# Properties

 Label 1764.4.f.a Level $1764$ Weight $4$ Character orbit 1764.f Analytic conductor $104.079$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$104.079369250$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{14}\cdot 3^{18}\cdot 7^{4}$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} +O(q^{10})$$ $$q -\beta_{3} q^{5} + \beta_{8} q^{11} -\beta_{4} q^{13} + ( -3 \beta_{3} - \beta_{10} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{9} ) q^{19} + ( -\beta_{5} - 3 \beta_{8} ) q^{23} + ( 27 + 2 \beta_{6} - \beta_{12} ) q^{25} + ( \beta_{5} - 2 \beta_{8} - \beta_{15} ) q^{29} + ( -5 \beta_{1} - 7 \beta_{2} + 6 \beta_{4} + 4 \beta_{9} ) q^{31} + ( -10 - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{12} ) q^{37} + ( 2 \beta_{3} + 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{41} + ( 88 + 3 \beta_{6} + 4 \beta_{7} - 3 \beta_{12} ) q^{43} + ( -14 \beta_{3} + 4 \beta_{10} - \beta_{13} ) q^{47} + ( \beta_{5} + 6 \beta_{8} - \beta_{15} ) q^{53} + ( 7 \beta_{1} - 2 \beta_{2} - 9 \beta_{4} + 2 \beta_{9} ) q^{55} + ( -26 \beta_{3} - 3 \beta_{10} - \beta_{13} + \beta_{14} ) q^{59} + ( 9 \beta_{1} - 3 \beta_{2} - \beta_{9} ) q^{61} + ( 6 \beta_{8} + 3 \beta_{11} + 2 \beta_{15} ) q^{65} + ( 189 + 2 \beta_{6} - \beta_{7} - 6 \beta_{12} ) q^{67} + ( \beta_{5} + 11 \beta_{8} + 8 \beta_{11} - \beta_{15} ) q^{71} + ( -19 \beta_{1} - 9 \beta_{2} - 17 \beta_{4} + 3 \beta_{9} ) q^{73} + ( 44 + \beta_{6} + 8 \beta_{7} - 4 \beta_{12} ) q^{79} + ( -19 \beta_{3} - 4 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} ) q^{83} + ( 461 - \beta_{6} - 9 \beta_{7} - \beta_{12} ) q^{85} + ( 16 \beta_{3} - 10 \beta_{10} - 4 \beta_{14} ) q^{89} + ( -5 \beta_{5} - 9 \beta_{8} + 4 \beta_{11} - 3 \beta_{15} ) q^{95} + ( 48 \beta_{1} + 9 \beta_{2} + 33 \beta_{4} + \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 424q^{25} - 152q^{37} + 1408q^{43} + 3056q^{67} + 728q^{79} + 7392q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-8606933517 \nu^{15} + 17491457046 \nu^{14} + 2762073108256 \nu^{13} - 9225620697790 \nu^{12} - 338722846892395 \nu^{11} + 1516388281986214 \nu^{10} + 18962342025989460 \nu^{9} - 105038524605413770 \nu^{8} - 440378181482053668 \nu^{7} + 3029046816285277760 \nu^{6} + 2302641961781731248 \nu^{5} - 25851157990713132280 \nu^{4} - 13012479407317279344 \nu^{3} + 97218732178861288576 \nu^{2} + 22061423532217981184 \nu - 125590689624071796768$$$$)/ 6787326577694418304$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!75$$$$\nu^{15} +$$$$87\!\cdots\!44$$$$\nu^{14} +$$$$49\!\cdots\!72$$$$\nu^{13} -$$$$29\!\cdots\!84$$$$\nu^{12} -$$$$62\!\cdots\!73$$$$\nu^{11} +$$$$39\!\cdots\!76$$$$\nu^{10} +$$$$34\!\cdots\!68$$$$\nu^{9} -$$$$24\!\cdots\!36$$$$\nu^{8} -$$$$72\!\cdots\!16$$$$\nu^{7} +$$$$64\!\cdots\!44$$$$\nu^{6} +$$$$14\!\cdots\!48$$$$\nu^{5} -$$$$48\!\cdots\!52$$$$\nu^{4} -$$$$16\!\cdots\!60$$$$\nu^{3} +$$$$18\!\cdots\!64$$$$\nu^{2} +$$$$27\!\cdots\!72$$$$\nu -$$$$22\!\cdots\!68$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{3}$$ $$=$$ $$($$$$13\!\cdots\!96$$$$\nu^{15} -$$$$28\!\cdots\!71$$$$\nu^{14} -$$$$39\!\cdots\!79$$$$\nu^{13} +$$$$16\!\cdots\!25$$$$\nu^{12} +$$$$39\!\cdots\!36$$$$\nu^{11} -$$$$27\!\cdots\!69$$$$\nu^{10} -$$$$14\!\cdots\!13$$$$\nu^{9} +$$$$16\!\cdots\!47$$$$\nu^{8} -$$$$11\!\cdots\!24$$$$\nu^{7} -$$$$35\!\cdots\!20$$$$\nu^{6} +$$$$13\!\cdots\!36$$$$\nu^{5} -$$$$71\!\cdots\!72$$$$\nu^{4} -$$$$32\!\cdots\!64$$$$\nu^{3} +$$$$23\!\cdots\!56$$$$\nu^{2} +$$$$45\!\cdots\!32$$$$\nu -$$$$88\!\cdots\!72$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$57\!\cdots\!29$$$$\nu^{15} +$$$$24\!\cdots\!09$$$$\nu^{14} +$$$$17\!\cdots\!58$$$$\nu^{13} -$$$$10\!\cdots\!01$$$$\nu^{12} -$$$$18\!\cdots\!23$$$$\nu^{11} +$$$$14\!\cdots\!59$$$$\nu^{10} +$$$$73\!\cdots\!10$$$$\nu^{9} -$$$$85\!\cdots\!11$$$$\nu^{8} +$$$$12\!\cdots\!84$$$$\nu^{7} +$$$$18\!\cdots\!00$$$$\nu^{6} -$$$$54\!\cdots\!64$$$$\nu^{5} +$$$$62\!\cdots\!96$$$$\nu^{4} +$$$$14\!\cdots\!44$$$$\nu^{3} -$$$$52\!\cdots\!92$$$$\nu^{2} -$$$$28\!\cdots\!12$$$$\nu +$$$$29\!\cdots\!84$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$68\!\cdots\!01$$$$\nu^{15} +$$$$66\!\cdots\!94$$$$\nu^{14} +$$$$19\!\cdots\!35$$$$\nu^{13} -$$$$23\!\cdots\!29$$$$\nu^{12} -$$$$17\!\cdots\!35$$$$\nu^{11} +$$$$29\!\cdots\!90$$$$\nu^{10} +$$$$16\!\cdots\!81$$$$\nu^{9} -$$$$15\!\cdots\!43$$$$\nu^{8} +$$$$47\!\cdots\!88$$$$\nu^{7} +$$$$26\!\cdots\!24$$$$\nu^{6} -$$$$17\!\cdots\!12$$$$\nu^{5} +$$$$22\!\cdots\!64$$$$\nu^{4} +$$$$32\!\cdots\!08$$$$\nu^{3} -$$$$60\!\cdots\!08$$$$\nu^{2} -$$$$88\!\cdots\!20$$$$\nu +$$$$14\!\cdots\!24$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!49$$$$\nu^{15} +$$$$12\!\cdots\!76$$$$\nu^{14} +$$$$50\!\cdots\!60$$$$\nu^{13} -$$$$44\!\cdots\!24$$$$\nu^{12} -$$$$52\!\cdots\!07$$$$\nu^{11} +$$$$57\!\cdots\!36$$$$\nu^{10} +$$$$17\!\cdots\!80$$$$\nu^{9} -$$$$32\!\cdots\!24$$$$\nu^{8} +$$$$34\!\cdots\!96$$$$\nu^{7} +$$$$66\!\cdots\!24$$$$\nu^{6} -$$$$24\!\cdots\!48$$$$\nu^{5} +$$$$69\!\cdots\!92$$$$\nu^{4} +$$$$66\!\cdots\!20$$$$\nu^{3} -$$$$26\!\cdots\!68$$$$\nu^{2} -$$$$12\!\cdots\!48$$$$\nu +$$$$12\!\cdots\!04$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!99$$$$\nu^{15} +$$$$25\!\cdots\!04$$$$\nu^{14} +$$$$53\!\cdots\!68$$$$\nu^{13} -$$$$11\!\cdots\!90$$$$\nu^{12} -$$$$57\!\cdots\!37$$$$\nu^{11} +$$$$24\!\cdots\!80$$$$\nu^{10} +$$$$25\!\cdots\!88$$$$\nu^{9} -$$$$16\!\cdots\!90$$$$\nu^{8} -$$$$20\!\cdots\!00$$$$\nu^{7} +$$$$40\!\cdots\!96$$$$\nu^{6} -$$$$96\!\cdots\!12$$$$\nu^{5} -$$$$15\!\cdots\!88$$$$\nu^{4} +$$$$26\!\cdots\!52$$$$\nu^{3} -$$$$24\!\cdots\!00$$$$\nu^{2} -$$$$52\!\cdots\!96$$$$\nu -$$$$52\!\cdots\!96$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{8}$$ $$=$$ $$($$$$23\!\cdots\!95$$$$\nu^{15} -$$$$48\!\cdots\!86$$$$\nu^{14} -$$$$68\!\cdots\!98$$$$\nu^{13} +$$$$26\!\cdots\!10$$$$\nu^{12} +$$$$72\!\cdots\!81$$$$\nu^{11} -$$$$43\!\cdots\!46$$$$\nu^{10} -$$$$29\!\cdots\!98$$$$\nu^{9} +$$$$27\!\cdots\!70$$$$\nu^{8} +$$$$70\!\cdots\!24$$$$\nu^{7} -$$$$59\!\cdots\!20$$$$\nu^{6} +$$$$17\!\cdots\!80$$$$\nu^{5} -$$$$49\!\cdots\!96$$$$\nu^{4} -$$$$42\!\cdots\!24$$$$\nu^{3} +$$$$22\!\cdots\!80$$$$\nu^{2} +$$$$94\!\cdots\!36$$$$\nu -$$$$10\!\cdots\!84$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{9}$$ $$=$$ $$($$$$15\!\cdots\!67$$$$\nu^{15} -$$$$56\!\cdots\!56$$$$\nu^{14} -$$$$45\!\cdots\!36$$$$\nu^{13} +$$$$25\!\cdots\!08$$$$\nu^{12} +$$$$48\!\cdots\!57$$$$\nu^{11} -$$$$36\!\cdots\!64$$$$\nu^{10} -$$$$18\!\cdots\!08$$$$\nu^{9} +$$$$22\!\cdots\!68$$$$\nu^{8} -$$$$94\!\cdots\!40$$$$\nu^{7} -$$$$47\!\cdots\!88$$$$\nu^{6} +$$$$15\!\cdots\!08$$$$\nu^{5} -$$$$47\!\cdots\!52$$$$\nu^{4} -$$$$43\!\cdots\!00$$$$\nu^{3} +$$$$32\!\cdots\!24$$$$\nu^{2} +$$$$84\!\cdots\!92$$$$\nu -$$$$10\!\cdots\!48$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$36\!\cdots\!39$$$$\nu^{15} +$$$$90\!\cdots\!56$$$$\nu^{14} +$$$$10\!\cdots\!74$$$$\nu^{13} -$$$$46\!\cdots\!66$$$$\nu^{12} -$$$$11\!\cdots\!25$$$$\nu^{11} +$$$$73\!\cdots\!36$$$$\nu^{10} +$$$$47\!\cdots\!58$$$$\nu^{9} -$$$$45\!\cdots\!14$$$$\nu^{8} -$$$$94\!\cdots\!16$$$$\nu^{7} +$$$$10\!\cdots\!32$$$$\nu^{6} -$$$$29\!\cdots\!28$$$$\nu^{5} -$$$$45\!\cdots\!24$$$$\nu^{4} +$$$$91\!\cdots\!44$$$$\nu^{3} -$$$$21\!\cdots\!16$$$$\nu^{2} -$$$$20\!\cdots\!36$$$$\nu +$$$$12\!\cdots\!28$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{11}$$ $$=$$ $$($$$$1508211484212094895295 \nu^{15} - 2042933163117031611288 \nu^{14} - 439253308654693863881418 \nu^{13} + 1450495914027627203357166 \nu^{12} + 46994815669119101240110065 \nu^{11} - 247874106788934307120833084 \nu^{10} - 1988938317293089652193634458 \nu^{9} + 16127251053578459040451501386 \nu^{8} + 10266896078569080179037095436 \nu^{7} - 364055953972190375764005961032 \nu^{6} + 981304131032841111857426210616 \nu^{5} - 92670014753316466982364742920 \nu^{4} - 2436280186130887921098606063792 \nu^{3} + 856097779910477052351091216032 \nu^{2} + 5323619205744740232273004797216 \nu - 5489463328251957179851909284576$$$$)/$$$$23\!\cdots\!84$$ $$\beta_{12}$$ $$=$$ $$($$$$52\!\cdots\!07$$$$\nu^{15} -$$$$12\!\cdots\!12$$$$\nu^{14} -$$$$15\!\cdots\!80$$$$\nu^{13} +$$$$64\!\cdots\!54$$$$\nu^{12} +$$$$16\!\cdots\!57$$$$\nu^{11} -$$$$10\!\cdots\!84$$$$\nu^{10} -$$$$66\!\cdots\!68$$$$\nu^{9} +$$$$64\!\cdots\!26$$$$\nu^{8} +$$$$98\!\cdots\!60$$$$\nu^{7} -$$$$14\!\cdots\!48$$$$\nu^{6} +$$$$41\!\cdots\!64$$$$\nu^{5} -$$$$32\!\cdots\!72$$$$\nu^{4} -$$$$11\!\cdots\!52$$$$\nu^{3} +$$$$29\!\cdots\!88$$$$\nu^{2} +$$$$21\!\cdots\!68$$$$\nu -$$$$17\!\cdots\!76$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{13}$$ $$=$$ $$($$$$30\!\cdots\!29$$$$\nu^{15} -$$$$12\!\cdots\!22$$$$\nu^{14} -$$$$89\!\cdots\!50$$$$\nu^{13} +$$$$52\!\cdots\!54$$$$\nu^{12} +$$$$94\!\cdots\!63$$$$\nu^{11} -$$$$75\!\cdots\!34$$$$\nu^{10} -$$$$35\!\cdots\!58$$$$\nu^{9} +$$$$45\!\cdots\!66$$$$\nu^{8} -$$$$24\!\cdots\!04$$$$\nu^{7} -$$$$97\!\cdots\!28$$$$\nu^{6} +$$$$32\!\cdots\!16$$$$\nu^{5} -$$$$42\!\cdots\!52$$$$\nu^{4} -$$$$95\!\cdots\!20$$$$\nu^{3} +$$$$22\!\cdots\!48$$$$\nu^{2} +$$$$19\!\cdots\!00$$$$\nu -$$$$15\!\cdots\!12$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{14}$$ $$=$$ $$($$$$65\!\cdots\!01$$$$\nu^{15} -$$$$71\!\cdots\!58$$$$\nu^{14} -$$$$19\!\cdots\!76$$$$\nu^{13} +$$$$59\!\cdots\!96$$$$\nu^{12} +$$$$20\!\cdots\!31$$$$\nu^{11} -$$$$10\!\cdots\!82$$$$\nu^{10} -$$$$86\!\cdots\!00$$$$\nu^{9} +$$$$69\!\cdots\!80$$$$\nu^{8} +$$$$49\!\cdots\!92$$$$\nu^{7} -$$$$16\!\cdots\!48$$$$\nu^{6} +$$$$41\!\cdots\!80$$$$\nu^{5} +$$$$79\!\cdots\!80$$$$\nu^{4} -$$$$12\!\cdots\!28$$$$\nu^{3} +$$$$29\!\cdots\!56$$$$\nu^{2} +$$$$28\!\cdots\!36$$$$\nu -$$$$18\!\cdots\!56$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{15}$$ $$=$$ $$($$$$17\!\cdots\!73$$$$\nu^{15} -$$$$50\!\cdots\!70$$$$\nu^{14} -$$$$52\!\cdots\!52$$$$\nu^{13} +$$$$11\!\cdots\!52$$$$\nu^{12} +$$$$57\!\cdots\!39$$$$\nu^{11} -$$$$23\!\cdots\!22$$$$\nu^{10} -$$$$26\!\cdots\!56$$$$\nu^{9} +$$$$16\!\cdots\!88$$$$\nu^{8} +$$$$30\!\cdots\!72$$$$\nu^{7} -$$$$39\!\cdots\!12$$$$\nu^{6} +$$$$71\!\cdots\!32$$$$\nu^{5} +$$$$69\!\cdots\!40$$$$\nu^{4} -$$$$21\!\cdots\!84$$$$\nu^{3} -$$$$99\!\cdots\!32$$$$\nu^{2} +$$$$40\!\cdots\!28$$$$\nu -$$$$22\!\cdots\!64$$$$)/$$$$70\!\cdots\!88$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{14} + \beta_{13} - 21 \beta_{12} + 7 \beta_{11} - \beta_{10} - 15 \beta_{7} - 12 \beta_{6} + 7 \beta_{3} + 78$$$$)/378$$ $$\nu^{2}$$ $$=$$ $$($$$$15 \beta_{15} + 3 \beta_{14} - 20 \beta_{13} + 21 \beta_{12} + \beta_{11} - 43 \beta_{10} - 57 \beta_{8} + 195 \beta_{7} + 30 \beta_{6} - 21 \beta_{5} + 301 \beta_{3} + 54 \beta_{1} + 14106$$$$)/378$$ $$\nu^{3}$$ $$=$$ $$($$$$-207 \beta_{15} + 801 \beta_{14} + 204 \beta_{13} - 2520 \beta_{12} + 1655 \beta_{11} + 1119 \beta_{10} - 378 \beta_{9} - 423 \beta_{8} - 2952 \beta_{7} - 2412 \beta_{6} + 441 \beta_{5} - 756 \beta_{4} - 1029 \beta_{3} + 756 \beta_{2} + 162 \beta_{1} - 79956$$$$)/756$$ $$\nu^{4}$$ $$=$$ $$($$$$1890 \beta_{15} - 966 \beta_{14} - 3304 \beta_{13} + 4851 \beta_{12} - 3458 \beta_{11} - 7154 \beta_{10} + 378 \beta_{9} - 7686 \beta_{8} + 16929 \beta_{7} + 3690 \beta_{6} - 3654 \beta_{5} + 2268 \beta_{4} + 30422 \beta_{3} - 6048 \beta_{2} + 10746 \beta_{1} + 1002852$$$$)/378$$ $$\nu^{5}$$ $$=$$ $$($$$$-38925 \beta_{15} + 89691 \beta_{14} + 36932 \beta_{13} - 194082 \beta_{12} + 201581 \beta_{11} + 161833 \beta_{10} - 73080 \beta_{9} - 14625 \beta_{8} - 283926 \beta_{7} - 196548 \beta_{6} + 72135 \beta_{5} - 196560 \beta_{4} - 365491 \beta_{3} + 194040 \beta_{2} - 88920 \beta_{1} - 10523976$$$$)/756$$ $$\nu^{6}$$ $$=$$ $$($$$$247251 \beta_{15} - 218997 \beta_{14} - 375588 \beta_{13} + 575946 \beta_{12} - 671387 \beta_{11} - 906399 \beta_{10} + 214515 \beta_{9} - 837897 \beta_{8} + 1532430 \beta_{7} + 462636 \beta_{6} - 497553 \beta_{5} + 670950 \beta_{4} + 3253509 \beta_{3} - 1372140 \beta_{2} + 1793205 \beta_{1} + 82934064$$$$)/378$$ $$\nu^{7}$$ $$=$$ $$($$$$-5457375 \beta_{15} + 9167565 \beta_{14} + 5256004 \beta_{13} - 16598400 \beta_{12} + 24293255 \beta_{11} + 19349591 \beta_{10} - 11248146 \beta_{9} + 2556477 \beta_{8} - 27504192 \beta_{7} - 16546848 \beta_{6} + 10081701 \beta_{5} - 32036004 \beta_{4} - 51500141 \beta_{3} + 35283528 \beta_{2} - 24950142 \beta_{1} - 1143777696$$$$)/756$$ $$\nu^{8}$$ $$=$$ $$($$$$31761492 \beta_{15} - 30732828 \beta_{14} - 38908016 \beta_{13} + 61087341 \beta_{12} - 99578612 \beta_{11} - 102373012 \beta_{10} + 44463384 \beta_{9} - 89225724 \beta_{8} + 143167695 \beta_{7} + 52162518 \beta_{6} - 63447132 \beta_{5} + 131107536 \beta_{4} + 341214748 \beta_{3} - 225763776 \beta_{2} + 258792984 \beta_{1} + 7342936392$$$$)/378$$ $$\nu^{9}$$ $$=$$ $$($$$$-705685329 \beta_{15} + 905768487 \beta_{14} + 630291996 \beta_{13} - 1491982674 \beta_{12} + 2914570457 \beta_{11} + 2109655557 \beta_{10} - 1553081040 \beta_{9} + 690659955 \beta_{8} - 2667928134 \beta_{7} - 1451788764 \beta_{6} + 1320955083 \beta_{5} - 4478646816 \beta_{4} - 5995144071 \beta_{3} + 5381818848 \beta_{2} - 4415367024 \beta_{1} - 117465566928$$$$)/756$$ $$\nu^{10}$$ $$=$$ $$($$$$3983508633 \beta_{15} - 3593813727 \beta_{14} - 3873978556 \beta_{13} + 6187643826 \beta_{12} - 13391555297 \beta_{11} - 10779333005 \beta_{10} + 7062339375 \beta_{9} - 9641062059 \beta_{8} + 13519561638 \beta_{7} + 5484433020 \beta_{6} - 7865474883 \beta_{5} + 20610765630 \beta_{4} + 34535195807 \beta_{3} - 32288423580 \beta_{2} + 34454087721 \beta_{1} + 670379739216$$$$)/378$$ $$\nu^{11}$$ $$=$$ $$($$$$-88019325471 \beta_{15} + 87423864621 \beta_{14} + 68068170740 \beta_{13} - 136607395848 \beta_{12} + 348057970967 \beta_{11} + 216338895511 \beta_{10} - 201306893634 \beta_{9} + 114069661629 \beta_{8} - 256189015128 \beta_{7} - 130393288752 \beta_{6} + 166608558501 \beta_{5} - 582527009988 \beta_{4} - 635602694125 \beta_{3} + 740695680648 \beta_{2} - 652161282894 \beta_{1} - 11644698741504$$$$)/756$$ $$\nu^{12}$$ $$=$$ $$($$$$69826854858 \beta_{15} - 54148201326 \beta_{14} - 53369153208 \beta_{13} + 86322951231 \beta_{12} - 244021801370 \beta_{11} - 153640517418 \beta_{10} + 140859532440 \beta_{9} - 151770071838 \beta_{8} + 181047726741 \beta_{7} + 78162369810 \beta_{6} - 136670744334 \beta_{5} + 409904987184 \beta_{4} + 481197260142 \beta_{3} - 609337507680 \beta_{2} + 623361422376 \beta_{1} + 8788001773704$$$$)/54$$ $$\nu^{13}$$ $$=$$ $$($$$$-10715369975505 \beta_{15} + 8205381038199 \beta_{14} + 6819476356444 \beta_{13} - 12444995066322 \beta_{12} + 41256721282841 \beta_{11} + 21060178184021 \beta_{10} - 25034840334504 \beta_{9} + 15991071712083 \beta_{8} - 24038443432518 \beta_{7} - 11724814079292 \beta_{6} + 20434048828011 \beta_{5} - 72543618293328 \beta_{4} - 63021735353591 \beta_{3} + 95498829657504 \beta_{2} - 87324431058072 \beta_{1} - 1113034475400816$$$$)/756$$ $$\nu^{14}$$ $$=$$ $$($$$$58781025379719 \beta_{15} - 37030892703297 \beta_{14} - 34684564933124 \beta_{13} + 56607602007642 \beta_{12} - 210142914779519 \beta_{11} - 101893537055827 \beta_{10} + 127569023016435 \beta_{9} - 118772375499189 \beta_{8} + 115884521667102 \beta_{7} + 51884296051884 \beta_{6} - 114394442978781 \beta_{5} + 370792334073798 \beta_{4} + 314924159092417 \beta_{3} - 535058753448492 \beta_{2} + 533742420199077 \beta_{1} + 5550942388854192$$$$)/378$$ $$\nu^{15}$$ $$=$$ $$($$$$-1277826382488615 \beta_{15} + 739525552505109 \beta_{14} + 638478540273492 \beta_{13} - 1102930179866256 \beta_{12} + 4839594059223839 \beta_{11} + 1939900022482959 \beta_{10} - 3020067066896082 \beta_{9} + 2060641280479125 \beta_{8} - 2169287020890672 \beta_{7} - 1030495300713696 \beta_{6} + 2448113612899773 \beta_{5} - 8757454706027748 \beta_{4} - 5866179196536981 \beta_{3} + 11772210945075336 \beta_{2} - 10996020069922494 \beta_{1} - 101543997959861376$$$$)/756$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 4.65022 − 0.707107i 4.65022 + 0.707107i −10.4548 − 0.707107i −10.4548 + 0.707107i −1.35249 − 0.707107i −1.35249 + 0.707107i 8.15703 + 0.707107i 8.15703 − 0.707107i 5.70754 + 0.707107i 5.70754 − 0.707107i 1.09700 − 0.707107i 1.09700 + 0.707107i −8.00527 − 0.707107i −8.00527 + 0.707107i 2.20073 − 0.707107i 2.20073 + 0.707107i
0 0 0 −20.2518 0 0 0 0 0
881.2 0 0 0 −20.2518 0 0 0 0 0
881.3 0 0 0 −8.73625 0 0 0 0 0
881.4 0 0 0 −8.73625 0 0 0 0 0
881.5 0 0 0 −8.54212 0 0 0 0 0
881.6 0 0 0 −8.54212 0 0 0 0 0
881.7 0 0 0 −6.82452 0 0 0 0 0
881.8 0 0 0 −6.82452 0 0 0 0 0
881.9 0 0 0 6.82452 0 0 0 0 0
881.10 0 0 0 6.82452 0 0 0 0 0
881.11 0 0 0 8.54212 0 0 0 0 0
881.12 0 0 0 8.54212 0 0 0 0 0
881.13 0 0 0 8.73625 0 0 0 0 0
881.14 0 0 0 8.73625 0 0 0 0 0
881.15 0 0 0 20.2518 0 0 0 0 0
881.16 0 0 0 20.2518 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.f.a 16
3.b odd 2 1 inner 1764.4.f.a 16
7.b odd 2 1 inner 1764.4.f.a 16
7.c even 3 1 252.4.t.a 16
7.c even 3 1 1764.4.t.b 16
7.d odd 6 1 252.4.t.a 16
7.d odd 6 1 1764.4.t.b 16
21.c even 2 1 inner 1764.4.f.a 16
21.g even 6 1 252.4.t.a 16
21.g even 6 1 1764.4.t.b 16
21.h odd 6 1 252.4.t.a 16
21.h odd 6 1 1764.4.t.b 16
28.f even 6 1 1008.4.bt.b 16
28.g odd 6 1 1008.4.bt.b 16
84.j odd 6 1 1008.4.bt.b 16
84.n even 6 1 1008.4.bt.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a 16 7.c even 3 1
252.4.t.a 16 7.d odd 6 1
252.4.t.a 16 21.g even 6 1
252.4.t.a 16 21.h odd 6 1
1008.4.bt.b 16 28.f even 6 1
1008.4.bt.b 16 28.g odd 6 1
1008.4.bt.b 16 84.j odd 6 1
1008.4.bt.b 16 84.n even 6 1
1764.4.f.a 16 1.a even 1 1 trivial
1764.4.f.a 16 3.b odd 2 1 inner
1764.4.f.a 16 7.b odd 2 1 inner
1764.4.f.a 16 21.c even 2 1 inner
1764.4.t.b 16 7.c even 3 1
1764.4.t.b 16 7.d odd 6 1
1764.4.t.b 16 21.g even 6 1
1764.4.t.b 16 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 606 T_{5}^{6} + 92853 T_{5}^{4} - 5395140 T_{5}^{2} + 106378596$$ acting on $$S_{4}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 106378596 - 5395140 T^{2} + 92853 T^{4} - 606 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 84182380164 + 1463257548 T^{2} + 5194125 T^{4} + 4806 T^{6} + T^{8} )^{2}$$
$13$ $$( 22573860516 + 426062052 T^{2} + 2267613 T^{4} + 2826 T^{6} + T^{8} )^{2}$$
$17$ $$( 70607585620224 - 141379508160 T^{2} + 91834020 T^{4} - 21396 T^{6} + T^{8} )^{2}$$
$19$ $$( 46677071171844 + 379172228220 T^{2} + 262304325 T^{4} + 30858 T^{6} + T^{8} )^{2}$$
$23$ $$( 22211376380153856 + 10406803131648 T^{2} + 1497771972 T^{4} + 71460 T^{6} + T^{8} )^{2}$$
$29$ $$( 416548977121475136 + 86086124811216 T^{2} + 5723596161 T^{4} + 133110 T^{6} + T^{8} )^{2}$$
$31$ $$( 17938763430477561 + 336040273495584 T^{2} + 14869021014 T^{4} + 213456 T^{6} + T^{8} )^{2}$$
$37$ $$( -424087754 - 14086516 T - 109317 T^{2} + 38 T^{3} + T^{4} )^{4}$$
$41$ $$( 3423608728179335424 - 403281542442816 T^{2} + 16046933220 T^{4} - 240060 T^{6} + T^{8} )^{2}$$
$43$ $$( -6254809742 + 68765576 T - 146451 T^{2} - 352 T^{3} + T^{4} )^{4}$$
$47$ $$( 68567622309159789456 - 4436251230240000 T^{2} + 83116153944 T^{4} - 522912 T^{6} + T^{8} )^{2}$$
$53$ $$( 1998740432850076224 + 378603077947344 T^{2} + 18621022689 T^{4} + 252054 T^{6} + T^{8} )^{2}$$
$59$ $$( 236957268508674624 - 4728735561416688 T^{2} + 134996565465 T^{4} - 705414 T^{6} + T^{8} )^{2}$$
$61$ $$( 10077652565550144 + 15350015516832 T^{2} + 2414692692 T^{4} + 93852 T^{6} + T^{8} )^{2}$$
$67$ $$( 30097773916 + 189716044 T - 345693 T^{2} - 764 T^{3} + T^{4} )^{4}$$
$71$ $$($$$$17\!\cdots\!44$$$$+ 42297868577973888 T^{2} + 337035071448 T^{4} + 1006848 T^{6} + T^{8} )^{2}$$
$73$ $$( 11798164103488007184 + 5973381293368680 T^{2} + 408979621881 T^{4} + 1634910 T^{6} + T^{8} )^{2}$$
$79$ $$( 1237158301 + 173275438 T - 635574 T^{2} - 182 T^{3} + T^{4} )^{4}$$
$83$ $$($$$$10\!\cdots\!84$$$$- 79735840708215204 T^{2} + 734790325365 T^{4} - 1589970 T^{6} + T^{8} )^{2}$$
$89$ $$($$$$31\!\cdots\!64$$$$- 534309537696420096 T^{2} + 2392343285328 T^{4} - 3076488 T^{6} + T^{8} )^{2}$$
$97$ $$($$$$47\!\cdots\!36$$$$+ 3276863917344744120 T^{2} + 6899032044873 T^{4} + 5135454 T^{6} + T^{8} )^{2}$$