2.p.e 12
21.c even 2 1 819.2.y.g 12
39.f even 4 1 819.2.y.g 12
91.i even 4 1 inner 273.2.p.f yes 12
273.o odd 4 1 819.2.y.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.e 12 7.b odd 2 1
273.2.p.e 12 13.d odd 4 1
273.2.p.f yes 12 1.a even 1 1 trivial
273.2.p.f yes 12 91.i even 4 1 inner
819.2.y.f 12 3.b odd 2 1
819.2.y.f 12 273.o odd 4 1
819.2.y.g 12 21.c even 2 1
819.2.y.g 12 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{12} + \cdots$$ $$T_{5}^{12} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1369 - 148 T + 8 T^{2} + 188 T^{3} + 1247 T^{4} + 52 T^{5} - 4 T^{7} + 75 T^{8} + T^{12}$$
$3$ $$( 1 + T^{2} )^{6}$$
$5$ $$16 - 160 T + 800 T^{2} - 1600 T^{3} + 1760 T^{4} - 848 T^{5} + 480 T^{6} - 520 T^{7} + 480 T^{8} - 240 T^{9} + 72 T^{10} - 12 T^{11} + T^{12}$$
$7$ $$117649 + 67228 T + 19208 T^{2} + 1372 T^{3} + 931 T^{4} - 896 T^{5} - 528 T^{6} - 128 T^{7} + 19 T^{8} + 4 T^{9} + 8 T^{10} + 4 T^{11} + T^{12}$$
$11$ $$1936 - 5280 T + 7200 T^{2} - 992 T^{3} - 1360 T^{4} + 704 T^{5} + 3392 T^{6} + 1720 T^{7} + 436 T^{8} + 8 T^{10} + 4 T^{11} + T^{12}$$
$13$ $$4826809 - 285610 T^{2} + 26871 T^{4} + 3744 T^{5} - 3020 T^{6} + 288 T^{7} + 159 T^{8} - 10 T^{10} + T^{12}$$
$17$ $$( -696 + 768 T + 236 T^{2} - 136 T^{3} - 34 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$19$ $$295936 - 731136 T + 903168 T^{2} - 593920 T^{3} + 220928 T^{4} - 32512 T^{5} + 2048 T^{6} - 1472 T^{7} + 1104 T^{8} - 64 T^{9} + T^{12}$$
$23$ $$10816 + 88128 T^{2} + 161968 T^{4} + 36288 T^{6} + 2876 T^{8} + 92 T^{10} + T^{12}$$
$29$ $$( -352 + 1952 T + 992 T^{2} - 256 T^{3} - 72 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$31$ $$65536 - 327680 T + 819200 T^{2} - 819200 T^{3} + 450560 T^{4} - 108544 T^{5} + 30720 T^{6} - 16640 T^{7} + 7680 T^{8} - 1920 T^{9} + 288 T^{10} - 24 T^{11} + T^{12}$$
$37$ $$17676234304 - 1104033408 T + 34478208 T^{2} + 22387008 T^{3} + 33012656 T^{4} - 1452608 T^{5} + 40512 T^{6} + 21600 T^{7} + 11212 T^{8} - 296 T^{9} + 8 T^{10} + 4 T^{11} + T^{12}$$
$41$ $$112444816 + 157363360 T + 110112800 T^{2} + 6263424 T^{3} + 1017504 T^{4} + 975344 T^{5} + 543008 T^{6} + 41896 T^{7} + 2304 T^{8} + 576 T^{9} + 200 T^{10} + 20 T^{11} + T^{12}$$
$43$ $$75759616 + 65601536 T^{2} + 15790080 T^{4} + 1005568 T^{6} + 25536 T^{8} + 272 T^{10} + T^{12}$$
$47$ $$6961566096 - 1563924384 T + 175668768 T^{2} - 28275840 T^{3} + 35712304 T^{4} - 8952256 T^{5} + 1167392 T^{6} - 72936 T^{7} + 18344 T^{8} - 4056 T^{9} + 512 T^{10} - 32 T^{11} + T^{12}$$
$53$ $$( -192 - 768 T - 880 T^{2} - 320 T^{3} - 20 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$59$ $$123032464 + 94503840 T + 36295200 T^{2} - 43459776 T^{3} + 33963248 T^{4} + 3370528 T^{5} + 245472 T^{6} - 72312 T^{7} + 14632 T^{8} + 520 T^{9} + 32 T^{10} - 8 T^{11} + T^{12}$$
$61$ $$11283538176 + 2518968576 T^{2} + 193435456 T^{4} + 6158496 T^{6} + 85456 T^{8} + 496 T^{10} + T^{12}$$
$67$ $$2359296 + 393216 T^{3} + 815104 T^{4} + 229376 T^{5} + 32768 T^{6} + 4096 T^{7} + 16576 T^{8} + 4096 T^{9} + 512 T^{10} + 32 T^{11} + T^{12}$$
$71$ $$47169424 + 18406240 T + 3591200 T^{2} + 5289888 T^{3} + 8913136 T^{4} + 5122144 T^{5} + 1616768 T^{6} + 258296 T^{7} + 21236 T^{8} + 304 T^{9} + 72 T^{10} + 12 T^{11} + T^{12}$$
$73$ $$127872038464 + 55761466112 T + 12158018048 T^{2} + 286882304 T^{3} + 103984688 T^{4} + 35007360 T^{5} + 5700736 T^{6} + 300224 T^{7} + 18972 T^{8} + 3408 T^{9} + 512 T^{10} + 32 T^{11} + T^{12}$$
$79$ $$( 49424 - 28480 T + 656 T^{2} + 1448 T^{3} - 108 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$83$ $$2417098896 - 7932709728 T + 13017233952 T^{2} - 3662531424 T^{3} + 528538192 T^{4} - 22015296 T^{5} + 663552 T^{6} - 179640 T^{7} + 42920 T^{8} - 1152 T^{9} + T^{12}$$
$89$ $$169744 + 3424544 T + 34544672 T^{2} + 144584960 T^{3} + 304148192 T^{4} + 15870096 T^{5} + 316000 T^{6} - 188248 T^{7} + 75360 T^{8} + 576 T^{9} + 8 T^{10} - 4 T^{11} + T^{12}$$
$97$ $$3655744 + 13644032 T + 25461248 T^{2} + 24235776 T^{3} + 12994672 T^{4} + 2730368 T^{5} + 21632 T^{6} - 65792 T^{7} + 72620 T^{8} - 208 T^{9} + T^{12}$$