# Properties

 Label 2013.2.a.c Level 2013 Weight 2 Character orbit 2013.a Self dual yes Analytic conductor 16.074 Analytic rank 1 Dimension 12 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2013 = 3 \cdot 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2013.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.0738859269$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} - 173 x^{7} + 29 x^{6} + 281 x^{5} - 41 x^{4} - 201 x^{3} + 8 x^{2} + 49 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} - 173 x^{7} + 29 x^{6} + 281 x^{5} - 41 x^{4} - 201 x^{3} + 8 x^{2} + 49 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-18 \nu^{11} + 63 \nu^{10} + 341 \nu^{9} - 1448 \nu^{8} - 920 \nu^{7} + 7681 \nu^{6} - 1364 \nu^{5} - 15242 \nu^{4} + 6045 \nu^{3} + 11277 \nu^{2} - 4356 \nu - 2408$$$$)/313$$ $$\beta_{4}$$ $$=$$ $$($$$$14 \nu^{11} - 49 \nu^{10} + 13 \nu^{9} - 404 \nu^{8} + 646 \nu^{7} + 4494 \nu^{6} - 5060 \nu^{5} - 12420 \nu^{4} + 8653 \nu^{3} + 11261 \nu^{2} - 4124 \nu - 2370$$$$)/313$$ $$\beta_{5}$$ $$=$$ $$($$$$-41 \nu^{11} - 13 \nu^{10} + 1281 \nu^{9} - 1142 \nu^{8} - 8599 \nu^{7} + 6871 \nu^{6} + 23359 \nu^{5} - 10130 \nu^{4} - 26034 \nu^{3} + 1742 \nu^{2} + 9797 \nu + 2201$$$$)/313$$ $$\beta_{6}$$ $$=$$ $$($$$$-110 \nu^{11} + 385 \nu^{10} + 1284 \nu^{9} - 4293 \nu^{8} - 5970 \nu^{7} + 15396 \nu^{6} + 14583 \nu^{5} - 19521 \nu^{4} - 15851 \nu^{3} + 7254 \nu^{2} + 5306 \nu + 65$$$$)/313$$ $$\beta_{7}$$ $$=$$ $$($$$$98 \nu^{11} - 656 \nu^{10} + 404 \nu^{9} + 4997 \nu^{8} - 7059 \nu^{7} - 13927 \nu^{6} + 22798 \nu^{5} + 17915 \nu^{4} - 26756 \nu^{3} - 11317 \nu^{2} + 9631 \nu + 3442$$$$)/313$$ $$\beta_{8}$$ $$=$$ $$($$$$121 \nu^{11} - 580 \nu^{10} - 536 \nu^{9} + 4691 \nu^{8} + 620 \nu^{7} - 13117 \nu^{6} - 1925 \nu^{5} + 13116 \nu^{4} + 4384 \nu^{3} - 2721 \nu^{2} - 2331 \nu - 854$$$$)/313$$ $$\beta_{9}$$ $$=$$ $$($$$$-134 \nu^{11} + 782 \nu^{10} - 35 \nu^{9} - 6015 \nu^{8} + 4593 \nu^{7} + 16769 \nu^{6} - 13319 \nu^{5} - 19916 \nu^{4} + 10989 \nu^{3} + 8831 \nu^{2} - 1128 \nu - 1059$$$$)/313$$ $$\beta_{10}$$ $$=$$ $$($$$$154 \nu^{11} - 539 \nu^{10} - 1735 \nu^{9} + 5572 \nu^{8} + 8984 \nu^{7} - 20052 \nu^{6} - 24047 \nu^{5} + 26766 \nu^{4} + 26323 \nu^{3} - 10719 \nu^{2} - 8430 \nu - 404$$$$)/313$$ $$\beta_{11}$$ $$=$$ $$($$$$-236 \nu^{11} + 1139 \nu^{10} + 1167 \nu^{9} - 10047 \nu^{8} - 1142 \nu^{7} + 32229 \nu^{6} + 27 \nu^{5} - 42957 \nu^{4} - 454 \nu^{3} + 21402 \nu^{2} + 167 \nu - 1767$$$$)/313$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 3 \beta_{2} + 4 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} - \beta_{3} + 12 \beta_{2} + 8 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{11} + 13 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} - 12 \beta_{6} + 16 \beta_{5} + \beta_{4} - 6 \beta_{3} + 38 \beta_{2} + 26 \beta_{1} + 36$$ $$\nu^{6}$$ $$=$$ $$46 \beta_{11} + 42 \beta_{10} - 48 \beta_{9} - 22 \beta_{8} - 20 \beta_{7} - 40 \beta_{6} + 58 \beta_{5} + 5 \beta_{4} - 27 \beta_{3} + 130 \beta_{2} + 68 \beta_{1} + 132$$ $$\nu^{7}$$ $$=$$ $$166 \beta_{11} + 148 \beta_{10} - 177 \beta_{9} - 75 \beta_{8} - 75 \beta_{7} - 137 \beta_{6} + 203 \beta_{5} + 26 \beta_{4} - 110 \beta_{3} + 421 \beta_{2} + 210 \beta_{1} + 364$$ $$\nu^{8}$$ $$=$$ $$549 \beta_{11} + 483 \beta_{10} - 606 \beta_{9} - 237 \beta_{8} - 287 \beta_{7} - 458 \beta_{6} + 697 \beta_{5} + 109 \beta_{4} - 414 \beta_{3} + 1392 \beta_{2} + 612 \beta_{1} + 1192$$ $$\nu^{9}$$ $$=$$ $$1852 \beta_{11} + 1613 \beta_{10} - 2093 \beta_{9} - 797 \beta_{8} - 1020 \beta_{7} - 1526 \beta_{6} + 2355 \beta_{5} + 441 \beta_{4} - 1507 \beta_{3} + 4537 \beta_{2} + 1893 \beta_{1} + 3611$$ $$\nu^{10}$$ $$=$$ $$6095 \beta_{11} + 5266 \beta_{10} - 7064 \beta_{9} - 2636 \beta_{8} - 3600 \beta_{7} - 5065 \beta_{6} + 7896 \beta_{5} + 1672 \beta_{4} - 5323 \beta_{3} + 14873 \beta_{2} + 5798 \beta_{1} + 11557$$ $$\nu^{11}$$ $$=$$ $$20133 \beta_{11} + 17302 \beta_{10} - 23796 \beta_{9} - 8850 \beta_{8} - 12387 \beta_{7} - 16746 \beta_{6} + 26291 \beta_{5} + 6167 \beta_{4} - 18491 \beta_{3} + 48577 \beta_{2} + 18190 \beta_{1} + 36321$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73351 −1.62825 −1.31792 −1.03538 −0.456328 −0.199068 0.858571 1.10918 1.23554 2.22712 2.66112 3.27893
−2.73351 1.00000 5.47208 −3.64694 −2.73351 −3.96128 −9.49098 1.00000 9.96895
1.2 −2.62825 1.00000 4.90768 2.26825 −2.62825 −0.677820 −7.64209 1.00000 −5.96152
1.3 −2.31792 1.00000 3.37276 −2.47951 −2.31792 3.26859 −3.18195 1.00000 5.74732
1.4 −2.03538 1.00000 2.14278 3.64320 −2.03538 −4.68983 −0.290605 1.00000 −7.41529
1.5 −1.45633 1.00000 0.120890 −0.271377 −1.45633 −0.415334 2.73660 1.00000 0.395214
1.6 −1.19907 1.00000 −0.562235 0.908382 −1.19907 −2.10536 3.07229 1.00000 −1.08921
1.7 −0.141429 1.00000 −1.98000 −4.38024 −0.141429 3.47902 0.562887 1.00000 0.619492
1.8 0.109182 1.00000 −1.98808 0.0420221 0.109182 −3.35497 −0.435426 1.00000 0.00458805
1.9 0.235537 1.00000 −1.94452 2.04419 0.235537 −0.723848 −0.929081 1.00000 0.481482
1.10 1.22712 1.00000 −0.494178 −1.92536 1.22712 2.05062 −3.06065 1.00000 −2.36265
1.11 1.66112 1.00000 0.759316 −1.47319 1.66112 −2.81455 −2.06092 1.00000 −2.44715
1.12 2.27893 1.00000 3.19351 −1.72942 2.27893 −5.05524 2.71992 1.00000 −3.94122
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$
$$61$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2013.2.a.c 12
3.b odd 2 1 6039.2.a.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.c 12 1.a even 1 1 trivial
6039.2.a.f 12 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2013))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T + 30 T^{2} + 97 T^{3} + 261 T^{4} + 611 T^{5} + 1286 T^{6} + 2482 T^{7} + 4459 T^{8} + 7521 T^{9} + 12001 T^{10} + 18203 T^{11} + 26347 T^{12} + 36406 T^{13} + 48004 T^{14} + 60168 T^{15} + 71344 T^{16} + 79424 T^{17} + 82304 T^{18} + 78208 T^{19} + 66816 T^{20} + 49664 T^{21} + 30720 T^{22} + 14336 T^{23} + 4096 T^{24}$$
$3$ $$( 1 - T )^{12}$$
$5$ $$1 + 7 T + 49 T^{2} + 225 T^{3} + 972 T^{4} + 3480 T^{5} + 11659 T^{6} + 35101 T^{7} + 99430 T^{8} + 262799 T^{9} + 659489 T^{10} + 1575031 T^{11} + 3598106 T^{12} + 7875155 T^{13} + 16487225 T^{14} + 32849875 T^{15} + 62143750 T^{16} + 109690625 T^{17} + 182171875 T^{18} + 271875000 T^{19} + 379687500 T^{20} + 439453125 T^{21} + 478515625 T^{22} + 341796875 T^{23} + 244140625 T^{24}$$
$7$ $$1 + 15 T + 139 T^{2} + 948 T^{3} + 5305 T^{4} + 25518 T^{5} + 109397 T^{6} + 426094 T^{7} + 1529603 T^{8} + 5096399 T^{9} + 15850126 T^{10} + 46134877 T^{11} + 126000578 T^{12} + 322944139 T^{13} + 776656174 T^{14} + 1748064857 T^{15} + 3672576803 T^{16} + 7161361858 T^{17} + 12870447653 T^{18} + 21015170274 T^{19} + 30582269305 T^{20} + 38255219436 T^{21} + 39264059611 T^{22} + 29659901145 T^{23} + 13841287201 T^{24}$$
$11$ $$( 1 - T )^{12}$$
$13$ $$1 + 11 T + 97 T^{2} + 662 T^{3} + 4145 T^{4} + 22669 T^{5} + 115669 T^{6} + 537401 T^{7} + 2387791 T^{8} + 9912352 T^{9} + 39737149 T^{10} + 150753874 T^{11} + 556680330 T^{12} + 1959800362 T^{13} + 6715578181 T^{14} + 21777437344 T^{15} + 68197698751 T^{16} + 199533229493 T^{17} + 558312170221 T^{18} + 1422446131873 T^{19} + 3381203838545 T^{20} + 7020178584926 T^{21} + 13372273709353 T^{22} + 19713764334407 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 + 33 T + 597 T^{2} + 7431 T^{3} + 70404 T^{4} + 532853 T^{5} + 3320531 T^{6} + 17322548 T^{7} + 76479658 T^{8} + 288030690 T^{9} + 946301382 T^{10} + 2935275070 T^{11} + 10476783848 T^{12} + 49899676190 T^{13} + 273481099398 T^{14} + 1415094779970 T^{15} + 6387657515818 T^{16} + 24595541035636 T^{17} + 80149546129139 T^{18} + 218650192924069 T^{19} + 491121226876164 T^{20} + 881226510249207 T^{21} + 1203548358568053 T^{22} + 1130972578151889 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 + 24 T + 371 T^{2} + 4280 T^{3} + 41156 T^{4} + 341152 T^{5} + 2518088 T^{6} + 16766116 T^{7} + 102173034 T^{8} + 573298456 T^{9} + 2983989899 T^{10} + 14439165481 T^{11} + 65182491500 T^{12} + 274344144139 T^{13} + 1077220353539 T^{14} + 3932254109704 T^{15} + 13315291963914 T^{16} + 41514563061484 T^{17} + 118465668395528 T^{18} + 304946131503328 T^{19} + 698975520515396 T^{20} + 1381103346494120 T^{21} + 2274625581644171 T^{22} + 2795766213557256 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 + 9 T + 122 T^{2} + 871 T^{3} + 8659 T^{4} + 52566 T^{5} + 409672 T^{6} + 2213499 T^{7} + 15173441 T^{8} + 73639616 T^{9} + 449170145 T^{10} + 2008306342 T^{11} + 11293340992 T^{12} + 46191045866 T^{13} + 237611006705 T^{14} + 895973207872 T^{15} + 4246150902881 T^{16} + 14246838794157 T^{17} + 60646158718408 T^{18} + 178978054447002 T^{19} + 678094821548179 T^{20} + 1568803968134273 T^{21} + 5054034368065178 T^{22} + 8575287821225343 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 + 16 T + 250 T^{2} + 2714 T^{3} + 27246 T^{4} + 229891 T^{5} + 1814447 T^{6} + 12769588 T^{7} + 85498737 T^{8} + 527376390 T^{9} + 3151032337 T^{10} + 17723986898 T^{11} + 97961804350 T^{12} + 513995620042 T^{13} + 2650018195417 T^{14} + 12862182775710 T^{15} + 60471632204097 T^{16} + 261918922136612 T^{17} + 1079275390318487 T^{18} + 3965591314552319 T^{19} + 13629713767535406 T^{20} + 39372394178508466 T^{21} + 105176808325050250 T^{22} + 195208156251293264 T^{23} + 353814783205469041 T^{24}$$
$31$ $$1 - T + 129 T^{2} - 543 T^{3} + 9764 T^{4} - 58349 T^{5} + 613100 T^{6} - 3780663 T^{7} + 31015429 T^{8} - 186699029 T^{9} + 1266384968 T^{10} - 7274116808 T^{11} + 42944835044 T^{12} - 225497621048 T^{13} + 1216995954248 T^{14} - 5561950772939 T^{15} + 28643400005509 T^{16} - 108237171907113 T^{17} + 544128506821100 T^{18} - 1605333520762739 T^{19} + 8327628089573924 T^{20} - 14356714833244353 T^{21} + 105732049020523329 T^{22} - 25408476896404831 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 + 6 T + 212 T^{2} + 777 T^{3} + 20725 T^{4} + 41254 T^{5} + 1358539 T^{6} + 1119107 T^{7} + 72741900 T^{8} + 23069488 T^{9} + 3440579340 T^{10} + 894799187 T^{11} + 139567844104 T^{12} + 33107569919 T^{13} + 4710153116460 T^{14} + 1168538775664 T^{15} + 136330032045900 T^{16} + 77603307686399 T^{17} + 3485639389956451 T^{18} + 3916319659244782 T^{19} + 72796136682512725 T^{20} + 100980271820774829 T^{21} + 1019419886952583988 T^{22} + 1067505730676762478 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 + 21 T + 409 T^{2} + 5323 T^{3} + 65528 T^{4} + 669694 T^{5} + 6574879 T^{6} + 57551319 T^{7} + 484655090 T^{8} + 3742091431 T^{9} + 27770380575 T^{10} + 191467662475 T^{11} + 1269193856456 T^{12} + 7850174161475 T^{13} + 46682009746575 T^{14} + 257908683515951 T^{15} + 1369519451773490 T^{16} + 6667677181879119 T^{17} + 31231360621961839 T^{18} + 130425768692462414 T^{19} + 523236180413840888 T^{20} + 1742654036779054403 T^{21} + 5489867657852332009 T^{22} + 11556909666041217261 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$1 + 39 T + 916 T^{2} + 15723 T^{3} + 221865 T^{4} + 2696997 T^{5} + 29240744 T^{6} + 287210810 T^{7} + 2589998088 T^{8} + 21591484480 T^{9} + 167516124308 T^{10} + 1213165093292 T^{11} + 8223006197160 T^{12} + 52166099011556 T^{13} + 309737313845492 T^{14} + 1716674156551360 T^{15} + 8854688053252488 T^{16} + 42222413990868830 T^{17} + 184841358646868456 T^{18} + 733093978699745679 T^{19} + 2593202554589945865 T^{20} + 7902263637482982489 T^{21} + 19796117798968372084 T^{22} + 36242455839377685573 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 18 T + 379 T^{2} + 4654 T^{3} + 57946 T^{4} + 554755 T^{5} + 5371745 T^{6} + 44514567 T^{7} + 383459216 T^{8} + 2951440873 T^{9} + 23624359086 T^{10} + 168738502880 T^{11} + 1228757363190 T^{12} + 7930709635360 T^{13} + 52186209220974 T^{14} + 306427445757479 T^{15} + 1871158650590096 T^{16} + 10209193680216969 T^{17} + 57903196047479105 T^{18} + 281051709192451565 T^{19} + 1379768816902402906 T^{20} + 5208433221820277618 T^{21} + 19935071117379588571 T^{22} + 44498865871512221454 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 + 14 T + 404 T^{2} + 4409 T^{3} + 75815 T^{4} + 695281 T^{5} + 9143452 T^{6} + 73386372 T^{7} + 812507082 T^{8} + 5855303311 T^{9} + 57239299361 T^{10} + 375557294264 T^{11} + 3321887781522 T^{12} + 19904536595992 T^{13} + 160785191905049 T^{14} + 871719991031747 T^{15} + 6411071692886442 T^{16} + 30689850018021396 T^{17} + 202658772093677308 T^{18} + 816754336017009197 T^{19} + 4720218428537334215 T^{20} + 14548657676255604397 T^{21} + 70654538027667271796 T^{22} +$$$$12\!\cdots\!58$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 23 T + 515 T^{2} + 7688 T^{3} + 104142 T^{4} + 1187224 T^{5} + 12217738 T^{6} + 114370298 T^{7} + 980246121 T^{8} + 7957083742 T^{9} + 61521602112 T^{10} + 468570698606 T^{11} + 3582366333314 T^{12} + 27645671217754 T^{13} + 214156696951872 T^{14} + 1634217901848218 T^{15} + 11877996117006681 T^{16} + 81766105124071102 T^{17} + 515350708725924058 T^{18} + 2954586770412752456 T^{19} + 15291215432989197582 T^{20} + 66601111853819171032 T^{21} +$$$$26\!\cdots\!15$$$$T^{22} +$$$$69\!\cdots\!57$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$( 1 + T )^{12}$$
$67$ $$1 + 449 T^{2} + 173 T^{3} + 90315 T^{4} + 119182 T^{5} + 10802408 T^{6} + 33210707 T^{7} + 873198976 T^{8} + 5211503508 T^{9} + 54633204281 T^{10} + 521654265811 T^{11} + 3370799768398 T^{12} + 34950835809337 T^{13} + 245248454017409 T^{14} + 1567427429576604 T^{15} + 17595938222452096 T^{16} + 44838609341920649 T^{17} + 977168351209462952 T^{18} + 722327730545605786 T^{19} + 36674002298528031915 T^{20} + 4706730450559025831 T^{21} +$$$$81\!\cdots\!01$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 19 T + 608 T^{2} + 9307 T^{3} + 179552 T^{4} + 2303574 T^{5} + 33989917 T^{6} + 377566421 T^{7} + 4624247615 T^{8} + 45164214709 T^{9} + 476984599751 T^{10} + 4124585113695 T^{11} + 38252651163656 T^{12} + 292845543072345 T^{13} + 2404479367344791 T^{14} + 16164769250712899 T^{15} + 117509905257390815 T^{16} + 681216418720222771 T^{17} + 4354118018151224557 T^{18} + 20951282323745389434 T^{19} +$$$$11\!\cdots\!72$$$$T^{20} +$$$$42\!\cdots\!17$$$$T^{21} +$$$$19\!\cdots\!08$$$$T^{22} +$$$$43\!\cdots\!49$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 42 T + 1043 T^{2} + 17998 T^{3} + 245747 T^{4} + 2754720 T^{5} + 26880038 T^{6} + 230692403 T^{7} + 1799301587 T^{8} + 12486762683 T^{9} + 80041477558 T^{10} + 485176939647 T^{11} + 3589099987174 T^{12} + 35417916594231 T^{13} + 426541033906582 T^{14} + 4857562958652611 T^{15} + 51097000099308467 T^{16} + 478241867380207979 T^{17} + 4067869753348918982 T^{18} + 30432489648526887840 T^{19} +$$$$19\!\cdots\!07$$$$T^{20} +$$$$10\!\cdots\!74$$$$T^{21} +$$$$44\!\cdots\!07$$$$T^{22} +$$$$13\!\cdots\!34$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 11 T + 375 T^{2} + 1971 T^{3} + 61812 T^{4} + 141297 T^{5} + 8030226 T^{6} + 5418989 T^{7} + 921051700 T^{8} + 1159310 T^{9} + 93618602276 T^{10} + 97393057 T^{11} + 8156189470360 T^{12} + 7694051503 T^{13} + 584273696804516 T^{14} + 571585043090 T^{15} + 35875038320187700 T^{16} + 16674534778560611 T^{17} + 1952047205598577746 T^{18} + 2713454728017308223 T^{19} + 93775529757944348532 T^{20} +$$$$23\!\cdots\!49$$$$T^{21} +$$$$35\!\cdots\!75$$$$T^{22} +$$$$82\!\cdots\!69$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 + 56 T + 1951 T^{2} + 49386 T^{3} + 1022271 T^{4} + 17968694 T^{5} + 278807356 T^{6} + 3878361170 T^{7} + 49264947230 T^{8} + 575419961940 T^{9} + 6239757087976 T^{10} + 62947612589627 T^{11} + 593568643259770 T^{12} + 5224651844939041 T^{13} + 42985686579066664 T^{14} + 329017653777786780 T^{15} + 2338031679689400830 T^{16} + 15277022276863032310 T^{17} + 91153381068663702364 T^{18} +$$$$48\!\cdots\!38$$$$T^{19} +$$$$23\!\cdots\!11$$$$T^{20} +$$$$92\!\cdots\!58$$$$T^{21} +$$$$30\!\cdots\!99$$$$T^{22} +$$$$72\!\cdots\!52$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 + 55 T + 2149 T^{2} + 60868 T^{3} + 1441797 T^{4} + 28799664 T^{5} + 507606343 T^{6} + 7923948344 T^{7} + 111878225657 T^{8} + 1429878368183 T^{9} + 16734332409804 T^{10} + 179077113801815 T^{11} + 1764114163881438 T^{12} + 15937863128361535 T^{13} + 132552647018057484 T^{14} + 1008019923339601327 T^{15} + 7019490596823877337 T^{16} + 44247798623701102456 T^{17} +$$$$25\!\cdots\!23$$$$T^{18} +$$$$12\!\cdots\!56$$$$T^{19} +$$$$56\!\cdots\!57$$$$T^{20} +$$$$21\!\cdots\!12$$$$T^{21} +$$$$67\!\cdots\!49$$$$T^{22} +$$$$15\!\cdots\!95$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 + 7 T + 483 T^{2} + 3384 T^{3} + 135472 T^{4} + 1005978 T^{5} + 27299194 T^{6} + 202779564 T^{7} + 4278918623 T^{8} + 31268940433 T^{9} + 548132714887 T^{10} + 3783249792977 T^{11} + 58040314464774 T^{12} + 366975229918769 T^{13} + 5157380714371783 T^{14} + 28538317671807409 T^{15} + 378809589151700063 T^{16} + 1741337113234107948 T^{17} + 22739464359125727226 T^{18} + 81281296622723159514 T^{19} +$$$$10\!\cdots\!92$$$$T^{20} +$$$$25\!\cdots\!28$$$$T^{21} +$$$$35\!\cdots\!67$$$$T^{22} +$$$$50\!\cdots\!71$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$