# Properties

 Label 210.2.n.b Level 210 Weight 2 Character orbit 210.n Analytic conductor 1.677 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.7652750400000000.1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} - 112 x^{3} + 288 x^{2} - 192 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$10967 \nu^{11} + 32246 \nu^{10} - 320538 \nu^{9} + 1200782 \nu^{8} + 26475 \nu^{7} + 192316 \nu^{6} - 5356872 \nu^{5} + 25802216 \nu^{4} + 4101028 \nu^{3} - 2644000 \nu^{2} - 7541936 \nu + 11668096$$$$)/6321728$$ $$\beta_{3}$$ $$=$$ $$($$$$-11057 \nu^{11} + 124978 \nu^{10} - 554210 \nu^{9} + 1179630 \nu^{8} - 841061 \nu^{7} + 1776760 \nu^{6} - 10333816 \nu^{5} + 23519360 \nu^{4} - 8799372 \nu^{3} - 6514400 \nu^{2} - 11214000 \nu + 15302400$$$$)/6321728$$ $$\beta_{4}$$ $$=$$ $$($$$$-25665 \nu^{11} + 172184 \nu^{10} - 569522 \nu^{9} + 611754 \nu^{8} - 480425 \nu^{7} + 2503214 \nu^{6} - 11800108 \nu^{5} + 10615144 \nu^{4} - 68948 \nu^{3} - 8958456 \nu^{2} - 19968496 \nu + 4417824$$$$)/6321728$$ $$\beta_{5}$$ $$=$$ $$($$$$-34905 \nu^{11} + 255324 \nu^{10} - 866962 \nu^{9} + 1148906 \nu^{8} - 1016073 \nu^{7} + 5063202 \nu^{6} - 17291492 \nu^{5} + 20727744 \nu^{4} - 3071364 \nu^{3} + 16388488 \nu^{2} - 17161936 \nu + 6873568$$$$)/6321728$$ $$\beta_{6}$$ $$=$$ $$($$$$43053 \nu^{11} - 242212 \nu^{10} + 665338 \nu^{9} - 227874 \nu^{8} + 407037 \nu^{7} - 4046658 \nu^{6} + 13863204 \nu^{5} - 1847616 \nu^{4} - 6494364 \nu^{3} + 2417208 \nu^{2} + 10569008 \nu - 2193056$$$$)/6321728$$ $$\beta_{7}$$ $$=$$ $$($$$$-65619 \nu^{11} + 412146 \nu^{10} - 1257142 \nu^{9} + 1078106 \nu^{8} - 1258503 \nu^{7} + 7769956 \nu^{6} - 26209616 \nu^{5} + 17934344 \nu^{4} - 1580084 \nu^{3} + 15774688 \nu^{2} - 25667824 \nu + 17709568$$$$)/6321728$$ $$\beta_{8}$$ $$=$$ $$($$$$-65883 \nu^{11} + 372468 \nu^{10} - 1055638 \nu^{9} + 532062 \nu^{8} - 1079355 \nu^{7} + 6443046 \nu^{6} - 21848556 \nu^{5} + 6245376 \nu^{4} - 1136412 \nu^{3} - 251880 \nu^{2} - 16808976 \nu + 6409568$$$$)/6321728$$ $$\beta_{9}$$ $$=$$ $$($$$$112417 \nu^{11} - 805130 \nu^{10} + 2628966 \nu^{9} - 2969054 \nu^{8} + 1600117 \nu^{7} - 14110440 \nu^{6} + 52377004 \nu^{5} - 54894792 \nu^{4} - 17165628 \nu^{3} - 21185216 \nu^{2} + 39098384 \nu - 33288832$$$$)/6321728$$ $$\beta_{10}$$ $$=$$ $$($$$$-227 \nu^{11} + 1290 \nu^{10} - 3622 \nu^{9} + 1778 \nu^{8} - 3655 \nu^{7} + 23804 \nu^{6} - 75424 \nu^{5} + 20464 \nu^{4} - 228 \nu^{3} + 36720 \nu^{2} - 54704 \nu + 16768$$$$)/8768$$ $$\beta_{11}$$ $$=$$ $$($$$$-213873 \nu^{11} + 1049078 \nu^{10} - 2530334 \nu^{9} - 669562 \nu^{8} - 2913901 \nu^{7} + 19696548 \nu^{6} - 54686860 \nu^{5} - 30221320 \nu^{4} - 2505732 \nu^{3} + 31536720 \nu^{2} - 25274576 \nu - 13456256$$$$)/6321728$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{10} - 4 \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} - 6 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{11} + 4 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} - 11 \beta_{4} + 11 \beta_{3} + 12 \beta_{2} - 14 \beta_{1} - 19$$ $$\nu^{5}$$ $$=$$ $$30 \beta_{11} - 62 \beta_{10} + 21 \beta_{8} + 40 \beta_{7} - 20 \beta_{6} - 35 \beta_{4} + 15 \beta_{3} + 37 \beta_{2} - 55 \beta_{1} - 38$$ $$\nu^{6}$$ $$=$$ $$70 \beta_{11} - 334 \beta_{10} - 70 \beta_{9} + 338 \beta_{8} + 127 \beta_{7} + 46 \beta_{6} + 127 \beta_{5} - 55 \beta_{4} - 55 \beta_{3} - 72 \beta_{2} - 46 \beta_{1} - 97$$ $$\nu^{7}$$ $$=$$ $$-820 \beta_{10} - 394 \beta_{9} + 1305 \beta_{8} + 590 \beta_{6} + 566 \beta_{5} + 197 \beta_{4} - 399 \beta_{3} - 815 \beta_{2} + 283 \beta_{1} + 92$$ $$\nu^{8}$$ $$=$$ $$-960 \beta_{11} - 960 \beta_{9} + 2716 \beta_{8} - 1521 \beta_{7} + 2098 \beta_{6} + 1521 \beta_{5} + 1469 \beta_{4} - 1469 \beta_{3} - 3104 \beta_{2} + 2098 \beta_{1} + 1741$$ $$\nu^{9}$$ $$=$$ $$-4962 \beta_{11} + 10290 \beta_{10} - 2183 \beta_{8} - 7248 \beta_{7} + 3624 \beta_{6} + 4653 \beta_{4} - 2481 \beta_{3} - 5523 \beta_{2} + 6777 \beta_{1} + 6482$$ $$\nu^{10}$$ $$=$$ $$-12210 \beta_{11} + 51306 \beta_{10} + 12210 \beta_{9} - 44234 \beta_{8} - 18405 \beta_{7} - 6950 \beta_{6} - 18405 \beta_{5} + 5325 \beta_{4} + 5325 \beta_{3} + 13080 \beta_{2} + 6950 \beta_{1} + 13443$$ $$\nu^{11}$$ $$=$$ $$126436 \beta_{10} + 61230 \beta_{9} - 189895 \beta_{8} - 80174 \beta_{6} - 89626 \beta_{5} - 30615 \beta_{4} + 55165 \beta_{3} + 132501 \beta_{2} - 44813 \beta_{1} - 13900$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$\beta_{10}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.593239 + 0.593239i 2.45845 + 2.45845i −0.685661 − 0.685661i 0.406761 − 0.406761i −1.45845 + 1.45845i 1.68566 − 1.68566i 0.593239 − 0.593239i 2.45845 − 2.45845i −0.685661 + 0.685661i 0.406761 + 0.406761i −1.45845 − 1.45845i 1.68566 + 1.68566i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.40280 + 1.74131i 1.00000 −1.45550 2.20942i 1.00000i 0.500000 0.866025i 2.08551 0.806615i
79.2 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.806615 2.08551i 1.00000 2.24325 + 1.40280i 1.00000i 0.500000 0.866025i −0.344208 + 2.20942i
79.3 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.20942 + 0.344208i 1.00000 −2.51980 + 0.806615i 1.00000i 0.500000 0.866025i −1.74131 1.40280i
79.4 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.40280 1.74131i 1.00000 2.51980 0.806615i 1.00000i 0.500000 0.866025i −0.344208 2.20942i
79.5 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.806615 + 2.08551i 1.00000 1.45550 + 2.20942i 1.00000i 0.500000 0.866025i −1.74131 + 1.40280i
79.6 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 2.20942 0.344208i 1.00000 −2.24325 1.40280i 1.00000i 0.500000 0.866025i 2.08551 + 0.806615i
109.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.40280 1.74131i 1.00000 −1.45550 + 2.20942i 1.00000i 0.500000 + 0.866025i 2.08551 + 0.806615i
109.2 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −0.806615 + 2.08551i 1.00000 2.24325 1.40280i 1.00000i 0.500000 + 0.866025i −0.344208 2.20942i
109.3 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 2.20942 0.344208i 1.00000 −2.51980 0.806615i 1.00000i 0.500000 + 0.866025i −1.74131 + 1.40280i
109.4 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −1.40280 + 1.74131i 1.00000 2.51980 + 0.806615i 1.00000i 0.500000 + 0.866025i −0.344208 + 2.20942i
109.5 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.806615 2.08551i 1.00000 1.45550 2.20942i 1.00000i 0.500000 + 0.866025i −1.74131 1.40280i
109.6 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 2.20942 + 0.344208i 1.00000 −2.24325 + 1.40280i 1.00000i 0.500000 + 0.866025i 2.08551 0.806615i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.n.b 12
3.b odd 2 1 630.2.u.f 12
4.b odd 2 1 1680.2.di.c 12
5.b even 2 1 inner 210.2.n.b 12
5.c odd 4 1 1050.2.i.u 6
5.c odd 4 1 1050.2.i.v 6
7.b odd 2 1 1470.2.n.j 12
7.c even 3 1 inner 210.2.n.b 12
7.c even 3 1 1470.2.g.i 6
7.d odd 6 1 1470.2.g.h 6
7.d odd 6 1 1470.2.n.j 12
15.d odd 2 1 630.2.u.f 12
20.d odd 2 1 1680.2.di.c 12
21.h odd 6 1 630.2.u.f 12
28.g odd 6 1 1680.2.di.c 12
35.c odd 2 1 1470.2.n.j 12
35.i odd 6 1 1470.2.g.h 6
35.i odd 6 1 1470.2.n.j 12
35.j even 6 1 inner 210.2.n.b 12
35.j even 6 1 1470.2.g.i 6
35.k even 12 1 7350.2.a.do 3
35.k even 12 1 7350.2.a.dp 3
35.l odd 12 1 1050.2.i.u 6
35.l odd 12 1 1050.2.i.v 6
35.l odd 12 1 7350.2.a.dn 3
35.l odd 12 1 7350.2.a.dq 3
105.o odd 6 1 630.2.u.f 12
140.p odd 6 1 1680.2.di.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 1.a even 1 1 trivial
210.2.n.b 12 5.b even 2 1 inner
210.2.n.b 12 7.c even 3 1 inner
210.2.n.b 12 35.j even 6 1 inner
630.2.u.f 12 3.b odd 2 1
630.2.u.f 12 15.d odd 2 1
630.2.u.f 12 21.h odd 6 1
630.2.u.f 12 105.o odd 6 1
1050.2.i.u 6 5.c odd 4 1
1050.2.i.u 6 35.l odd 12 1
1050.2.i.v 6 5.c odd 4 1
1050.2.i.v 6 35.l odd 12 1
1470.2.g.h 6 7.d odd 6 1
1470.2.g.h 6 35.i odd 6 1
1470.2.g.i 6 7.c even 3 1
1470.2.g.i 6 35.j even 6 1
1470.2.n.j 12 7.b odd 2 1
1470.2.n.j 12 7.d odd 6 1
1470.2.n.j 12 35.c odd 2 1
1470.2.n.j 12 35.i odd 6 1
1680.2.di.c 12 4.b odd 2 1
1680.2.di.c 12 20.d odd 2 1
1680.2.di.c 12 28.g odd 6 1
1680.2.di.c 12 140.p odd 6 1
7350.2.a.dn 3 35.l odd 12 1
7350.2.a.do 3 35.k even 12 1
7350.2.a.dp 3 35.k even 12 1
7350.2.a.dq 3 35.l odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{6} + 3 T_{11}^{5} + 36 T_{11}^{4} + 17 T_{11}^{3} + 876 T_{11}^{2} + 1323 T_{11} + 2401$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$( 1 - 20 T^{3} + 125 T^{6} )^{2}$$
$7$ $$1 - 12 T^{2} + 120 T^{4} - 790 T^{6} + 5880 T^{8} - 28812 T^{10} + 117649 T^{12}$$
$11$ $$( 1 + 3 T + 3 T^{2} - 16 T^{3} - 81 T^{4} + 69 T^{5} + 1466 T^{6} + 759 T^{7} - 9801 T^{8} - 21296 T^{9} + 43923 T^{10} + 483153 T^{11} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 187083 T^{8} - 1285245 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$1 - 30 T^{2} - 27 T^{4} + 2790 T^{6} + 132426 T^{8} - 614550 T^{10} - 41149843 T^{12} - 177604950 T^{14} + 11060351946 T^{16} + 67343817510 T^{18} - 188345450907 T^{20} - 60479817013470 T^{22} + 582622237229761 T^{24}$$
$19$ $$( 1 - 3 T - 36 T^{2} + 45 T^{3} + 900 T^{4} - 3 T^{5} - 19906 T^{6} - 57 T^{7} + 324900 T^{8} + 308655 T^{9} - 4691556 T^{10} - 7428297 T^{11} + 47045881 T^{12} )^{2}$$
$23$ $$1 + 81 T^{2} + 3582 T^{4} + 90087 T^{6} + 1174464 T^{8} - 7396947 T^{10} - 518277076 T^{12} - 3912984963 T^{14} + 328663180224 T^{16} + 13336109132343 T^{18} + 280509949276542 T^{20} + 3355547408305569 T^{22} + 21914624432020321 T^{24}$$
$29$ $$( 1 + 12 T + 120 T^{2} + 720 T^{3} + 3480 T^{4} + 10092 T^{5} + 24389 T^{6} )^{4}$$
$31$ $$( 1 - 78 T^{2} - 20 T^{3} + 3666 T^{4} + 780 T^{5} - 128922 T^{6} + 24180 T^{7} + 3523026 T^{8} - 595820 T^{9} - 72034638 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$1 + 165 T^{2} + 14838 T^{4} + 911635 T^{6} + 43071816 T^{8} + 1725566145 T^{10} + 64531645932 T^{12} + 2362300052505 T^{14} + 80723517746376 T^{16} + 2339005994868715 T^{18} + 52118170137279798 T^{20} + 793416421448945085 T^{22} + 6582952005840035281 T^{24}$$
$41$ $$( 1 + 9 T + 75 T^{2} + 610 T^{3} + 3075 T^{4} + 15129 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$( 1 - 82 T^{2} + 1849 T^{4} )^{6}$$
$47$ $$1 + 9 T^{2} - 1578 T^{4} - 306297 T^{6} - 2357976 T^{8} + 244410597 T^{10} + 40880244764 T^{12} + 539903008773 T^{14} - 11506170685656 T^{16} - 3301641317626713 T^{18} - 37574210352258858 T^{20} + 473392190122470441 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 + 51 T^{2} - 6213 T^{4} - 119588 T^{6} + 38415969 T^{8} + 385077393 T^{10} - 113446795386 T^{12} + 1081682396937 T^{14} + 303120473491089 T^{16} - 2650591618694852 T^{18} - 386819456525785893 T^{20} + 8919260988641165499 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 + 12 T - 6 T^{2} - 160 T^{3} + 1890 T^{4} - 23628 T^{5} - 457606 T^{6} - 1394052 T^{7} + 6579090 T^{8} - 32860640 T^{9} - 72704166 T^{10} + 8579091588 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 + 6 T - 39 T^{2} + 410 T^{3} + 930 T^{4} - 31074 T^{5} - 22719 T^{6} - 1895514 T^{7} + 3460530 T^{8} + 93062210 T^{9} - 539987799 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} )^{2}$$
$67$ $$1 + 150 T^{2} + 6333 T^{4} - 110030 T^{6} - 1332774 T^{8} + 1742888430 T^{10} + 160875783717 T^{12} + 7823826162270 T^{14} - 26856890139654 T^{16} - 9953135790055070 T^{18} + 2571626601966207453 T^{20} +$$$$27\!\cdots\!50$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{12}$$
$73$ $$( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} )^{3}$$
$79$ $$( 1 - 24 T + 162 T^{2} - 1548 T^{3} + 38034 T^{4} - 316212 T^{5} + 1440614 T^{6} - 24980748 T^{7} + 237370194 T^{8} - 763224372 T^{9} + 6309913122 T^{10} - 73849353576 T^{11} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 278260488 T^{8} - 14237496300 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 6 T - 123 T^{2} - 398 T^{3} + 7074 T^{4} - 14802 T^{5} - 668251 T^{6} - 1317378 T^{7} + 56033154 T^{8} - 280577662 T^{9} - 7717295643 T^{10} + 33504356694 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 157619568 T^{8} - 21247027440 T^{10} + 832972004929 T^{12} )^{2}$$