# Properties

 Label 3234.2.e.b Level 3234 Weight 2 Character orbit 3234.e Analytic conductor 25.824 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 462) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + 493030 x^{4} - 386266 x^{3} + 223844 x^{2} - 82874 x + 13417$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$6983 \nu^{14} - 48881 \nu^{13} + 474100 \nu^{12} - 2209147 \nu^{11} + 11140707 \nu^{10} - 36618018 \nu^{9} + 118372755 \nu^{8} - 276086391 \nu^{7} + 593880549 \nu^{6} - 946931332 \nu^{5} + 1294712939 \nu^{4} - 1271162965 \nu^{3} + 915620088 \nu^{2} - 401151387 \nu + 32362633$$$$)/22807968$$ $$\beta_{2}$$ $$=$$ $$($$$$-3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} - 4890679917 \nu^{10} + 15780182142 \nu^{9} - 48699356733 \nu^{8} + 110238860505 \nu^{7} - 222125567883 \nu^{6} + 336742486204 \nu^{5} - 415275809861 \nu^{4} + 371398223899 \nu^{3} - 230411582664 \nu^{2} + 86434816533 \nu - 13304183911$$$$)/ 3261539424$$ $$\beta_{3}$$ $$=$$ $$($$$$-189803 \nu^{14} + 1328621 \nu^{13} - 12040024 \nu^{12} + 54968071 \nu^{11} - 260705691 \nu^{10} + 831319938 \nu^{9} - 2534621871 \nu^{8} + 5700932979 \nu^{7} - 11573124741 \nu^{6} + 17710001596 \nu^{5} - 22848161615 \nu^{4} + 21441640105 \nu^{3} - 14950384260 \nu^{2} + 6439036695 \nu - 1021023169$$$$)/ 148251792$$ $$\beta_{4}$$ $$=$$ $$($$$$-1252325 \nu^{14} + 8766275 \nu^{13} - 81315274 \nu^{12} + 373930069 \nu^{11} - 1818246999 \nu^{10} + 5872472250 \nu^{9} - 18432565677 \nu^{8} + 42250495017 \nu^{7} - 88964564157 \nu^{6} + 139944185800 \nu^{5} - 189878074001 \nu^{4} + 185929817443 \nu^{3} - 140966904954 \nu^{2} + 65763256533 \nu - 14998863325$$$$)/ 815384856$$ $$\beta_{5}$$ $$=$$ $$($$$$-250035 \nu^{14} + 1750245 \nu^{13} - 16051742 \nu^{12} + 73557267 \nu^{11} - 351447497 \nu^{10} + 1124676710 \nu^{9} - 3429869109 \nu^{8} + 7709036103 \nu^{7} - 15429510803 \nu^{6} + 23296503804 \nu^{5} - 28630334953 \nu^{4} + 25544328593 \nu^{3} - 15305197790 \nu^{2} + 5412809207 \nu + 128107011$$$$)/ 135897476$$ $$\beta_{6}$$ $$=$$ $$($$$$31288770 \nu^{15} - 16748485 \nu^{14} + 797619205 \nu^{13} + 2631204646 \nu^{12} - 4029956479 \nu^{11} + 98833917597 \nu^{10} - 253528285620 \nu^{9} + 1180057442883 \nu^{8} - 2297035581639 \nu^{7} + 5676299627439 \nu^{6} - 7256018389690 \nu^{5} + 10030408696559 \nu^{4} - 6580080409159 \nu^{3} + 4269219214770 \nu^{2} - 43198382379 \nu - 181576943117$$$$)/ 158184662064$$ $$\beta_{7}$$ $$=$$ $$($$$$-31288770 \nu^{15} + 452583065 \nu^{14} - 3848461265 \nu^{13} + 25712532526 \nu^{12} - 126371519773 \nu^{11} + 533477938419 \nu^{10} - 1785395864580 \nu^{9} + 5160916096233 \nu^{8} - 12143081222517 \nu^{7} + 24141818008761 \nu^{6} - 38917872020806 \nu^{5} + 50481947842229 \nu^{4} - 50908209830125 \nu^{3} + 37436097284178 \nu^{2} - 18719983335993 \nu + 4642794315301$$$$)/ 158184662064$$ $$\beta_{8}$$ $$=$$ $$($$$$-61841878 \nu^{15} + 463814085 \nu^{14} - 4332228469 \nu^{13} + 21124971426 \nu^{12} - 103973734141 \nu^{11} + 354956782059 \nu^{10} - 1130840178408 \nu^{9} + 2742004258845 \nu^{8} - 5867164985181 \nu^{7} + 9805433466473 \nu^{6} - 13472523722594 \nu^{5} + 14104341127729 \nu^{4} - 10389357329569 \nu^{3} + 4991902261014 \nu^{2} - 438472848665 \nu - 306749906363$$$$)/ 158184662064$$ $$\beta_{9}$$ $$=$$ $$($$$$124647224 \nu^{15} - 792790405 \nu^{14} + 7684579947 \nu^{13} - 32966828504 \nu^{12} + 164837589985 \nu^{11} - 499636537173 \nu^{10} + 1593858572646 \nu^{9} - 3402670735473 \nu^{8} + 7208963912421 \nu^{7} - 10318496609323 \nu^{6} + 14096862838860 \nu^{5} - 12094824539073 \nu^{4} + 10307867859415 \nu^{3} - 4833384190332 \nu^{2} + 3394648790233 \nu - 1247802075679$$$$)/ 158184662064$$ $$\beta_{10}$$ $$=$$ $$($$$$124647224 \nu^{15} - 1076917955 \nu^{14} + 9673472797 \nu^{13} - 51503270872 \nu^{12} + 250200637143 \nu^{11} - 916288554507 \nu^{10} + 2942026006626 \nu^{9} - 7649444175615 \nu^{8} + 16964786108547 \nu^{7} - 30953383474645 \nu^{6} + 46664778955628 \nu^{5} - 56590192827103 \nu^{4} + 54130606016001 \nu^{3} - 38151775447740 \nu^{2} + 18943545384919 \nu - 4344274484769$$$$)/ 158184662064$$ $$\beta_{11}$$ $$=$$ $$($$$$15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} - 96035346 \nu^{10} + 317610396 \nu^{9} - 791678301 \nu^{8} + 1812360285 \nu^{7} - 3211318028 \nu^{6} + 5028988886 \nu^{5} - 5993427763 \nu^{4} + 5979185161 \nu^{3} - 4172380467 \nu^{2} + 2140166891 \nu - 518381770$$$$)/18627492$$ $$\beta_{12}$$ $$=$$ $$($$$$-13342 \nu^{15} + 100065 \nu^{14} - 928497 \nu^{13} + 4517578 \nu^{12} - 22245945 \nu^{11} + 75998175 \nu^{10} - 244879016 \nu^{9} + 599354649 \nu^{8} - 1324712505 \nu^{7} + 2282122421 \nu^{6} - 3405656298 \nu^{5} + 3894289773 \nu^{4} - 3651650205 \nu^{3} + 2418391182 \nu^{2} - 1148282565 \nu + 261797265$$$$)/7787936$$ $$\beta_{13}$$ $$=$$ $$($$$$15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} - 96035346 \nu^{10} + 317610396 \nu^{9} - 791678301 \nu^{8} + 1812360285 \nu^{7} - 3211318028 \nu^{6} + 5028988886 \nu^{5} - 5993427763 \nu^{4} + 5979185161 \nu^{3} - 4172380467 \nu^{2} + 2121539399 \nu - 509068024$$$$)/9313746$$ $$\beta_{14}$$ $$=$$ $$($$$$575875964 \nu^{15} - 4436099745 \nu^{14} + 41319402047 \nu^{13} - 205410283332 \nu^{12} + 1021087579937 \nu^{11} - 3562268688861 \nu^{10} + 11652271091298 \nu^{9} - 29183922744189 \nu^{8} + 65936755256061 \nu^{7} - 116732328141163 \nu^{6} + 179642141127088 \nu^{5} - 212376054391829 \nu^{4} + 207235157443403 \nu^{3} - 142699154525280 \nu^{2} + 71422281850849 \nu - 16963072781711$$$$)/ 158184662064$$ $$\beta_{15}$$ $$=$$ $$($$$$158954387 \nu^{15} - 1172652900 \nu^{14} + 11041151095 \nu^{13} - 53296410441 \nu^{12} + 265930611712 \nu^{11} - 905854992267 \nu^{10} + 2960788040621 \nu^{9} - 7259574921286 \nu^{8} + 16337255299198 \nu^{7} - 28374154378543 \nu^{6} + 43439436758339 \nu^{5} - 50569373700088 \nu^{4} + 49211364580575 \nu^{3} - 33565389721049 \nu^{2} + 16733640083382 \nu - 3970555117103$$$$)/ 26364110344$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{13} + 2 \beta_{11} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{13} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} + \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 17 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 14$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{15} + 2 \beta_{14} + 19 \beta_{13} - 4 \beta_{12} - 36 \beta_{11} - 22 \beta_{10} + 28 \beta_{9} - 4 \beta_{8} + 20 \beta_{7} + 30 \beta_{6} - 22 \beta_{5} + 46 \beta_{4} + 26 \beta_{3} + 42 \beta_{2} - 10 \beta_{1} + 71$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$19 \beta_{15} + 3 \beta_{14} - 83 \beta_{13} + 34 \beta_{12} + 145 \beta_{11} - 69 \beta_{10} + 35 \beta_{9} + 22 \beta_{8} + 87 \beta_{7} + 17 \beta_{6} - 60 \beta_{5} + 94 \beta_{4} + 70 \beta_{3} + 115 \beta_{2} - 25 \beta_{1} + 201$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$67 \beta_{15} + 4 \beta_{14} - 297 \beta_{13} + 112 \beta_{12} + 526 \beta_{11} + 176 \beta_{10} - 293 \beta_{9} + 76 \beta_{8} - 97 \beta_{7} - 332 \beta_{6} + 207 \beta_{5} - 393 \beta_{4} - 259 \beta_{3} - 361 \beta_{2} + 141 \beta_{1} - 586$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-150 \beta_{15} - 121 \beta_{14} + 707 \beta_{13} - 360 \beta_{12} - 1095 \beta_{11} + 1072 \beta_{10} - 747 \beta_{9} - 175 \beta_{8} - 1169 \beta_{7} - 510 \beta_{6} + 938 \beta_{5} - 1518 \beta_{4} - 1155 \beta_{3} - 1673 \beta_{2} + 581 \beta_{1} - 2771$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-922 \beta_{15} - 498 \beta_{14} + 4259 \beta_{13} - 1972 \beta_{12} - 6920 \beta_{11} - 826 \beta_{10} + 2686 \beta_{9} - 1064 \beta_{8} - 482 \beta_{7} + 3274 \beta_{6} - 1508 \beta_{5} + 2644 \beta_{4} + 1964 \beta_{3} + 2428 \beta_{2} - 1340 \beta_{1} + 3975$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$927 \beta_{15} + 1353 \beta_{14} - 4574 \beta_{13} + 2602 \beta_{12} + 5587 \beta_{11} - 14119 \beta_{10} + 11807 \beta_{9} + 992 \beta_{8} + 12937 \beta_{7} + 8909 \beta_{6} - 12672 \beta_{5} + 20172 \beta_{4} + 16068 \beta_{3} + 21459 \beta_{2} - 9621 \beta_{1} + 35386$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$12039 \beta_{15} + 10518 \beta_{14} - 56988 \beta_{13} + 28604 \beta_{12} + 83700 \beta_{11} - 6230 \beta_{10} - 20129 \beta_{9} + 13492 \beta_{8} + 21505 \beta_{7} - 28500 \beta_{6} + 4915 \beta_{5} - 8061 \beta_{4} - 7043 \beta_{3} - 6263 \beta_{2} + 7359 \beta_{1} - 11089$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-50 \beta_{15} - 6198 \beta_{14} + 796 \beta_{13} - 3010 \beta_{12} + 18558 \beta_{11} + 166047 \beta_{10} - 163611 \beta_{9} + 1489 \beta_{8} - 124121 \beta_{7} - 128625 \beta_{6} + 154187 \beta_{5} - 241383 \beta_{4} - 199518 \beta_{3} - 250855 \beta_{2} + 135333 \beta_{1} - 418131$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-74741 \beta_{15} - 80579 \beta_{14} + 354385 \beta_{13} - 183886 \beta_{12} - 467854 \beta_{11} + 130150 \beta_{10} + 38814 \beta_{9} - 79386 \beta_{8} - 195571 \beta_{7} + 99599 \beta_{6} + 48888 \beta_{5} - 81749 \beta_{4} - 59284 \beta_{3} - 89044 \beta_{2} + 24892 \beta_{1} - 141469$$ $$\nu^{13}$$ $$=$$ $$($$$$-144088 \beta_{15} - 108010 \beta_{14} + 697161 \beta_{13} - 343180 \beta_{12} - 1133136 \beta_{11} - 1750154 \beta_{10} + 2068612 \beta_{9} - 175276 \beta_{8} + 997188 \beta_{7} + 1658494 \beta_{6} - 1711866 \beta_{5} + 2645958 \beta_{4} + 2256254 \beta_{3} + 2688400 \beta_{2} - 1689844 \beta_{1} + 4562117$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$1733162 \beta_{15} + 2074966 \beta_{14} - 8106763 \beta_{13} + 4282548 \beta_{12} + 9602618 \beta_{11} - 5068490 \beta_{10} + 1147786 \beta_{9} + 1715280 \beta_{8} + 5625272 \beta_{7} - 660412 \beta_{6} - 2864062 \beta_{5} + 4680930 \beta_{4} + 3676122 \beta_{3} + 4707416 \beta_{2} - 2421852 \beta_{1} + 7785995$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$3522094 \beta_{15} + 3795721 \beta_{14} - 16818405 \beta_{13} + 8683710 \beta_{12} + 22482349 \beta_{11} + 16006457 \beta_{10} - 24041258 \beta_{9} + 3807178 \beta_{8} - 5454972 \beta_{7} - 19608667 \beta_{6} + 17217975 \beta_{5} - 26269633 \beta_{4} - 23170387 \beta_{3} - 26098192 \beta_{2} + 18842846 \beta_{1} - 45325614$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$1079$$ $$2059$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2155.1
 0.5 + 2.40229i 0.5 + 3.32851i 0.5 + 1.35798i 0.5 − 1.56688i 0.5 + 0.0286340i 0.5 − 0.921602i 0.5 − 3.19339i 0.5 − 3.43554i 0.5 + 3.43554i 0.5 + 3.19339i 0.5 + 0.921602i 0.5 − 0.0286340i 0.5 + 1.56688i 0.5 − 1.35798i 0.5 − 3.32851i 0.5 − 2.40229i
1.00000i 1.00000i −1.00000 3.26832i 1.00000 0 1.00000i −1.00000 −3.26832
2155.2 1.00000i 1.00000i −1.00000 2.46248i 1.00000 0 1.00000i −1.00000 −2.46248
2155.3 1.00000i 1.00000i −1.00000 2.22400i 1.00000 0 1.00000i −1.00000 −2.22400
2155.4 1.00000i 1.00000i −1.00000 0.700858i 1.00000 0 1.00000i −1.00000 0.700858
2155.5 1.00000i 1.00000i −1.00000 0.837391i 1.00000 0 1.00000i −1.00000 0.837391
2155.6 1.00000i 1.00000i −1.00000 1.78763i 1.00000 0 1.00000i −1.00000 1.78763
2155.7 1.00000i 1.00000i −1.00000 2.32736i 1.00000 0 1.00000i −1.00000 2.32736
2155.8 1.00000i 1.00000i −1.00000 4.30156i 1.00000 0 1.00000i −1.00000 4.30156
2155.9 1.00000i 1.00000i −1.00000 4.30156i 1.00000 0 1.00000i −1.00000 4.30156
2155.10 1.00000i 1.00000i −1.00000 2.32736i 1.00000 0 1.00000i −1.00000 2.32736
2155.11 1.00000i 1.00000i −1.00000 1.78763i 1.00000 0 1.00000i −1.00000 1.78763
2155.12 1.00000i 1.00000i −1.00000 0.837391i 1.00000 0 1.00000i −1.00000 0.837391
2155.13 1.00000i 1.00000i −1.00000 0.700858i 1.00000 0 1.00000i −1.00000 0.700858
2155.14 1.00000i 1.00000i −1.00000 2.22400i 1.00000 0 1.00000i −1.00000 −2.22400
2155.15 1.00000i 1.00000i −1.00000 2.46248i 1.00000 0 1.00000i −1.00000 −2.46248
2155.16 1.00000i 1.00000i −1.00000 3.26832i 1.00000 0 1.00000i −1.00000 −3.26832
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2155.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.e.b 16
7.b odd 2 1 3234.2.e.a 16
7.c even 3 1 462.2.p.b yes 16
7.d odd 6 1 462.2.p.a 16
11.b odd 2 1 3234.2.e.a 16
21.g even 6 1 1386.2.bk.a 16
21.h odd 6 1 1386.2.bk.b 16
77.b even 2 1 inner 3234.2.e.b 16
77.h odd 6 1 462.2.p.a 16
77.i even 6 1 462.2.p.b yes 16
231.k odd 6 1 1386.2.bk.b 16
231.l even 6 1 1386.2.bk.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.p.a 16 7.d odd 6 1
462.2.p.a 16 77.h odd 6 1
462.2.p.b yes 16 7.c even 3 1
462.2.p.b yes 16 77.i even 6 1
1386.2.bk.a 16 21.g even 6 1
1386.2.bk.a 16 231.l even 6 1
1386.2.bk.b 16 21.h odd 6 1
1386.2.bk.b 16 231.k odd 6 1
3234.2.e.a 16 7.b odd 2 1
3234.2.e.a 16 11.b odd 2 1
3234.2.e.b 16 1.a even 1 1 trivial
3234.2.e.b 16 77.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3234, [\chi])$$:

 $$T_{5}^{16} + \cdots$$ $$T_{13}^{8} - 64 T_{13}^{6} + 8 T_{13}^{5} + 836 T_{13}^{4} - 1168 T_{13}^{3} - 592 T_{13}^{2} + 1216 T_{13} - 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ $$1 - 30 T^{2} + 471 T^{4} - 4930 T^{6} + 37781 T^{8} - 222460 T^{10} + 1052266 T^{12} - 4383184 T^{14} + 19642634 T^{16} - 109579600 T^{18} + 657666250 T^{20} - 3475937500 T^{22} + 14758203125 T^{24} - 48144531250 T^{26} + 114990234375 T^{28} - 183105468750 T^{30} + 152587890625 T^{32}$$
$7$ 1
$11$ $$1 - 8 T + 33 T^{2} - 56 T^{3} - 206 T^{4} + 1320 T^{5} - 1177 T^{6} - 15528 T^{7} + 86618 T^{8} - 170808 T^{9} - 142417 T^{10} + 1756920 T^{11} - 3016046 T^{12} - 9018856 T^{13} + 58461513 T^{14} - 155897368 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 + 40 T^{2} + 8 T^{3} + 576 T^{4} - 648 T^{5} + 3672 T^{6} - 30816 T^{7} + 19294 T^{8} - 400608 T^{9} + 620568 T^{10} - 1423656 T^{11} + 16451136 T^{12} + 2970344 T^{13} + 193072360 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 83 T^{2} - 68 T^{3} + 3413 T^{4} - 4044 T^{5} + 95086 T^{6} - 108260 T^{7} + 1908750 T^{8} - 1840420 T^{9} + 27479854 T^{10} - 19868172 T^{11} + 285057173 T^{12} - 96550276 T^{13} + 2003418227 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 + 10 T + 145 T^{2} + 1030 T^{3} + 8870 T^{4} + 49582 T^{5} + 317127 T^{6} + 1444786 T^{7} + 7373490 T^{8} + 27450934 T^{9} + 114482847 T^{10} + 340082938 T^{11} + 1155947270 T^{12} + 2550381970 T^{13} + 6821652745 T^{14} + 8938717390 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 4 T + 57 T^{2} - 124 T^{3} + 2103 T^{4} - 4828 T^{5} + 65020 T^{6} - 100872 T^{7} + 1438254 T^{8} - 2320056 T^{9} + 34395580 T^{10} - 58742276 T^{11} + 588505623 T^{12} - 798106532 T^{13} + 8438045673 T^{14} - 13619301788 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$1 - 220 T^{2} + 25766 T^{4} - 2101752 T^{6} + 132425201 T^{8} - 6784185928 T^{10} + 290786240918 T^{12} - 10599475438788 T^{14} + 331431007100868 T^{16} - 8914158844020708 T^{18} + 205667583262723958 T^{20} - 4035392003974426888 T^{22} + 66245231785889430161 T^{24} -$$$$88\!\cdots\!52$$$$T^{26} +$$$$91\!\cdots\!06$$$$T^{28} -$$$$65\!\cdots\!20$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 - 386 T^{2} + 71901 T^{4} - 8595466 T^{6} + 739336166 T^{8} - 48582533718 T^{10} + 2525237875987 T^{12} - 105975133331486 T^{14} + 3630081582174546 T^{16} - 101842103131558046 T^{18} + 2332110208469390227 T^{20} - 43117177507031615958 T^{22} +$$$$63\!\cdots\!06$$$$T^{24} -$$$$70\!\cdots\!66$$$$T^{26} +$$$$56\!\cdots\!61$$$$T^{28} -$$$$29\!\cdots\!06$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$( 1 + 14 T + 229 T^{2} + 2126 T^{3} + 20986 T^{4} + 154618 T^{5} + 1182051 T^{6} + 7476394 T^{7} + 49108426 T^{8} + 276626578 T^{9} + 1618227819 T^{10} + 7831865554 T^{11} + 39331142746 T^{12} + 147425252582 T^{13} + 587551347661 T^{14} + 1329046279862 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 16 T + 330 T^{2} + 3640 T^{3} + 42745 T^{4} + 362832 T^{5} + 3123194 T^{6} + 21761352 T^{7} + 152229812 T^{8} + 892215432 T^{9} + 5250089114 T^{10} + 25006744272 T^{11} + 120787153945 T^{12} + 421716571640 T^{13} + 1567534399530 T^{14} + 3116068382096 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 - 370 T^{2} + 69205 T^{4} - 8714850 T^{6} + 829915302 T^{8} - 63533346462 T^{10} + 4048663635627 T^{12} - 219017168933710 T^{14} + 10156745084830162 T^{16} - 404962745358429790 T^{18} + 13841575286145223227 T^{20} -$$$$40\!\cdots\!38$$$$T^{22} +$$$$97\!\cdots\!02$$$$T^{24} -$$$$18\!\cdots\!50$$$$T^{26} +$$$$27\!\cdots\!05$$$$T^{28} -$$$$27\!\cdots\!30$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 62 T^{2} + 2031 T^{4} - 368922 T^{6} + 20631365 T^{8} - 625319636 T^{10} + 67952187226 T^{12} - 3345437111480 T^{14} + 84733582253162 T^{16} - 7390070579259320 T^{18} + 331584996915154906 T^{20} - 6740455005895900244 T^{22} +$$$$49\!\cdots\!65$$$$T^{24} -$$$$19\!\cdots\!78$$$$T^{26} +$$$$23\!\cdots\!71$$$$T^{28} -$$$$15\!\cdots\!78$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 + 189 T^{2} - 116 T^{3} + 18282 T^{4} - 21964 T^{5} + 1293871 T^{6} - 2238960 T^{7} + 74773842 T^{8} - 118664880 T^{9} + 3634483639 T^{10} - 3269934428 T^{11} + 144253773642 T^{12} - 48510677188 T^{13} + 4189064253381 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 - 412 T^{2} + 84726 T^{4} - 11812648 T^{6} + 1278873825 T^{8} - 115825341832 T^{10} + 9135071092582 T^{12} - 639128365018580 T^{14} + 39901635887620324 T^{16} - 2224805838629676980 T^{18} +$$$$11\!\cdots\!02$$$$T^{20} -$$$$48\!\cdots\!12$$$$T^{22} +$$$$18\!\cdots\!25$$$$T^{24} -$$$$60\!\cdots\!48$$$$T^{26} +$$$$15\!\cdots\!06$$$$T^{28} -$$$$25\!\cdots\!32$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 28 T + 652 T^{2} - 10788 T^{3} + 153837 T^{4} - 1841484 T^{5} + 19466748 T^{6} - 180831988 T^{7} + 1498320772 T^{8} - 11030751268 T^{9} + 72435769308 T^{10} - 417981879804 T^{11} + 2130002641917 T^{12} - 9111504895188 T^{13} + 33591284083372 T^{14} - 87996799408588 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 16 T + 442 T^{2} - 5580 T^{3} + 88929 T^{4} - 911040 T^{5} + 10786038 T^{6} - 91408468 T^{7} + 872545804 T^{8} - 6124367356 T^{9} + 48418524582 T^{10} - 274007123520 T^{11} + 1792019039409 T^{12} - 7533698097060 T^{13} + 39982604918698 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 28 T + 665 T^{2} + 9556 T^{3} + 124346 T^{4} + 1164764 T^{5} + 10825387 T^{6} + 79684020 T^{7} + 712940082 T^{8} + 5657565420 T^{9} + 54570775867 T^{10} + 416881848004 T^{11} + 3159840885626 T^{12} + 17241215678156 T^{13} + 85186688807465 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 44 T + 1204 T^{2} + 23988 T^{3} + 389092 T^{4} + 5297980 T^{5} + 62491788 T^{6} + 643636388 T^{7} + 5856468854 T^{8} + 46985456324 T^{9} + 333018738252 T^{10} + 2061004285660 T^{11} + 11049528387172 T^{12} + 49728841372884 T^{13} + 182206408451956 T^{14} + 486085534840268 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$1 - 538 T^{2} + 147807 T^{4} - 27647974 T^{6} + 3965128581 T^{8} - 467618207428 T^{10} + 47603630292250 T^{12} - 4323347326389920 T^{14} + 357043246909249546 T^{16} - 26982010663999490720 T^{18} +$$$$18\!\cdots\!50$$$$T^{20} -$$$$11\!\cdots\!88$$$$T^{22} +$$$$60\!\cdots\!41$$$$T^{24} -$$$$26\!\cdots\!74$$$$T^{26} +$$$$87\!\cdots\!87$$$$T^{28} -$$$$19\!\cdots\!78$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$( 1 + 4 T + 523 T^{2} + 1544 T^{3} + 123705 T^{4} + 263372 T^{5} + 17791866 T^{6} + 28394068 T^{7} + 1752220430 T^{8} + 2356707644 T^{9} + 122568164874 T^{10} + 150592685764 T^{11} + 5870831599305 T^{12} + 6081878752792 T^{13} + 170989815271987 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 640 T^{2} + 210552 T^{4} - 47499904 T^{6} + 8295415452 T^{8} - 1194559990144 T^{10} + 146915919459400 T^{12} - 15750756401654144 T^{14} + 1489545499043303110 T^{16} -$$$$12\!\cdots\!24$$$$T^{18} +$$$$92\!\cdots\!00$$$$T^{20} -$$$$59\!\cdots\!84$$$$T^{22} +$$$$32\!\cdots\!12$$$$T^{24} -$$$$14\!\cdots\!04$$$$T^{26} +$$$$52\!\cdots\!92$$$$T^{28} -$$$$12\!\cdots\!40$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 608 T^{2} + 187588 T^{4} - 40328000 T^{6} + 6818823050 T^{8} - 967972100768 T^{10} + 120672937256208 T^{12} - 13517338511335328 T^{14} + 1373778662576096019 T^{16} -$$$$12\!\cdots\!52$$$$T^{18} +$$$$10\!\cdots\!48$$$$T^{20} -$$$$80\!\cdots\!72$$$$T^{22} +$$$$53\!\cdots\!50$$$$T^{24} -$$$$29\!\cdots\!00$$$$T^{26} +$$$$13\!\cdots\!08$$$$T^{28} -$$$$39\!\cdots\!52$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$