Properties

 Label 390.2.x.b Level $390$ Weight $2$ Character orbit 390.x Analytic conductor $3.114$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.x (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} - 4513282461 \nu^{7} + 6489146722 \nu^{6} + 4655211661 \nu^{5} - 5740233327 \nu^{4} - 8165110694 \nu^{3} - 33307875431 \nu^{2} + 64277898945 \nu + 81183629852$$$$)/ 63907274600$$ $$\beta_{3}$$ $$=$$ $$($$$$60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} - 55879543947 \nu^{7} - 372787906 \nu^{6} + 135156205247 \nu^{5} - 272345430479 \nu^{4} + 414518069862 \nu^{3} - 554153957987 \nu^{2} - 833086363785 \nu + 1269272187404$$$$)/ 830794569800$$ $$\beta_{4}$$ $$=$$ $$($$$$-120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + 2513656023 \nu^{7} + 29967292429 \nu^{6} - 18436259198 \nu^{5} + 35262141211 \nu^{4} + 8309942667 \nu^{3} + 28687118858 \nu^{2} + 98318215515 \nu + 618192439839$$$$)/ 830794569800$$ $$\beta_{5}$$ $$=$$ $$($$$$16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + 1065744289 \nu^{7} - 511723478 \nu^{6} + 4136894311 \nu^{5} + 1601173623 \nu^{4} + 3964105106 \nu^{3} - 3499453881 \nu^{2} + 44426935995 \nu + 20321135952$$$$)/ 63907274600$$ $$\beta_{6}$$ $$=$$ $$($$$$4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + 35557 \nu^{6} + 1844066 \nu^{5} - 940887 \nu^{4} + 77961 \nu^{3} - 7481036 \nu^{2} - 4439955 \nu + 11867687$$$$)/11796200$$ $$\beta_{7}$$ $$=$$ $$($$$$-20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} - 1390803 \nu^{6} + 3193024 \nu^{5} - 2792113 \nu^{4} - 7615595 \nu^{3} - 12957312 \nu^{2} + 2476201 \nu + 21764327$$$$)/47610004$$ $$\beta_{8}$$ $$=$$ $$($$$$5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + 32851611 \nu^{6} + 1496635908 \nu^{5} - 1228577871 \nu^{4} - 1963928377 \nu^{3} + 3156976042 \nu^{2} + 8732837145 \nu - 40106339$$$$)/ 12781454920$$ $$\beta_{9}$$ $$=$$ $$($$$$-680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} - 56431077566 \nu^{7} + 138431742257 \nu^{6} - 46377269759 \nu^{5} - 85505629812 \nu^{4} - 58320816489 \nu^{3} - 633066289511 \nu^{2} + 550784663520 \nu + 1108736562037$$$$)/ 830794569800$$ $$\beta_{10}$$ $$=$$ $$($$$$-1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + 75140023171 \nu^{7} - 22116401667 \nu^{6} - 93428783546 \nu^{5} + 206023103197 \nu^{4} - 574822497041 \nu^{3} + 859741244016 \nu^{2} + 783010043855 \nu - 1182431604497$$$$)/ 830794569800$$ $$\beta_{11}$$ $$=$$ $$($$$$-78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} - 695335944 \nu^{7} + 7168384238 \nu^{6} - 1904719281 \nu^{5} - 3795941083 \nu^{4} - 26424465301 \nu^{3} - 15605355549 \nu^{2} + 99935137380 \nu + 16572567908$$$$)/ 31953637300$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_{1} - 5$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{11} + 7 \beta_{10} + 12 \beta_{9} + \beta_{8} + 2 \beta_{7} + 9 \beta_{5} - 6 \beta_{4} + \beta_{2} - 3$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + 47 \beta_{4} - \beta_{3} + 12 \beta_{2} - 42$$ $$\nu^{6}$$ $$=$$ $$-6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + 6 \beta_{3} + 38 \beta_{2} - 32 \beta_{1} - 7$$ $$\nu^{7}$$ $$=$$ $$50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + 202 \beta_{4} - 8 \beta_{3} + 76 \beta_{2} - 11 \beta_{1} - 104$$ $$\nu^{8}$$ $$=$$ $$81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + 113 \beta_{5} - 491 \beta_{4} + 73 \beta_{3} + 64 \beta_{2} - 112 \beta_{1} + 388$$ $$\nu^{9}$$ $$=$$ $$404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} - 365 \beta_{5} + 328 \beta_{4} + 7 \beta_{3} - 231 \beta_{2} + 163 \beta_{1} + 249$$ $$\nu^{10}$$ $$=$$ $$327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + 1181 \beta_{5} - 3528 \beta_{4} + 248 \beta_{3} - 1421 \beta_{2} - 202 \beta_{1} + 2981$$ $$\nu^{11}$$ $$=$$ $$1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} - 1755 \beta_{5} + 3931 \beta_{4} - 1427 \beta_{3} - 3608 \beta_{2} + 1392 \beta_{1} - 288$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.40719 − 0.536449i −1.44229 + 0.433312i −0.330925 + 1.46916i 2.00607 − 1.30680i 1.75374 + 1.62986i −2.39378 + 0.0429626i 1.40719 + 0.536449i −1.44229 − 0.433312i −0.330925 − 1.46916i 2.00607 + 1.30680i 1.75374 − 1.62986i −2.39378 − 0.0429626i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −2.03420 0.928463i −0.866025 0.500000i −1.40247 + 2.42916i −1.00000 0.500000 0.866025i −0.213026 2.22590i
49.2 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0.230377 + 2.22417i −0.866025 0.500000i −0.432713 + 0.749482i −1.00000 0.500000 0.866025i −1.81100 + 1.31160i
49.3 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0.571769 2.16173i −0.866025 0.500000i 0.603137 1.04466i −1.00000 0.500000 0.866025i 2.15800 0.585699i
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −1.26873 1.84128i 0.866025 + 0.500000i 2.17283 3.76344i −1.00000 0.500000 0.866025i 0.960230 2.01940i
49.5 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 1.40066 + 1.74303i 0.866025 + 0.500000i −0.763837 + 1.32301i −1.00000 0.500000 0.866025i −0.809179 + 2.08452i
49.6 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 2.10012 0.767774i 0.866025 + 0.500000i 0.823063 1.42559i −1.00000 0.500000 0.866025i 1.71497 + 1.43487i
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i −2.03420 + 0.928463i −0.866025 + 0.500000i −1.40247 2.42916i −1.00000 0.500000 + 0.866025i −0.213026 + 2.22590i
199.2 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0.230377 2.22417i −0.866025 + 0.500000i −0.432713 0.749482i −1.00000 0.500000 + 0.866025i −1.81100 1.31160i
199.3 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0.571769 + 2.16173i −0.866025 + 0.500000i 0.603137 + 1.04466i −1.00000 0.500000 + 0.866025i 2.15800 + 0.585699i
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −1.26873 + 1.84128i 0.866025 0.500000i 2.17283 + 3.76344i −1.00000 0.500000 + 0.866025i 0.960230 + 2.01940i
199.5 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 1.40066 1.74303i 0.866025 0.500000i −0.763837 1.32301i −1.00000 0.500000 + 0.866025i −0.809179 2.08452i
199.6 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 2.10012 + 0.767774i 0.866025 0.500000i 0.823063 + 1.42559i −1.00000 0.500000 + 0.866025i 1.71497 1.43487i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.x.b yes 12
3.b odd 2 1 1170.2.bj.c 12
5.b even 2 1 390.2.x.a 12
5.c odd 4 1 1950.2.bc.i 12
5.c odd 4 1 1950.2.bc.j 12
13.e even 6 1 390.2.x.a 12
15.d odd 2 1 1170.2.bj.d 12
39.h odd 6 1 1170.2.bj.d 12
65.l even 6 1 inner 390.2.x.b yes 12
65.r odd 12 1 1950.2.bc.i 12
65.r odd 12 1 1950.2.bc.j 12
195.y odd 6 1 1170.2.bj.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 5.b even 2 1
390.2.x.a 12 13.e even 6 1
390.2.x.b yes 12 1.a even 1 1 trivial
390.2.x.b yes 12 65.l even 6 1 inner
1170.2.bj.c 12 3.b odd 2 1
1170.2.bj.c 12 195.y odd 6 1
1170.2.bj.d 12 15.d odd 2 1
1170.2.bj.d 12 39.h odd 6 1
1950.2.bc.i 12 5.c odd 4 1
1950.2.bc.i 12 65.r odd 12 1
1950.2.bc.j 12 5.c odd 4 1
1950.2.bc.j 12 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{6}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$15625 - 6250 T + 4375 T^{2} - 750 T^{3} + 375 T^{4} - 160 T^{5} + 114 T^{6} - 32 T^{7} + 15 T^{8} - 6 T^{9} + 7 T^{10} - 2 T^{11} + T^{12}$$
$7$ $$1024 + 512 T + 1664 T^{2} + 64 T^{3} + 1648 T^{4} + 112 T^{5} + 708 T^{6} - 20 T^{7} + 205 T^{8} + 6 T^{9} + 19 T^{10} - 2 T^{11} + T^{12}$$
$11$ $$16 - 96 T - 4 T^{2} + 1176 T^{3} + 2585 T^{4} + 1446 T^{5} - 330 T^{6} - 444 T^{7} + 87 T^{8} + 84 T^{9} - 2 T^{10} - 6 T^{11} + T^{12}$$
$13$ $$4826809 - 2970344 T + 1028196 T^{2} - 87880 T^{3} - 39208 T^{4} + 26520 T^{5} - 7838 T^{6} + 2040 T^{7} - 232 T^{8} - 40 T^{9} + 36 T^{10} - 8 T^{11} + T^{12}$$
$17$ $$65536 - 491520 T + 1269760 T^{2} - 307200 T^{3} - 300288 T^{4} + 85248 T^{5} + 75744 T^{6} + 4632 T^{7} - 2695 T^{8} - 234 T^{9} + 95 T^{10} + 18 T^{11} + T^{12}$$
$19$ $$1982464 - 3649536 T + 837120 T^{2} + 2581632 T^{3} + 672400 T^{4} - 254496 T^{5} - 43876 T^{6} + 14568 T^{7} + 2701 T^{8} - 390 T^{9} - 53 T^{10} + 6 T^{11} + T^{12}$$
$23$ $$190660864 + 284334336 T + 171279232 T^{2} + 44643456 T^{3} + 1290352 T^{4} - 1617792 T^{5} - 129352 T^{6} + 53748 T^{7} + 7441 T^{8} - 630 T^{9} - 93 T^{10} + 6 T^{11} + T^{12}$$
$29$ $$21904 - 490768 T + 11075036 T^{2} + 1644412 T^{3} + 1737745 T^{4} - 196610 T^{5} + 148926 T^{6} - 15668 T^{7} + 6703 T^{8} - 960 T^{9} + 190 T^{10} - 14 T^{11} + T^{12}$$
$31$ $$177209344 + 93061120 T^{2} + 11339776 T^{4} + 592640 T^{6} + 15433 T^{8} + 198 T^{10} + T^{12}$$
$37$ $$227195329 - 175510012 T + 129327441 T^{2} - 32084244 T^{3} + 11678146 T^{4} - 1319436 T^{5} + 734609 T^{6} - 60444 T^{7} + 14658 T^{8} - 1028 T^{9} + 209 T^{10} - 12 T^{11} + T^{12}$$
$41$ $$65536 - 491520 T + 1269760 T^{2} - 307200 T^{3} - 300288 T^{4} + 85248 T^{5} + 75744 T^{6} + 4632 T^{7} - 2695 T^{8} - 234 T^{9} + 95 T^{10} + 18 T^{11} + T^{12}$$
$43$ $$349241344 - 627916800 T + 494054400 T^{2} - 211680000 T^{3} + 51560464 T^{4} - 6267168 T^{5} + 54524 T^{6} + 57864 T^{7} + 6253 T^{8} - 3708 T^{9} + 535 T^{10} - 36 T^{11} + T^{12}$$
$47$ $$( 5956 - 2352 T - 4283 T^{2} - 1280 T^{3} - 98 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$53$ $$2473271824 + 989681000 T^{2} + 127872521 T^{4} + 5778420 T^{6} + 87438 T^{8} + 508 T^{10} + T^{12}$$
$59$ $$4983230464 - 752228352 T - 820266240 T^{2} + 129534336 T^{3} + 157498704 T^{4} + 37281744 T^{5} + 3235868 T^{6} - 77796 T^{7} - 26427 T^{8} + 972 T^{9} + 459 T^{10} + 36 T^{11} + T^{12}$$
$61$ $$89718784 - 41828352 T + 26851328 T^{2} - 7106048 T^{3} + 3388992 T^{4} - 835296 T^{5} + 256760 T^{6} - 39396 T^{7} + 7561 T^{8} - 762 T^{9} + 135 T^{10} - 10 T^{11} + T^{12}$$
$67$ $$83759104 - 2928640 T + 25142272 T^{2} + 2632704 T^{3} + 6100480 T^{4} + 320768 T^{5} + 394560 T^{6} - 35776 T^{7} + 18784 T^{8} - 784 T^{9} + 164 T^{10} + 4 T^{11} + T^{12}$$
$71$ $$4194304 + 25165824 T + 56295424 T^{2} + 35782656 T^{3} - 703488 T^{4} - 6873600 T^{5} + 1150080 T^{6} + 505152 T^{7} + 36080 T^{8} - 2640 T^{9} - 172 T^{10} + 12 T^{11} + T^{12}$$
$73$ $$( -230528 + 114240 T + 18280 T^{2} - 3068 T^{3} - 263 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$79$ $$( -29312 - 12576 T + 3892 T^{2} + 788 T^{3} - 179 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$83$ $$( -6912 - 51840 T - 16560 T^{2} - 144 T^{3} + 360 T^{4} + 36 T^{5} + T^{6} )^{2}$$
$89$ $$1341001056256 + 39576354816 T - 61874871296 T^{2} - 1837575168 T^{3} + 2363129536 T^{4} + 191431776 T^{5} - 15114712 T^{6} - 1612140 T^{7} + 96433 T^{8} + 7506 T^{9} - 309 T^{10} - 18 T^{11} + T^{12}$$
$97$ $$415519473664 + 141504348160 T + 88030961664 T^{2} - 14104406016 T^{3} + 3300888832 T^{4} - 348175872 T^{5} + 53039744 T^{6} - 5437824 T^{7} + 522672 T^{8} - 32704 T^{9} + 1640 T^{10} - 48 T^{11} + T^{12}$$