# Properties

 Label 210.3.h.a Level 210 Weight 3 Character orbit 210.h Analytic conductor 5.722 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 210.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} -\beta_{1} q^{3} -2 q^{4} + \beta_{8} q^{5} -\beta_{4} q^{6} + ( -\beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} -\beta_{1} q^{3} -2 q^{4} + \beta_{8} q^{5} -\beta_{4} q^{6} + ( -\beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9} + ( -\beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} ) q^{10} + ( 5 - \beta_{3} - \beta_{12} + \beta_{13} ) q^{11} + 2 \beta_{1} q^{12} + ( 3 \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( 1 - \beta_{1} - \beta_{3} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{14} + ( -2 - \beta_{3} + \beta_{5} + \beta_{13} ) q^{15} + 4 q^{16} + ( \beta_{1} - 2 \beta_{2} - \beta_{8} - \beta_{9} ) q^{17} -3 \beta_{5} q^{18} + ( -4 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{19} -2 \beta_{8} q^{20} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{21} + ( -6 \beta_{5} - \beta_{10} - \beta_{15} ) q^{22} + ( -2 \beta_{1} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} ) q^{23} + 2 \beta_{4} q^{24} + ( 2 + \beta_{1} - 2 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{25} + ( \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{26} -3 \beta_{1} q^{27} + ( 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{28} + ( 2 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{14} + 2 \beta_{15} ) q^{29} + ( 2 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{14} - \beta_{15} ) q^{30} + ( \beta_{1} - 9 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - 2 \beta_{15} ) q^{31} -4 \beta_{5} q^{32} + ( -6 \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{33} + ( \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{15} ) q^{34} + ( 1 + 5 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} + 2 \beta_{7} - 3 \beta_{11} + 5 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{35} -6 q^{36} + ( 7 \beta_{1} + 12 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 3 \beta_{10} - 4 \beta_{12} - 4 \beta_{13} - 3 \beta_{15} ) q^{37} + ( \beta_{1} + 4 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{38} + ( -9 - 2 \beta_{1} - 5 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{39} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{11} ) q^{40} + ( 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{42} + ( -4 \beta_{1} + 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{43} + ( -10 + 2 \beta_{3} + 2 \beta_{12} - 2 \beta_{13} ) q^{44} + 3 \beta_{8} q^{45} + ( -12 - 3 \beta_{1} - 6 \beta_{3} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} - 6 \beta_{14} - 3 \beta_{15} ) q^{46} + ( 13 \beta_{1} - 2 \beta_{2} - 5 \beta_{6} - 5 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{47} -4 \beta_{1} q^{48} + ( 17 - 3 \beta_{1} + 4 \beta_{3} - 12 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{49} + ( -8 - 3 \beta_{1} - 5 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{50} + ( -2 - 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} - \beta_{12} + \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{51} + ( -6 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( 9 \beta_{5} + 4 \beta_{10} - \beta_{12} - \beta_{13} + 4 \beta_{15} ) q^{53} -3 \beta_{4} q^{54} + ( -17 \beta_{1} + 5 \beta_{2} - 5 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 8 \beta_{8} - 3 \beta_{11} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{56} + ( 3 \beta_{1} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{10} - \beta_{12} - \beta_{13} - 3 \beta_{15} ) q^{57} + ( -2 \beta_{5} + 4 \beta_{12} + 4 \beta_{13} ) q^{58} + ( 10 \beta_{1} + 18 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} - 2 \beta_{15} ) q^{59} + ( 4 + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{13} ) q^{60} + ( 6 \beta_{10} - 6 \beta_{15} ) q^{61} + ( 13 \beta_{1} - 4 \beta_{2} - 3 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{62} + ( -3 \beta_{2} + 3 \beta_{5} - 3 \beta_{6} ) q^{63} -8 q^{64} + ( 24 - 4 \beta_{1} + 4 \beta_{3} + 21 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 5 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + 6 \beta_{15} ) q^{65} + ( -4 \beta_{1} - 8 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 4 \beta_{11} + \beta_{15} ) q^{66} + ( 9 \beta_{1} - 10 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 2 \beta_{10} - 6 \beta_{12} - 6 \beta_{13} - 2 \beta_{15} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{8} + 2 \beta_{9} ) q^{68} + ( \beta_{1} - 8 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - 4 \beta_{15} ) q^{69} + ( -3 - 10 \beta_{1} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + \beta_{8} + 5 \beta_{9} + 5 \beta_{10} + 4 \beta_{13} - 5 \beta_{14} ) q^{70} + ( -27 + 4 \beta_{1} + 11 \beta_{3} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{10} + \beta_{12} - \beta_{13} + 8 \beta_{14} + 4 \beta_{15} ) q^{71} + 6 \beta_{5} q^{72} + ( 3 \beta_{1} - 7 \beta_{2} - 7 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} ) q^{73} + ( 16 + 4 \beta_{1} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{10} + 6 \beta_{12} - 6 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} ) q^{74} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} - \beta_{8} + 5 \beta_{9} - 3 \beta_{11} ) q^{75} + ( 8 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - 4 \beta_{15} ) q^{76} + ( -9 \beta_{1} + 15 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{77} + ( \beta_{1} + 8 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - 4 \beta_{12} - 4 \beta_{13} - 2 \beta_{15} ) q^{78} + ( -40 + 2 \beta_{1} + 8 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{14} + 2 \beta_{15} ) q^{79} + 4 \beta_{8} q^{80} + 9 q^{81} + ( -7 \beta_{1} - 2 \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{82} + ( 5 \beta_{1} + 10 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{84} + ( -25 - 5 \beta_{1} + 10 \beta_{3} + 15 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + 10 \beta_{12} + 5 \beta_{14} + 5 \beta_{15} ) q^{85} + ( 28 + 2 \beta_{1} + 16 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{86} + ( -4 \beta_{1} - 6 \beta_{6} - 6 \beta_{7} ) q^{87} + ( 12 \beta_{5} + 2 \beta_{10} + 2 \beta_{15} ) q^{88} + ( \beta_{1} - 12 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} + 6 \beta_{10} + 12 \beta_{11} - 6 \beta_{15} ) q^{89} + ( -3 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{11} ) q^{90} + ( 22 + 6 \beta_{1} - 6 \beta_{3} + 15 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} - 8 \beta_{14} - 6 \beta_{15} ) q^{91} + ( 4 \beta_{1} + 12 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{10} - 6 \beta_{12} - 6 \beta_{13} - 4 \beta_{15} ) q^{92} + ( -6 \beta_{1} - 25 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + \beta_{12} + \beta_{13} ) q^{93} + ( -6 \beta_{1} + 14 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + 10 \beta_{8} - 10 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} + 2 \beta_{15} ) q^{94} + ( -35 - 13 \beta_{1} - 11 \beta_{3} - 34 \beta_{5} + 13 \beta_{6} - 13 \beta_{7} + 10 \beta_{10} + 8 \beta_{13} - 8 \beta_{14} + 2 \beta_{15} ) q^{95} -4 \beta_{4} q^{96} + ( 33 \beta_{1} - 5 \beta_{2} - 5 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{97} + ( 26 \beta_{1} - 11 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{98} + ( 15 - 3 \beta_{3} - 3 \beta_{12} + 3 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 32q^{4} + 48q^{9} + O(q^{10})$$ $$16q - 32q^{4} + 48q^{9} + 96q^{11} + 16q^{14} - 24q^{15} + 64q^{16} + 24q^{21} + 24q^{25} + 64q^{29} + 24q^{30} - 8q^{35} - 96q^{36} - 144q^{39} - 192q^{44} - 176q^{46} + 224q^{49} - 96q^{50} - 48q^{51} - 32q^{56} + 48q^{60} - 128q^{64} + 368q^{65} - 56q^{70} - 384q^{71} + 224q^{74} - 608q^{79} + 144q^{81} - 48q^{84} - 440q^{85} + 416q^{86} + 224q^{91} - 560q^{95} + 288q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + 1591101 x^{4} - 1183860 x^{3} + 619650 x^{2} - 202500 x + 33750$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1304 \nu^{14} + 9128 \nu^{13} - 115446 \nu^{12} + 574012 \nu^{11} - 3591842 \nu^{10} + 12914984 \nu^{9} - 49995515 \nu^{8} + 128433344 \nu^{7} - 320844898 \nu^{6} + 561338382 \nu^{5} - 866468654 \nu^{4} + 924341564 \nu^{3} - 755470905 \nu^{2} + 368877150 \nu - 99070875$$$$)/560625$$ $$\beta_{2}$$ $$=$$ $$($$$$-19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + 191153301 \nu^{9} - 739735340 \nu^{8} + 1899973946 \nu^{7} - 4742613417 \nu^{6} + 8293098988 \nu^{5} - 12773014036 \nu^{4} + 13602334971 \nu^{3} - 11039578230 \nu^{2} + 5354640900 \nu - 1406219625$$$$)/3027375$$ $$\beta_{3}$$ $$=$$ $$($$$$-181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + 1796001 \nu^{9} - 6961130 \nu^{8} + 17893811 \nu^{7} - 44790412 \nu^{6} + 78463393 \nu^{5} - 121536126 \nu^{4} + 129995121 \nu^{3} - 106932015 \nu^{2} + 52505910 \nu - 14271525$$$$)/26325$$ $$\beta_{4}$$ $$=$$ $$($$$$-1984 \nu^{15} + 14880 \nu^{14} - 171958 \nu^{13} + 892047 \nu^{12} - 4997626 \nu^{11} + 18170922 \nu^{10} - 58997141 \nu^{9} + 142860075 \nu^{8} - 233442698 \nu^{7} + 259430001 \nu^{6} + 304752040 \nu^{5} - 1154379162 \nu^{4} + 2711291745 \nu^{3} - 2854944621 \nu^{2} + 2020897980 \nu - 575687250$$$$)/85899825$$ $$\beta_{5}$$ $$=$$ $$($$$$5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + 15373400596 \nu^{11} - 58317525310 \nu^{10} + 223952035489 \nu^{9} - 608926339653 \nu^{8} + 1529887949640 \nu^{7} - 2869994011087 \nu^{6} + 4569918940190 \nu^{5} - 5417408634332 \nu^{4} + 4875031263099 \nu^{3} - 3000039760455 \nu^{2} + 1143288079650 \nu - 200352316500$$$$)/ 1213801875$$ $$\beta_{6}$$ $$=$$ $$($$$$-399728914 \nu^{15} + 3232898035 \nu^{14} - 38445891245 \nu^{13} + 214537975455 \nu^{12} - 1293560948733 \nu^{11} + 5161735252520 \nu^{10} - 19664320389392 \nu^{9} + 56145887333099 \nu^{8} - 141581224758025 \nu^{7} + 279967659388431 \nu^{6} - 454977456719385 \nu^{5} + 575402543557786 \nu^{4} - 544225821024222 \nu^{3} + 368154613738290 \nu^{2} - 155333941686450 \nu + 33279515835750$$$$)/ 83752329375$$ $$\beta_{7}$$ $$=$$ $$($$$$399728914 \nu^{15} - 2568229811 \nu^{14} + 33793213677 \nu^{13} - 155694952029 \nu^{12} + 1000987616561 \nu^{11} - 3331119827068 \nu^{10} + 13082276658338 \nu^{9} - 30671013312184 \nu^{8} + 76145779518861 \nu^{7} - 116580706899693 \nu^{6} + 169214780706043 \nu^{5} - 134875076277512 \nu^{4} + 74756402522688 \nu^{3} + 14002064235165 \nu^{2} - 30555343699200 \nu + 16014851920125$$$$)/ 83752329375$$ $$\beta_{8}$$ $$=$$ $$($$$$-447494447 \nu^{15} + 3483298596 \nu^{14} - 42073363272 \nu^{13} + 228052000574 \nu^{12} - 1387056367957 \nu^{11} + 5398872205078 \nu^{10} - 20631957152977 \nu^{9} + 57534505613712 \nu^{8} - 144656309564376 \nu^{7} + 279021438726355 \nu^{6} - 448287901742858 \nu^{5} + 550515081942494 \nu^{4} - 507632737721937 \nu^{3} + 330359822998215 \nu^{2} - 133431969127950 \nu + 27118230861375$$$$)/ 83752329375$$ $$\beta_{9}$$ $$=$$ $$($$$$447494447 \nu^{15} - 3034312245 \nu^{14} + 38930458815 \nu^{13} - 188284929725 \nu^{12} + 1189311700804 \nu^{11} - 4160752761880 \nu^{10} + 16179113496331 \nu^{9} - 40281239595702 \nu^{8} + 100313526189915 \nu^{7} - 168073717530868 \nu^{6} + 253982897346575 \nu^{5} - 249628471095518 \nu^{4} + 185854130658771 \nu^{3} - 65010730303320 \nu^{2} + 2797068934350 \nu + 8909332197750$$$$)/ 83752329375$$ $$\beta_{10}$$ $$=$$ $$($$$$-483208394 \nu^{15} + 3624062955 \nu^{14} - 44453885935 \nu^{13} + 233985303760 \nu^{12} - 1435876781253 \nu^{11} + 5444406856130 \nu^{10} - 20874093726577 \nu^{9} + 56702007359379 \nu^{8} - 142064690854595 \nu^{7} + 265942990325966 \nu^{6} - 421119834507795 \nu^{5} + 496610692750426 \nu^{4} - 440957572810407 \nu^{3} + 267075981404040 \nu^{2} - 98295840754200 \nu + 16389579233250$$$$)/ 83752329375$$ $$\beta_{11}$$ $$=$$ $$($$$$-567713312 \nu^{15} + 4257849840 \nu^{14} - 52266459230 \nu^{13} + 275154595755 \nu^{12} - 1690195150544 \nu^{11} + 6411443031015 \nu^{10} - 24618050055196 \nu^{9} + 66931017082167 \nu^{8} - 168101562328660 \nu^{7} + 315263632646343 \nu^{6} - 501506314001410 \nu^{5} + 593939025308073 \nu^{4} - 532857064795086 \nu^{3} + 326697174195345 \nu^{2} - 123640459408350 \nu + 21472387601625$$$$)/ 83752329375$$ $$\beta_{12}$$ $$=$$ $$($$$$34162567 \nu^{15} - 260736947 \nu^{14} + 3176839329 \nu^{13} - 16958406448 \nu^{12} + 103706772500 \nu^{11} - 398328867651 \nu^{10} + 1526616776483 \nu^{9} - 4202958874492 \nu^{8} + 10569496137537 \nu^{7} - 20109267310697 \nu^{6} + 32195169162241 \nu^{5} - 38877338622975 \nu^{4} + 35479953139413 \nu^{3} - 22492379099160 \nu^{2} + 8834413324050 \nu - 1667961659250$$$$)/ 3641405625$$ $$\beta_{13}$$ $$=$$ $$($$$$34162567 \nu^{15} - 251701558 \nu^{14} + 3113591606 \nu^{13} - 16157410637 \nu^{12} + 99723018033 \nu^{11} - 373359357129 \nu^{10} + 1436779569089 \nu^{9} - 3854434623527 \nu^{8} + 9673185200158 \nu^{7} - 17863512405754 \nu^{6} + 28258784429004 \nu^{5} - 32775290830761 \nu^{4} + 28949936815989 \nu^{3} - 17118376991730 \nu^{2} + 6194900930400 \nu - 947112736500$$$$)/ 3641405625$$ $$\beta_{14}$$ $$=$$ $$($$$$-1650986060 \nu^{15} + 13415356753 \nu^{14} - 159200027621 \nu^{13} + 891510559422 \nu^{12} - 5368094554979 \nu^{11} + 21484837848419 \nu^{10} - 81782053320718 \nu^{9} + 234176501346290 \nu^{8} - 590312762276383 \nu^{7} + 1170962757989076 \nu^{6} - 1903359133567699 \nu^{5} + 2416998273055543 \nu^{4} - 2288527543048278 \nu^{3} + 1559133346137285 \nu^{2} - 660049681351050 \nu + 143830007146125$$$$)/ 83752329375$$ $$\beta_{15}$$ $$=$$ $$($$$$2019305898 \nu^{15} - 15144794235 \nu^{14} + 185881965095 \nu^{13} - 978536727220 \nu^{12} + 6009893091901 \nu^{11} - 22795839307010 \nu^{10} + 87512193219009 \nu^{9} - 237899077894293 \nu^{8} + 597373089091315 \nu^{7} - 1120169176032722 \nu^{6} + 1781788381868015 \nu^{5} - 2110164042242842 \nu^{4} + 1894947099433719 \nu^{3} - 1163391720079530 \nu^{2} + 442484852310900 \nu - 77444936604000$$$$)/ 83752329375$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{5} - 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 4 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 32$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-19 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 32 \beta_{9} - 32 \beta_{8} + \beta_{7} + 5 \beta_{6} + 45 \beta_{5} + 19 \beta_{4} - 9 \beta_{3} + 3 \beta_{2} + 30 \beta_{1} - 49$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} + 68 \beta_{14} - 36 \beta_{13} + 12 \beta_{12} - 4 \beta_{11} - 13 \beta_{10} + 70 \beta_{9} - 62 \beta_{8} - 24 \beta_{7} - 96 \beta_{6} + 92 \beta_{5} + 38 \beta_{4} + 112 \beta_{3} - 48 \beta_{2} + 42 \beta_{1} + 498$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$344 \beta_{15} + 180 \beta_{14} + 85 \beta_{13} + 215 \beta_{12} - 4 \beta_{11} - 244 \beta_{10} - 566 \beta_{9} + 586 \beta_{8} - 13 \beta_{7} - 297 \beta_{6} - 989 \beta_{5} - 507 \beta_{4} + 295 \beta_{3} - 125 \beta_{2} - 584 \beta_{1} + 1327$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$120 \beta_{15} - 605 \beta_{14} + 365 \beta_{13} + 115 \beta_{12} - \beta_{11} + 45 \beta_{10} - 931 \beta_{9} + 963 \beta_{8} + 489 \beta_{7} + 943 \beta_{6} - 1599 \beta_{5} - 808 \beta_{4} - 1040 \beta_{3} + 494 \beta_{2} - 49 \beta_{1} - 4337$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-6370 \beta_{15} - 4872 \beta_{14} - 2107 \beta_{13} - 4319 \beta_{12} + 846 \beta_{11} + 5292 \beta_{10} + 10854 \beta_{9} - 10700 \beta_{8} + 879 \beta_{7} + 10237 \beta_{6} + 19715 \beta_{5} + 11633 \beta_{4} - 8323 \beta_{3} + 3899 \beta_{2} + 14498 \beta_{1} - 35061$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-8803 \beta_{15} + 21908 \beta_{14} - 16804 \beta_{13} - 13436 \beta_{12} + 3384 \beta_{11} + 2923 \beta_{10} + 49054 \beta_{9} - 55150 \beta_{8} - 29010 \beta_{7} - 31574 \beta_{6} + 94000 \beta_{5} + 54162 \beta_{4} + 38712 \beta_{3} - 18376 \beta_{2} - 6676 \beta_{1} + 154534$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$122447 \beta_{15} + 128586 \beta_{14} + 47961 \beta_{13} + 76947 \beta_{12} - 27482 \beta_{11} - 118643 \beta_{10} - 212674 \beta_{9} + 184402 \beta_{8} - 45245 \beta_{7} - 295393 \beta_{6} - 361575 \beta_{5} - 245591 \beta_{4} + 225399 \beta_{3} - 106617 \beta_{2} - 394764 \beta_{1} + 911441$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$254281 \beta_{15} - 381496 \beta_{14} + 415248 \beta_{13} + 442932 \beta_{12} - 162784 \beta_{11} - 188791 \beta_{10} - 1304300 \beta_{9} + 1489792 \beta_{8} + 762798 \beta_{7} + 389310 \beta_{6} - 2535724 \beta_{5} - 1645672 \beta_{4} - 689624 \beta_{3} + 312672 \beta_{2} + 112548 \beta_{1} - 2653032$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-2442238 \beta_{15} - 3332538 \beta_{14} - 1012319 \beta_{13} - 1151557 \beta_{12} + 638966 \beta_{11} + 2625080 \beta_{10} + 4021312 \beta_{9} - 2740472 \beta_{8} + 1822547 \beta_{7} + 7762281 \beta_{6} + 5946325 \beta_{5} + 4632093 \beta_{4} - 5953937 \beta_{3} + 2741299 \beta_{2} + 10726000 \beta_{1} - 23347091$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-3483639 \beta_{15} + 2928985 \beta_{14} - 5314483 \beta_{13} - 6157373 \beta_{12} + 2840117 \beta_{11} + 3619731 \beta_{10} + 17283722 \beta_{9} - 19271586 \beta_{8} - 9291408 \beta_{7} - 83516 \beta_{6} + 32596899 \beta_{5} + 23412401 \beta_{4} + 5408710 \beta_{3} - 2290588 \beta_{2} + 1662038 \beta_{1} + 20089762$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$49916468 \beta_{15} + 84671340 \beta_{14} + 19129253 \beta_{13} + 10768693 \beta_{12} - 11306262 \beta_{11} - 56045370 \beta_{10} - 68231238 \beta_{9} + 25058836 \beta_{8} - 63490515 \beta_{7} - 191572397 \beta_{6} - 77428447 \beta_{5} - 70485751 \beta_{4} + 153439325 \beta_{3} - 68120299 \beta_{2} - 280857766 \beta_{1} + 587892255$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$189382655 \beta_{15} - 60250246 \beta_{14} + 274871156 \beta_{13} + 313288780 \beta_{12} - 170967462 \beta_{11} - 233773547 \beta_{10} - 901930838 \beta_{9} + 961013426 \beta_{8} + 423040290 \beta_{7} - 215339516 \beta_{6} - 1622428532 \beta_{5} - 1262524236 \beta_{4} - 115088724 \beta_{3} + 42760214 \beta_{2} - 306867574 \beta_{1} - 403879466$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-1023202897 \beta_{15} - 2101665624 \beta_{14} - 288089139 \beta_{13} + 136058607 \beta_{12} + 111268180 \beta_{11} + 1130711311 \beta_{10} + 868910138 \beta_{9} + 391402090 \beta_{8} + 2010114319 \beta_{7} + 4490400413 \beta_{6} + 292095141 \beta_{5} + 502005055 \beta_{4} - 3847971231 \beta_{3} + 1651106253 \beta_{2} + 7018115652 \beta_{1} - 14495944639$$$$)/4$$

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 0.5 − 1.83656i 0.5 + 3.55177i 0.5 − 0.442923i 0.5 − 4.10071i 0.5 + 2.68650i 0.5 − 0.971291i 0.5 − 4.96598i 0.5 + 0.422343i 0.5 + 1.83656i 0.5 − 3.55177i 0.5 + 0.442923i 0.5 + 4.10071i 0.5 − 2.68650i 0.5 + 0.971291i 0.5 + 4.96598i 0.5 − 0.422343i
1.41421i −1.73205 −2.00000 −4.91728 0.905717i 2.44949i 1.91369 + 6.73333i 2.82843i 3.00000 −1.28088 + 6.95409i
139.2 1.41421i −1.73205 −2.00000 1.38028 + 4.80571i 2.44949i −5.24961 4.63050i 2.82843i 3.00000 6.79630 1.95201i
139.3 1.41421i −1.73205 −2.00000 2.40341 4.38447i 2.44949i −6.94781 0.853218i 2.82843i 3.00000 −6.20058 3.39894i
139.4 1.41421i −1.73205 −2.00000 4.59769 1.96501i 2.44949i 6.81963 + 1.57881i 2.82843i 3.00000 −2.77894 6.50212i
139.5 1.41421i 1.73205 −2.00000 −4.59769 + 1.96501i 2.44949i −6.81963 + 1.57881i 2.82843i 3.00000 2.77894 + 6.50212i
139.6 1.41421i 1.73205 −2.00000 −2.40341 + 4.38447i 2.44949i 6.94781 0.853218i 2.82843i 3.00000 6.20058 + 3.39894i
139.7 1.41421i 1.73205 −2.00000 −1.38028 4.80571i 2.44949i 5.24961 4.63050i 2.82843i 3.00000 −6.79630 + 1.95201i
139.8 1.41421i 1.73205 −2.00000 4.91728 + 0.905717i 2.44949i −1.91369 + 6.73333i 2.82843i 3.00000 1.28088 6.95409i
139.9 1.41421i −1.73205 −2.00000 −4.91728 + 0.905717i 2.44949i 1.91369 6.73333i 2.82843i 3.00000 −1.28088 6.95409i
139.10 1.41421i −1.73205 −2.00000 1.38028 4.80571i 2.44949i −5.24961 + 4.63050i 2.82843i 3.00000 6.79630 + 1.95201i
139.11 1.41421i −1.73205 −2.00000 2.40341 + 4.38447i 2.44949i −6.94781 + 0.853218i 2.82843i 3.00000 −6.20058 + 3.39894i
139.12 1.41421i −1.73205 −2.00000 4.59769 + 1.96501i 2.44949i 6.81963 1.57881i 2.82843i 3.00000 −2.77894 + 6.50212i
139.13 1.41421i 1.73205 −2.00000 −4.59769 1.96501i 2.44949i −6.81963 1.57881i 2.82843i 3.00000 2.77894 6.50212i
139.14 1.41421i 1.73205 −2.00000 −2.40341 4.38447i 2.44949i 6.94781 + 0.853218i 2.82843i 3.00000 6.20058 3.39894i
139.15 1.41421i 1.73205 −2.00000 −1.38028 + 4.80571i 2.44949i 5.24961 + 4.63050i 2.82843i 3.00000 −6.79630 1.95201i
139.16 1.41421i 1.73205 −2.00000 4.91728 0.905717i 2.44949i −1.91369 6.73333i 2.82843i 3.00000 1.28088 + 6.95409i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.16 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.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.h.a 16
3.b odd 2 1 630.3.h.e 16
4.b odd 2 1 1680.3.bd.a 16
5.b even 2 1 inner 210.3.h.a 16
5.c odd 4 2 1050.3.f.e 16
7.b odd 2 1 inner 210.3.h.a 16
15.d odd 2 1 630.3.h.e 16
20.d odd 2 1 1680.3.bd.a 16
21.c even 2 1 630.3.h.e 16
28.d even 2 1 1680.3.bd.a 16
35.c odd 2 1 inner 210.3.h.a 16
35.f even 4 2 1050.3.f.e 16
105.g even 2 1 630.3.h.e 16
140.c even 2 1 1680.3.bd.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 1.a even 1 1 trivial
210.3.h.a 16 5.b even 2 1 inner
210.3.h.a 16 7.b odd 2 1 inner
210.3.h.a 16 35.c odd 2 1 inner
630.3.h.e 16 3.b odd 2 1
630.3.h.e 16 15.d odd 2 1
630.3.h.e 16 21.c even 2 1
630.3.h.e 16 105.g even 2 1
1050.3.f.e 16 5.c odd 4 2
1050.3.f.e 16 35.f even 4 2
1680.3.bd.a 16 4.b odd 2 1
1680.3.bd.a 16 20.d odd 2 1
1680.3.bd.a 16 28.d even 2 1
1680.3.bd.a 16 140.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{8}$$
$3$ $$( 1 - 3 T^{2} )^{8}$$
$5$ $$1 - 12 T^{2} - 376 T^{4} - 3300 T^{6} + 588750 T^{8} - 2062500 T^{10} - 146875000 T^{12} - 2929687500 T^{14} + 152587890625 T^{16}$$
$7$ $$1 - 112 T^{2} + 3932 T^{4} - 24720 T^{6} - 1183546 T^{8} - 59352720 T^{10} + 22667197532 T^{12} - 1550224166512 T^{14} + 33232930569601 T^{16}$$
$11$ $$( 1 - 24 T + 512 T^{2} - 6936 T^{3} + 89778 T^{4} - 839256 T^{5} + 7496192 T^{6} - 42517464 T^{7} + 214358881 T^{8} )^{4}$$
$13$ $$( 1 + 308 T^{2} + 54824 T^{4} + 3243420 T^{6} - 106725490 T^{8} + 92635318620 T^{10} + 44721621048104 T^{12} + 7175810217724148 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 1316 T^{2} + 881960 T^{4} + 398466540 T^{6} + 132906424526 T^{8} + 33280323887340 T^{10} + 6152339032664360 T^{12} + 766730864194365476 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$( 1 - 1092 T^{2} + 576680 T^{4} - 222586380 T^{6} + 81331407246 T^{8} - 29007679627980 T^{10} + 9794081134483880 T^{12} - 2416939891620247812 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$( 1 - 1300 T^{2} + 1274456 T^{4} - 947494620 T^{6} + 528617435630 T^{8} - 265147841955420 T^{10} + 99803905057282136 T^{12} - 28489011761626417300 T^{14} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 - 16 T + 1276 T^{2} - 55408 T^{3} + 751270 T^{4} - 46598128 T^{5} + 902490556 T^{6} - 9517173136 T^{7} + 500246412961 T^{8} )^{4}$$
$31$ $$( 1 - 3948 T^{2} + 5903432 T^{4} - 3942646212 T^{6} + 1976573418510 T^{8} - 3641116572352452 T^{10} + 5034984242942397512 T^{12} -$$$$31\!\cdots\!28$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 - 2416 T^{2} + 5835932 T^{4} - 9959371920 T^{6} + 14384225705606 T^{8} - 18665466436959120 T^{10} + 20498591244480089372 T^{12} -$$$$15\!\cdots\!96$$$$T^{14} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$( 1 - 12516 T^{2} + 70001768 T^{4} - 228356671020 T^{6} + 475054496880078 T^{8} - 645281375058146220 T^{10} +$$$$55\!\cdots\!28$$$$T^{12} -$$$$28\!\cdots\!96$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$( 1 - 1912 T^{2} + 2635964 T^{4} - 3280241736 T^{6} + 4054697588678 T^{8} - 11214493727278536 T^{10} + 30809675156546242364 T^{12} -$$$$76\!\cdots\!12$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 + 6768 T^{2} + 28283420 T^{4} + 81813152400 T^{6} + 198252999971526 T^{8} + 399222085316384400 T^{10} +$$$$67\!\cdots\!20$$$$T^{12} +$$$$78\!\cdots\!88$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$( 1 - 6868 T^{2} + 39779480 T^{4} - 162177493980 T^{6} + 501674308076846 T^{8} - 1279658434876804380 T^{10} +$$$$24\!\cdots\!80$$$$T^{12} -$$$$33\!\cdots\!88$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$( 1 - 11736 T^{2} + 79787516 T^{4} - 412369170408 T^{6} + 1633889740528710 T^{8} - 4996826103104253288 T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$20\!\cdots\!16$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$61$ $$( 1 - 15944 T^{2} + 96425692 T^{4} - 269693317496 T^{6} + 596407807846918 T^{8} - 3734130792812134136 T^{10} +$$$$18\!\cdots\!52$$$$T^{12} -$$$$42\!\cdots\!24$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 - 18688 T^{2} + 141351740 T^{4} - 579001624320 T^{6} + 2070725817126086 T^{8} - 11667531790868862720 T^{10} +$$$$57\!\cdots\!40$$$$T^{12} -$$$$15\!\cdots\!68$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 + 96 T + 14048 T^{2} + 646080 T^{3} + 72093138 T^{4} + 3256889280 T^{5} + 356983294688 T^{6} + 12297627256416 T^{7} + 645753531245761 T^{8} )^{4}$$
$73$ $$( 1 + 18612 T^{2} + 177429992 T^{4} + 1152928560348 T^{6} + 6336699932034510 T^{8} + 32741143112545547868 T^{10} +$$$$14\!\cdots\!52$$$$T^{12} +$$$$42\!\cdots\!52$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 + 152 T + 28916 T^{2} + 2541000 T^{3} + 270417254 T^{4} + 15858381000 T^{5} + 1126280542196 T^{6} + 36949293239192 T^{7} + 1517108809906561 T^{8} )^{4}$$
$83$ $$( 1 + 39632 T^{2} + 756012572 T^{4} + 9074582537520 T^{6} + 74694570513655814 T^{8} +$$$$43\!\cdots\!20$$$$T^{10} +$$$$17\!\cdots\!52$$$$T^{12} +$$$$42\!\cdots\!52$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 38508 T^{2} + 778838024 T^{4} - 10318651040964 T^{6} + 96273467949842958 T^{8} -$$$$64\!\cdots\!24$$$$T^{10} +$$$$30\!\cdots\!44$$$$T^{12} -$$$$95\!\cdots\!68$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 + 54804 T^{2} + 1439825768 T^{4} + 23691050976252 T^{6} + 266516143722806094 T^{8} +$$$$20\!\cdots\!12$$$$T^{10} +$$$$11\!\cdots\!48$$$$T^{12} +$$$$38\!\cdots\!64$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$