# Properties

 Label 1050.3.p.f Level $1050$ Weight $3$ Character orbit 1050.p Analytic conductor $28.610$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} + 56 x^{10} + 300 x^{9} + 1007 x^{8} + 12456 x^{7} + 209990 x^{6} - 250384 x^{5} + 4799806 x^{4} + 51487320 x^{3} - 123648876 x^{2} + 379489320 x + 6882692292$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{5} ) q^{2} + ( 2 - \beta_{3} ) q^{3} -2 \beta_{3} q^{4} + ( 2 \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{11} ) q^{7} + 2 \beta_{5} q^{8} + ( 3 - 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{5} ) q^{2} + ( 2 - \beta_{3} ) q^{3} -2 \beta_{3} q^{4} + ( 2 \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{11} ) q^{7} + 2 \beta_{5} q^{8} + ( 3 - 3 \beta_{3} ) q^{9} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{11} + ( -2 - 2 \beta_{3} ) q^{12} + ( 2 + \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{14} + ( -4 + 4 \beta_{3} ) q^{16} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{17} + 3 \beta_{4} q^{18} + ( -\beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{19} + ( 1 + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{11} ) q^{21} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{22} + ( -11 + 2 \beta_{1} + 11 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + ( -2 \beta_{4} + 4 \beta_{5} ) q^{24} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{26} + ( 3 - 6 \beta_{3} ) q^{27} + ( 2 - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{11} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{29} + ( -22 - 3 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{31} -4 \beta_{4} q^{32} + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{34} -6 q^{36} + ( 13 + \beta_{1} - \beta_{2} - 13 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{37} + ( -8 + \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{11} ) q^{38} + ( \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{39} + ( -4 + \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 5 \beta_{11} ) q^{41} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{42} + ( -9 + 2 \beta_{1} - \beta_{2} - \beta_{4} + 16 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 4 \beta_{11} ) q^{43} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{44} + ( 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{46} + ( -5 - \beta_{2} - 5 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{47} + ( -4 + 8 \beta_{3} ) q^{48} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 15 \beta_{5} + 3 \beta_{6} + \beta_{7} - 4 \beta_{9} - 3 \beta_{11} ) q^{49} + ( -3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{51} + ( -8 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} ) q^{53} + ( 3 \beta_{4} + 3 \beta_{5} ) q^{54} + ( 6 - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{9} ) q^{56} + ( \beta_{1} - \beta_{2} - \beta_{4} + 4 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{57} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{58} + ( -4 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 16 \beta_{5} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{59} + ( 11 - 2 \beta_{2} + 11 \beta_{3} + 14 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} + 4 \beta_{8} - 6 \beta_{9} + 2 \beta_{10} ) q^{61} + ( 5 + \beta_{1} - \beta_{2} - 10 \beta_{3} - 21 \beta_{4} + 10 \beta_{5} - 5 \beta_{6} + 6 \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{11} ) q^{62} + ( 3 - 3 \beta_{4} + 3 \beta_{7} ) q^{63} + 8 q^{64} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{66} + ( \beta_{1} + 21 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 8 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{67} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{68} + ( -11 + 5 \beta_{1} + \beta_{2} + 22 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{69} + ( -10 - 8 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{71} + ( -6 \beta_{4} + 6 \beta_{5} ) q^{72} + ( 2 - \beta_{3} + 12 \beta_{4} - 24 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 \beta_{2} - 12 \beta_{3} + 11 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{76} + ( 13 - 6 \beta_{1} + \beta_{2} + 2 \beta_{3} + 39 \beta_{4} - 35 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 7 \beta_{10} - \beta_{11} ) q^{77} + ( -3 - 3 \beta_{1} - \beta_{2} - \beta_{4} + 7 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{78} + ( -14 + 4 \beta_{1} + 5 \beta_{2} + 14 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - \beta_{8} + 4 \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{79} -9 \beta_{3} q^{81} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} ) q^{82} + ( 10 + 5 \beta_{1} + 6 \beta_{2} - 20 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} + \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( 2 - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} + 4 \beta_{11} ) q^{84} + ( -37 + \beta_{1} + \beta_{2} + 37 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{86} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{87} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{88} + ( 13 - 5 \beta_{1} + 13 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 5 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} + \beta_{11} ) q^{89} + ( 19 - 4 \beta_{1} + 3 \beta_{2} + 15 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + 7 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{91} + ( 22 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 2 \beta_{10} + 4 \beta_{11} ) q^{92} + ( -33 - 3 \beta_{1} - \beta_{2} + 33 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{93} + ( -26 + \beta_{1} - \beta_{2} + 13 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + \beta_{6} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{94} + ( -4 \beta_{4} - 4 \beta_{5} ) q^{96} + ( 27 - 10 \beta_{1} - 4 \beta_{2} - 54 \beta_{3} - 42 \beta_{4} + 24 \beta_{5} + 6 \beta_{6} - 7 \beta_{7} + \beta_{8} + \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{97} + ( 13 + \beta_{1} + \beta_{2} - 22 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} - 6 \beta_{11} ) q^{98} + ( -3 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 18 q^{3} - 12 q^{4} + 8 q^{7} + 18 q^{9} + O(q^{10})$$ $$12 q + 18 q^{3} - 12 q^{4} + 8 q^{7} + 18 q^{9} - 4 q^{11} - 36 q^{12} + 8 q^{14} - 24 q^{16} - 24 q^{17} + 12 q^{19} + 18 q^{21} + 24 q^{22} - 60 q^{23} - 24 q^{26} + 4 q^{28} - 24 q^{29} - 198 q^{31} - 12 q^{33} - 72 q^{36} + 70 q^{37} - 60 q^{38} - 36 q^{39} + 36 q^{42} - 84 q^{43} - 8 q^{44} + 32 q^{46} - 60 q^{47} + 28 q^{49} - 24 q^{51} - 72 q^{52} + 44 q^{53} + 40 q^{56} + 24 q^{57} + 8 q^{58} - 48 q^{59} + 186 q^{61} + 30 q^{63} + 96 q^{64} + 36 q^{66} + 152 q^{67} + 48 q^{68} - 136 q^{71} + 18 q^{73} - 64 q^{74} + 132 q^{77} - 48 q^{78} - 70 q^{79} - 54 q^{81} - 84 q^{82} - 12 q^{84} - 208 q^{86} - 36 q^{87} - 24 q^{88} + 168 q^{89} + 292 q^{91} + 240 q^{92} - 198 q^{93} - 204 q^{94} + 48 q^{98} - 24 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 56 x^{10} + 300 x^{9} + 1007 x^{8} + 12456 x^{7} + 209990 x^{6} - 250384 x^{5} + 4799806 x^{4} + 51487320 x^{3} - 123648876 x^{2} + 379489320 x + 6882692292$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-7312640379351403241641847 \nu^{11} - 1220656239404306431507712517 \nu^{10} + 4805966189351074809209907371 \nu^{9} - 77082624202448882766051479335 \nu^{8} - 429689890669635037055974600508 \nu^{7} - 1687020719579756648102039790584 \nu^{6} - 15249047250122917892983496922486 \nu^{5} - 226048349092969743473019461171842 \nu^{4} + 240508681898520952709559210664608 \nu^{3} - 4554845829787427161940232655265160 \nu^{2} - 53531706331889152181397061344020988 \nu + 153749869847424591394514000575649004$$$$)/$$$$21\!\cdots\!72$$ $$\beta_{3}$$ $$=$$ $$($$$$-97053122943645 \nu^{11} + 103711575797434 \nu^{10} - 11848919253611537 \nu^{9} - 8793522094799700 \nu^{8} - 481962685768778140 \nu^{7} - 2640903328657569756 \nu^{6} - 27413887679403990002 \nu^{5} - 46390039494241191392 \nu^{4} - 1938952908037817336424 \nu^{3} - 1117799984051092801872 \nu^{2} - 17365236288539319211500 \nu - 128477218889527196728032$$$$)/$$$$24\!\cdots\!84$$ $$\beta_{4}$$ $$=$$ $$($$$$24253856643754123762567993 \nu^{11} - 332561852884708150820621961 \nu^{10} + 765684251710026347286627107 \nu^{9} - 20479146067519574219516427991 \nu^{8} - 49825312716093174099280139432 \nu^{7} - 1172994040482233237509999469768 \nu^{6} - 1788113610270522000593658743826 \nu^{5} - 85643806223010921556267095241582 \nu^{4} - 79939059340227051179227009423872 \nu^{3} - 1149213266765744428942761817781064 \nu^{2} - 9153428894317729744693308136177092 \nu - 24885514122548791790854235417378148$$$$)/$$$$43\!\cdots\!44$$ $$\beta_{5}$$ $$=$$ $$($$$$-173144824834609 \nu^{11} + 327589951576233 \nu^{10} - 12664102649046059 \nu^{9} - 78134392296611963 \nu^{8} - 396952612132669540 \nu^{7} - 3545035653960153094 \nu^{6} - 31953918746048973414 \nu^{5} - 56620595952907578158 \nu^{4} - 759501373771218234876 \nu^{3} - 6846417922576431521664 \nu^{2} + 21805736705260376845284 \nu - 44431041218766354370764$$$$)/$$$$16\!\cdots\!08$$ $$\beta_{6}$$ $$=$$ $$($$$$-284500915977146 \nu^{11} - 6413944368767417 \nu^{10} + 20322414788293800 \nu^{9} - 384230190964527625 \nu^{8} - 1432009629271527636 \nu^{7} - 7033702392467976452 \nu^{6} - 70690588629362801072 \nu^{5} - 1473116746214172403554 \nu^{4} + 3879205213947699279528 \nu^{3} - 29365745852810834804520 \nu^{2} - 91646595259766957356632 \nu + 667986781198753791884340$$$$)/$$$$24\!\cdots\!84$$ $$\beta_{7}$$ $$=$$ $$($$$$-34573176906724558286734222 \nu^{11} + 287816772078268966221697003 \nu^{10} - 1629959434672883964324375844 \nu^{9} + 8876158293531845208918950563 \nu^{8} - 17418784382383788955162125908 \nu^{7} + 259656230706074538972447955724 \nu^{6} - 1846424189350139900570864746088 \nu^{5} + 42713714980665430516120779185846 \nu^{4} - 39556052668414056501908170083000 \nu^{3} + 1900123897935594945037662736250616 \nu^{2} + 5236039425407566775202010513181448 \nu + 10366063356778215971568283462387140$$$$)/$$$$14\!\cdots\!48$$ $$\beta_{8}$$ $$=$$ $$($$$$-118280648117000274023242240 \nu^{11} + 200279396523248937328102845 \nu^{10} - 18461112871177641380964158090 \nu^{9} - 11805504472107487633101184847 \nu^{8} - 1008738152857369069410228828316 \nu^{7} - 7610024826275824060440331406428 \nu^{6} - 47301571440782087378988080407836 \nu^{5} - 207566169865897522064341653868262 \nu^{4} - 2614398983403626664029427899582328 \nu^{3} - 6061645507425319509179808242246472 \nu^{2} - 36780136858940076239439085508863296 \nu - 469625976510425130363922875414538548$$$$)/$$$$43\!\cdots\!44$$ $$\beta_{9}$$ $$=$$ $$($$$$55715980360172894158513661 \nu^{11} + 600474937779376077827963773 \nu^{10} - 1010072270379346174421169157 \nu^{9} + 57958122242501808941937811735 \nu^{8} + 103191502886315420672398546528 \nu^{7} + 1418017215246306671006804654168 \nu^{6} + 7781024118465489656540650313806 \nu^{5} + 48806887345368707515082309251358 \nu^{4} - 448902459171198711450459775037136 \nu^{3} + 3073916311371731956513911247347960 \nu^{2} + 3643552619017580556027238840624044 \nu - 120517735292122705278580942095915468$$$$)/$$$$14\!\cdots\!48$$ $$\beta_{10}$$ $$=$$ $$($$$$57881838624391780455102001 \nu^{11} + 6309224490772026379987727 \nu^{10} + 3573852232432001002361954107 \nu^{9} + 4817374183896997834172467121 \nu^{8} + 136695618168786957272320117088 \nu^{7} - 825688735233925518145935252032 \nu^{6} + 9690719868259484963200543458182 \nu^{5} - 51611724369422563222299944756438 \nu^{4} - 101802639969531533005584256113024 \nu^{3} - 2493146693631355276469124982405416 \nu^{2} - 4145573804800476991797114518713140 \nu - 100673414219259098703676674254331540$$$$)/$$$$14\!\cdots\!48$$ $$\beta_{11}$$ $$=$$ $$($$$$42339343767921687883000 \nu^{11} - 203699280429334160260229 \nu^{10} + 1786245712233590004493224 \nu^{9} + 12629788878516408575666453 \nu^{8} + 3042646095547488681820701 \nu^{7} + 251145208401682267882836088 \nu^{6} + 8457428145828394087111414285 \nu^{5} - 23448323671112683437457013994 \nu^{4} + 123047828192051148725379889098 \nu^{3} + 2327863592501780603020854653016 \nu^{2} - 5817177268621517903504482276236 \nu - 5662177116998536717207338192948$$$$)/$$$$91\!\cdots\!94$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} - \beta_{8} + 6 \beta_{7} - \beta_{6} - 6 \beta_{5} + 17 \beta_{4} + 4 \beta_{3} + \beta_{1} - 9$$ $$\nu^{3}$$ $$=$$ $$-15 \beta_{10} + 20 \beta_{8} - 8 \beta_{7} + 6 \beta_{6} + 11 \beta_{5} + 9 \beta_{4} - 266 \beta_{3} - 7 \beta_{2} - 18 \beta_{1} + 17$$ $$\nu^{4}$$ $$=$$ $$-14 \beta_{11} - 26 \beta_{10} - 105 \beta_{9} + 45 \beta_{8} - 103 \beta_{7} - 238 \beta_{6} - 272 \beta_{5} - 279 \beta_{4} - 100 \beta_{3} + 35 \beta_{2} + 52 \beta_{1} - 377$$ $$\nu^{5}$$ $$=$$ $$1778 \beta_{11} + 1152 \beta_{10} - 224 \beta_{9} - 732 \beta_{8} - 451 \beta_{7} - 29 \beta_{6} + 8554 \beta_{5} - 6606 \beta_{4} + 5102 \beta_{3} + 182 \beta_{2} - 687 \beta_{1} - 220$$ $$\nu^{6}$$ $$=$$ $$1344 \beta_{11} - 1574 \beta_{10} + 6643 \beta_{9} - 7355 \beta_{8} - 6859 \beta_{7} + 3450 \beta_{6} + 30546 \beta_{5} - 18015 \beta_{4} + 33136 \beta_{3} + 3521 \beta_{2} - 618 \beta_{1} - 47505$$ $$\nu^{7}$$ $$=$$ $$-49756 \beta_{11} - 12586 \beta_{10} - 12782 \beta_{9} - 21518 \beta_{8} + 16583 \beta_{7} + 67739 \beta_{6} - 267064 \beta_{5} + 323638 \beta_{4} + 256494 \beta_{3} - 69566 \beta_{2} - 77979 \beta_{1} + 81928$$ $$\nu^{8}$$ $$=$$ $$-129948 \beta_{11} - 46600 \beta_{10} - 128499 \beta_{9} + 451639 \beta_{8} - 164431 \beta_{7} + 549240 \beta_{6} - 2637690 \beta_{5} - 803835 \beta_{4} - 592392 \beta_{3} - 316099 \beta_{2} - 42244 \beta_{1} + 2699555$$ $$\nu^{9}$$ $$=$$ $$-1889356 \beta_{11} + 2356528 \beta_{10} - 760130 \beta_{9} - 53642 \beta_{8} + 5400723 \beta_{7} - 1514675 \beta_{6} - 15396006 \beta_{5} - 643074 \beta_{4} + 5838028 \beta_{3} + 2399894 \beta_{2} + 5337447 \beta_{1} - 9816566$$ $$\nu^{10}$$ $$=$$ $$-2920680 \beta_{11} + 11033388 \beta_{10} + 21939785 \beta_{9} - 2908245 \beta_{8} + 37427371 \beta_{7} - 18107978 \beta_{6} + 123956190 \beta_{5} + 27612733 \beta_{4} - 178279544 \beta_{3} + 20435065 \beta_{2} - 756046 \beta_{1} + 59634639$$ $$\nu^{11}$$ $$=$$ $$-45693032 \beta_{11} - 168135152 \beta_{10} + 124062764 \beta_{9} + 66719712 \beta_{8} - 232830447 \beta_{7} - 300859679 \beta_{6} + 309316910 \beta_{5} + 687584816 \beta_{4} - 1663689280 \beta_{3} + 78844584 \beta_{2} + 93807447 \beta_{1} + 585995148$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 3.11049 − 6.59331i −5.81071 + 4.13641i 4.40732 + 3.68164i −0.0663422 − 7.58000i −4.82374 + 1.03135i 5.18297 + 5.32390i 3.11049 + 6.59331i −5.81071 − 4.13641i 4.40732 − 3.68164i −0.0663422 + 7.58000i −4.82374 − 1.03135i 5.18297 − 5.32390i
−0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −2.94762 + 6.34914i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 1.88399 6.74171i 2.82843 1.50000 + 2.59808i 0
451.3 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.59916 + 2.33475i 2.82843 1.50000 + 2.59808i 0
451.4 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.80475 + 1.64177i −2.82843 1.50000 + 2.59808i 0
451.5 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −1.72580 6.78392i −2.82843 1.50000 + 2.59808i 0
451.6 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.99501 0.264136i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −2.94762 6.34914i 2.82843 1.50000 2.59808i 0
901.2 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 1.88399 + 6.74171i 2.82843 1.50000 2.59808i 0
901.3 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 6.59916 2.33475i 2.82843 1.50000 2.59808i 0
901.4 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −6.80475 1.64177i −2.82843 1.50000 2.59808i 0
901.5 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −1.72580 + 6.78392i −2.82843 1.50000 2.59808i 0
901.6 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 6.99501 + 0.264136i −2.82843 1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.f yes 12
5.b even 2 1 1050.3.p.e 12
5.c odd 4 2 1050.3.q.d 24
7.d odd 6 1 inner 1050.3.p.f yes 12
35.i odd 6 1 1050.3.p.e 12
35.k even 12 2 1050.3.q.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.e 12 5.b even 2 1
1050.3.p.e 12 35.i odd 6 1
1050.3.p.f yes 12 1.a even 1 1 trivial
1050.3.p.f yes 12 7.d odd 6 1 inner
1050.3.q.d 24 5.c odd 4 2
1050.3.q.d 24 35.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{12} + \cdots$$ $$42\!\cdots\!64$$">$$T_{17}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T^{2} + T^{4} )^{3}$$
$3$ $$( 3 - 3 T + T^{2} )^{6}$$
$5$ $$T^{12}$$
$7$ $$13841287201 - 2259801992 T + 103766418 T^{2} - 41883044 T^{3} - 633864 T^{4} + 1007048 T^{5} - 69678 T^{6} + 20552 T^{7} - 264 T^{8} - 356 T^{9} + 18 T^{10} - 8 T^{11} + T^{12}$$
$11$ $$463630257216 - 286780423104 T + 129337827696 T^{2} - 29825887728 T^{3} + 5383907860 T^{4} - 451837256 T^{5} + 41583508 T^{6} - 1027232 T^{7} + 227242 T^{8} - 2336 T^{9} + 562 T^{10} + 4 T^{11} + T^{12}$$
$13$ $$28563704064 + 13984786464 T^{2} + 1889099761 T^{4} + 45485292 T^{6} + 368342 T^{8} + 1068 T^{10} + T^{12}$$
$17$ $$4222235916864 + 985765370688 T - 213772663728 T^{2} - 67820278320 T^{3} + 19870112596 T^{4} - 1160246328 T^{5} - 69719924 T^{6} + 6657168 T^{7} + 311466 T^{8} - 17328 T^{9} - 530 T^{10} + 24 T^{11} + T^{12}$$
$19$ $$78337292544 - 128414853504 T + 43594294128 T^{2} + 43561525560 T^{3} + 9702497569 T^{4} + 544663884 T^{5} - 52581384 T^{6} - 4476120 T^{7} + 366383 T^{8} + 8352 T^{9} - 648 T^{10} - 12 T^{11} + T^{12}$$
$23$ $$876231988678656 - 479551066066944 T + 253371499107840 T^{2} - 9323399041536 T^{3} + 1268254246720 T^{4} + 39575452416 T^{5} + 4198387744 T^{6} + 95664384 T^{7} + 5055336 T^{8} + 112272 T^{9} + 4180 T^{10} + 60 T^{11} + T^{12}$$
$29$ $$( -5447232 + 448896 T + 220264 T^{2} - 10032 T^{3} - 1036 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$31$ $$8538318844344576 - 6127021275734016 T + 1758065895608064 T^{2} - 209895861460992 T^{3} + 4520197231216 T^{4} + 830509371744 T^{5} + 11894303568 T^{6} - 541244568 T^{7} - 10342855 T^{8} + 366498 T^{9} + 14919 T^{10} + 198 T^{11} + T^{12}$$
$37$ $$10632241068592081 + 1374236326700090 T + 346120628973385 T^{2} + 10274535808054 T^{3} + 4497113480398 T^{4} + 197952756890 T^{5} + 26896466861 T^{6} - 153382054 T^{7} + 14876974 T^{8} - 144746 T^{9} + 7273 T^{10} - 70 T^{11} + T^{12}$$
$41$ $$46617578071143395904 + 326223889918801056 T^{2} + 694124266310980 T^{4} + 471588066168 T^{6} + 140625656 T^{8} + 19308 T^{10} + T^{12}$$
$43$ $$( -4767344 + 598400 T + 787768 T^{2} - 85000 T^{3} - 3655 T^{4} + 42 T^{5} + T^{6} )^{2}$$
$47$ $$1027661018311324224 - 52904692827907776 T - 3650852092900176 T^{2} + 234685444858416 T^{3} + 13361339346196 T^{4} - 642047239320 T^{5} - 9247542972 T^{6} + 586453968 T^{7} + 9072458 T^{8} - 245160 T^{9} - 2886 T^{10} + 60 T^{11} + T^{12}$$
$53$ $$31577218407960105024 + 2137916069354419776 T + 233839019696684688 T^{2} - 1544256808471824 T^{3} + 343755765890740 T^{4} - 1482967036360 T^{5} + 299431774196 T^{6} - 3215519344 T^{7} + 113947786 T^{8} - 332480 T^{9} + 12530 T^{10} - 44 T^{11} + T^{12}$$
$59$ $$4729697242691770944 - 475199787100078656 T + 824281835731920 T^{2} + 1516154854789872 T^{3} + 10057764028564 T^{4} - 2171340383592 T^{5} - 8601605876 T^{6} + 1923153072 T^{7} + 30454554 T^{8} - 313440 T^{9} - 5762 T^{10} + 48 T^{11} + T^{12}$$
$61$ $$32\!\cdots\!89$$$$+$$$$14\!\cdots\!26$$$$T + 6380808370563310209 T^{2} - 670941875501451846 T^{3} - 2357828840121062 T^{4} + 170325393856818 T^{5} + 580797042177 T^{6} - 29347295850 T^{7} - 1244566 T^{8} + 2844126 T^{9} - 3759 T^{10} - 186 T^{11} + T^{12}$$
$67$ $$1377591905882966544 + 216737731822252752 T + 92569284446825484 T^{2} - 12348025134849900 T^{3} + 2237969292686433 T^{4} - 65738065571964 T^{5} + 1659117765660 T^{6} - 19557946092 T^{7} + 265466911 T^{8} - 2161832 T^{9} + 26532 T^{10} - 152 T^{11} + T^{12}$$
$71$ $$( -19178846208 + 2481128448 T + 67582336 T^{2} - 910400 T^{3} - 16444 T^{4} + 68 T^{5} + T^{6} )^{2}$$
$73$ $$133904078886249 - 255689956855494 T + 140001887465601 T^{2} + 43431302222322 T^{3} + 4184489224714 T^{4} + 96426452202 T^{5} - 6010417807 T^{6} - 196480866 T^{7} + 10320666 T^{8} + 63846 T^{9} - 3439 T^{10} - 18 T^{11} + T^{12}$$
$79$ $$353438403020018449 + 366946600555112494 T + 367203573035575171 T^{2} + 14820770397487502 T^{3} + 816159287744872 T^{4} + 2770963372954 T^{5} + 398277035813 T^{6} + 2041706674 T^{7} + 118690600 T^{8} + 149702 T^{9} + 15427 T^{10} + 70 T^{11} + T^{12}$$
$83$ $$50\!\cdots\!36$$$$+ 9041897034880820640 T^{2} + 6312081562432516 T^{4} + 2166411271464 T^{6} + 381161720 T^{8} + 32148 T^{10} + T^{12}$$
$89$ $$11\!\cdots\!96$$$$-$$$$92\!\cdots\!24$$$$T +$$$$28\!\cdots\!00$$$$T^{2} - 3409173887204069712 T^{3} - 14404491929660684 T^{4} + 601453576588872 T^{5} + 717014726220 T^{6} - 88416490464 T^{7} + 342306122 T^{8} + 4478544 T^{9} - 17250 T^{10} - 168 T^{11} + T^{12}$$
$97$ $$66\!\cdots\!01$$$$+$$$$15\!\cdots\!86$$$$T^{2} + 90643172097639823 T^{4} + 19816537157900 T^{6} + 1794419271 T^{8} + 70634 T^{10} + T^{12}$$