# Properties

 Label 252.4.t.a Level $252$ Weight $4$ Character orbit 252.t Analytic conductor $14.868$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{18}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{5} + ( 1 + 2 \beta_{1} - \beta_{6} - \beta_{8} ) q^{7} +O(q^{10})$$ $$q + \beta_{9} q^{5} + ( 1 + 2 \beta_{1} - \beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{5} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} ) q^{13} + ( -\beta_{3} + 3 \beta_{5} - 2 \beta_{9} + \beta_{11} - \beta_{14} ) q^{17} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{6} - \beta_{12} + \beta_{13} ) q^{19} + ( -\beta_{3} - 2 \beta_{5} + \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{14} ) q^{23} + ( -26 - 27 \beta_{1} + 5 \beta_{2} + 4 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + \beta_{12} - 2 \beta_{13} ) q^{25} + ( -4 \beta_{3} - 5 \beta_{5} + 12 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{14} - \beta_{15} ) q^{29} + ( -25 + 34 \beta_{1} + 6 \beta_{2} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 4 \beta_{12} ) q^{31} + ( -5 \beta_{3} + \beta_{4} + 9 \beta_{5} + 4 \beta_{9} + 2 \beta_{11} - \beta_{15} ) q^{35} + ( 7 - 6 \beta_{1} + 4 \beta_{2} - 9 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} ) q^{37} + ( -4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + \beta_{14} + \beta_{15} ) q^{41} + ( 92 + 8 \beta_{1} + 5 \beta_{2} - 5 \beta_{6} + 5 \beta_{7} - 11 \beta_{8} - 6 \beta_{12} + 3 \beta_{13} ) q^{43} + ( -9 \beta_{3} + \beta_{4} - 3 \beta_{5} + 17 \beta_{9} + 3 \beta_{10} - 3 \beta_{14} + \beta_{15} ) q^{47} + ( 25 - 2 \beta_{1} + 4 \beta_{6} - 2 \beta_{8} + 7 \beta_{13} ) q^{49} + ( -5 \beta_{3} + \beta_{4} - 11 \beta_{5} - 5 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{14} ) q^{53} + ( 60 + 111 \beta_{1} + 11 \beta_{2} + 11 \beta_{8} - 2 \beta_{13} ) q^{55} + ( -\beta_{4} + 24 \beta_{5} - 25 \beta_{9} + 4 \beta_{11} + 4 \beta_{14} + 2 \beta_{15} ) q^{59} + ( -136 - 70 \beta_{1} - 3 \beta_{2} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} ) q^{61} + ( -\beta_{4} - 22 \beta_{5} + 15 \beta_{9} - 8 \beta_{14} + \beta_{15} ) q^{65} + ( -194 - 188 \beta_{1} + 13 \beta_{2} + 7 \beta_{6} + 7 \beta_{8} + 6 \beta_{12} - 12 \beta_{13} ) q^{67} + ( 9 \beta_{3} + 3 \beta_{5} + 4 \beta_{9} + \beta_{10} + \beta_{11} + 10 \beta_{14} - 9 \beta_{15} ) q^{71} + ( -118 + 107 \beta_{1} - 17 \beta_{2} + 20 \beta_{6} - 26 \beta_{7} - 6 \beta_{8} + 3 \beta_{12} ) q^{73} + ( 11 \beta_{3} + 9 \beta_{4} + 40 \beta_{5} - 7 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 5 \beta_{14} - 8 \beta_{15} ) q^{77} + ( -13 + 39 \beta_{1} + 9 \beta_{2} + 17 \beta_{6} + 22 \beta_{7} + 22 \beta_{8} + 4 \beta_{12} + 4 \beta_{13} ) q^{79} + ( 3 \beta_{3} - 16 \beta_{4} - 32 \beta_{5} + 2 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} + 3 \beta_{14} + 8 \beta_{15} ) q^{83} + ( 452 - 20 \beta_{1} - 21 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} + 19 \beta_{8} - 2 \beta_{12} + \beta_{13} ) q^{85} + ( 18 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 20 \beta_{9} - 10 \beta_{10} + 4 \beta_{14} + 8 \beta_{15} ) q^{89} + ( -30 - 368 \beta_{1} - 7 \beta_{2} - 5 \beta_{6} - 5 \beta_{8} - 7 \beta_{12} ) q^{91} + ( 12 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} - 31 \beta_{9} + 10 \beta_{10} - 5 \beta_{11} + 5 \beta_{14} ) q^{95} + ( 294 + 611 \beta_{1} - 42 \beta_{2} - 10 \beta_{6} - 10 \beta_{7} - 42 \beta_{8} - \beta_{13} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{7} + O(q^{10})$$ $$16q - 4q^{7} - 72q^{19} - 212q^{25} - 708q^{31} + 76q^{37} + 1408q^{43} + 400q^{49} - 1632q^{61} - 1528q^{67} - 2700q^{73} - 364q^{79} + 7392q^{85} + 2472q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1229561931 \nu^{15} + 2498779578 \nu^{14} + 394581872608 \nu^{13} - 1317945813970 \nu^{12} - 48388978127485 \nu^{11} + 216626897426602 \nu^{10} + 2708906003712780 \nu^{9} - 15005503515059110 \nu^{8} - 62911168783150524 \nu^{7} + 432720973755039680 \nu^{6} + 328948851683104464 \nu^{5} - 3693022570101876040 \nu^{4} - 1858925629616754192 \nu^{3} + 13888390311265898368 \nu^{2} + 3151631933173997312 \nu - 24728853666847532128$$$$)/ 13574653155388836608$$ $$\beta_{2}$$ $$=$$ $$($$$$61\!\cdots\!81$$$$\nu^{15} -$$$$15\!\cdots\!75$$$$\nu^{14} -$$$$24\!\cdots\!20$$$$\nu^{13} +$$$$44\!\cdots\!00$$$$\nu^{12} +$$$$28\!\cdots\!71$$$$\nu^{11} -$$$$51\!\cdots\!97$$$$\nu^{10} -$$$$90\!\cdots\!44$$$$\nu^{9} +$$$$27\!\cdots\!00$$$$\nu^{8} -$$$$29\!\cdots\!56$$$$\nu^{7} -$$$$55\!\cdots\!72$$$$\nu^{6} +$$$$16\!\cdots\!04$$$$\nu^{5} +$$$$28\!\cdots\!96$$$$\nu^{4} -$$$$43\!\cdots\!48$$$$\nu^{3} +$$$$20\!\cdots\!32$$$$\nu^{2} +$$$$86\!\cdots\!56$$$$\nu -$$$$20\!\cdots\!52$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!52$$$$\nu^{15} -$$$$21\!\cdots\!81$$$$\nu^{14} +$$$$29\!\cdots\!22$$$$\nu^{13} +$$$$56\!\cdots\!88$$$$\nu^{12} -$$$$36\!\cdots\!28$$$$\nu^{11} -$$$$57\!\cdots\!55$$$$\nu^{10} +$$$$28\!\cdots\!54$$$$\nu^{9} +$$$$26\!\cdots\!20$$$$\nu^{8} -$$$$13\!\cdots\!20$$$$\nu^{7} -$$$$49\!\cdots\!12$$$$\nu^{6} +$$$$30\!\cdots\!56$$$$\nu^{5} +$$$$60\!\cdots\!16$$$$\nu^{4} -$$$$13\!\cdots\!12$$$$\nu^{3} -$$$$21\!\cdots\!72$$$$\nu^{2} +$$$$23\!\cdots\!88$$$$\nu -$$$$56\!\cdots\!12$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{4}$$ $$=$$ $$($$$$24\!\cdots\!91$$$$\nu^{15} +$$$$62\!\cdots\!62$$$$\nu^{14} -$$$$70\!\cdots\!74$$$$\nu^{13} -$$$$47\!\cdots\!02$$$$\nu^{12} +$$$$76\!\cdots\!05$$$$\nu^{11} -$$$$71\!\cdots\!38$$$$\nu^{10} -$$$$36\!\cdots\!50$$$$\nu^{9} +$$$$78\!\cdots\!74$$$$\nu^{8} +$$$$61\!\cdots\!96$$$$\nu^{7} -$$$$13\!\cdots\!44$$$$\nu^{6} +$$$$84\!\cdots\!92$$$$\nu^{5} -$$$$34\!\cdots\!32$$$$\nu^{4} +$$$$84\!\cdots\!76$$$$\nu^{3} +$$$$10\!\cdots\!48$$$$\nu^{2} -$$$$20\!\cdots\!08$$$$\nu -$$$$18\!\cdots\!08$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!96$$$$\nu^{15} +$$$$28\!\cdots\!71$$$$\nu^{14} +$$$$39\!\cdots\!79$$$$\nu^{13} -$$$$16\!\cdots\!25$$$$\nu^{12} -$$$$39\!\cdots\!36$$$$\nu^{11} +$$$$27\!\cdots\!69$$$$\nu^{10} +$$$$14\!\cdots\!13$$$$\nu^{9} -$$$$16\!\cdots\!47$$$$\nu^{8} +$$$$11\!\cdots\!24$$$$\nu^{7} +$$$$35\!\cdots\!20$$$$\nu^{6} -$$$$13\!\cdots\!36$$$$\nu^{5} +$$$$71\!\cdots\!72$$$$\nu^{4} +$$$$32\!\cdots\!64$$$$\nu^{3} -$$$$23\!\cdots\!56$$$$\nu^{2} -$$$$45\!\cdots\!32$$$$\nu +$$$$88\!\cdots\!72$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!72$$$$\nu^{15} +$$$$22\!\cdots\!97$$$$\nu^{14} +$$$$11\!\cdots\!66$$$$\nu^{13} -$$$$30\!\cdots\!92$$$$\nu^{12} -$$$$11\!\cdots\!96$$$$\nu^{11} +$$$$57\!\cdots\!71$$$$\nu^{10} +$$$$47\!\cdots\!98$$$$\nu^{9} -$$$$38\!\cdots\!36$$$$\nu^{8} -$$$$11\!\cdots\!72$$$$\nu^{7} +$$$$85\!\cdots\!80$$$$\nu^{6} -$$$$28\!\cdots\!36$$$$\nu^{5} +$$$$15\!\cdots\!64$$$$\nu^{4} +$$$$82\!\cdots\!44$$$$\nu^{3} -$$$$11\!\cdots\!48$$$$\nu^{2} -$$$$16\!\cdots\!80$$$$\nu +$$$$23\!\cdots\!48$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$59\!\cdots\!29$$$$\nu^{15} +$$$$37\!\cdots\!61$$$$\nu^{14} +$$$$17\!\cdots\!02$$$$\nu^{13} -$$$$14\!\cdots\!86$$$$\nu^{12} -$$$$18\!\cdots\!07$$$$\nu^{11} +$$$$18\!\cdots\!27$$$$\nu^{10} +$$$$62\!\cdots\!94$$$$\nu^{9} -$$$$10\!\cdots\!30$$$$\nu^{8} +$$$$11\!\cdots\!84$$$$\nu^{7} +$$$$21\!\cdots\!16$$$$\nu^{6} -$$$$83\!\cdots\!48$$$$\nu^{5} +$$$$37\!\cdots\!60$$$$\nu^{4} +$$$$23\!\cdots\!28$$$$\nu^{3} -$$$$18\!\cdots\!96$$$$\nu^{2} -$$$$43\!\cdots\!80$$$$\nu +$$$$59\!\cdots\!04$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{8}$$ $$=$$ $$($$$$10\!\cdots\!68$$$$\nu^{15} -$$$$33\!\cdots\!01$$$$\nu^{14} -$$$$32\!\cdots\!84$$$$\nu^{13} +$$$$15\!\cdots\!72$$$$\nu^{12} +$$$$34\!\cdots\!60$$$$\nu^{11} -$$$$23\!\cdots\!43$$$$\nu^{10} -$$$$14\!\cdots\!56$$$$\nu^{9} +$$$$14\!\cdots\!32$$$$\nu^{8} +$$$$78\!\cdots\!52$$$$\nu^{7} -$$$$31\!\cdots\!08$$$$\nu^{6} +$$$$92\!\cdots\!20$$$$\nu^{5} -$$$$22\!\cdots\!48$$$$\nu^{4} -$$$$25\!\cdots\!28$$$$\nu^{3} +$$$$76\!\cdots\!04$$$$\nu^{2} +$$$$48\!\cdots\!36$$$$\nu -$$$$37\!\cdots\!96$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{9}$$ $$=$$ $$($$$$12\!\cdots\!66$$$$\nu^{15} -$$$$16\!\cdots\!29$$$$\nu^{14} -$$$$37\!\cdots\!16$$$$\nu^{13} +$$$$76\!\cdots\!36$$$$\nu^{12} +$$$$41\!\cdots\!06$$$$\nu^{11} -$$$$15\!\cdots\!07$$$$\nu^{10} -$$$$19\!\cdots\!88$$$$\nu^{9} +$$$$11\!\cdots\!08$$$$\nu^{8} +$$$$24\!\cdots\!24$$$$\nu^{7} -$$$$28\!\cdots\!24$$$$\nu^{6} +$$$$45\!\cdots\!40$$$$\nu^{5} +$$$$58\!\cdots\!36$$$$\nu^{4} -$$$$14\!\cdots\!72$$$$\nu^{3} -$$$$92\!\cdots\!48$$$$\nu^{2} +$$$$28\!\cdots\!04$$$$\nu -$$$$11\!\cdots\!36$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{10}$$ $$=$$ $$($$$$13\!\cdots\!89$$$$\nu^{15} -$$$$71\!\cdots\!40$$$$\nu^{14} -$$$$39\!\cdots\!20$$$$\nu^{13} +$$$$28\!\cdots\!00$$$$\nu^{12} +$$$$38\!\cdots\!27$$$$\nu^{11} -$$$$39\!\cdots\!76$$$$\nu^{10} -$$$$10\!\cdots\!48$$$$\nu^{9} +$$$$22\!\cdots\!96$$$$\nu^{8} -$$$$38\!\cdots\!04$$$$\nu^{7} -$$$$41\!\cdots\!24$$$$\nu^{6} +$$$$20\!\cdots\!20$$$$\nu^{5} -$$$$21\!\cdots\!44$$$$\nu^{4} -$$$$39\!\cdots\!96$$$$\nu^{3} +$$$$61\!\cdots\!76$$$$\nu^{2} +$$$$87\!\cdots\!44$$$$\nu -$$$$16\!\cdots\!72$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{11}$$ $$=$$ $$($$$$14\!\cdots\!19$$$$\nu^{15} -$$$$90\!\cdots\!00$$$$\nu^{14} -$$$$43\!\cdots\!02$$$$\nu^{13} +$$$$34\!\cdots\!32$$$$\nu^{12} +$$$$42\!\cdots\!93$$$$\nu^{11} -$$$$47\!\cdots\!04$$$$\nu^{10} -$$$$11\!\cdots\!74$$$$\nu^{9} +$$$$26\!\cdots\!12$$$$\nu^{8} -$$$$47\!\cdots\!16$$$$\nu^{7} -$$$$52\!\cdots\!00$$$$\nu^{6} +$$$$24\!\cdots\!72$$$$\nu^{5} -$$$$18\!\cdots\!76$$$$\nu^{4} -$$$$60\!\cdots\!32$$$$\nu^{3} +$$$$52\!\cdots\!32$$$$\nu^{2} +$$$$14\!\cdots\!52$$$$\nu -$$$$16\!\cdots\!68$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{12}$$ $$=$$ $$($$$$15\!\cdots\!47$$$$\nu^{15} -$$$$87\!\cdots\!12$$$$\nu^{14} -$$$$43\!\cdots\!06$$$$\nu^{13} +$$$$11\!\cdots\!78$$$$\nu^{12} +$$$$46\!\cdots\!37$$$$\nu^{11} -$$$$21\!\cdots\!52$$$$\nu^{10} -$$$$19\!\cdots\!58$$$$\nu^{9} +$$$$14\!\cdots\!38$$$$\nu^{8} +$$$$12\!\cdots\!84$$$$\nu^{7} -$$$$33\!\cdots\!92$$$$\nu^{6} +$$$$89\!\cdots\!08$$$$\nu^{5} +$$$$32\!\cdots\!48$$$$\nu^{4} -$$$$24\!\cdots\!52$$$$\nu^{3} -$$$$50\!\cdots\!88$$$$\nu^{2} +$$$$45\!\cdots\!84$$$$\nu -$$$$21\!\cdots\!56$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!21$$$$\nu^{15} +$$$$36\!\cdots\!98$$$$\nu^{14} +$$$$31\!\cdots\!08$$$$\nu^{13} -$$$$16\!\cdots\!34$$$$\nu^{12} -$$$$32\!\cdots\!63$$$$\nu^{11} +$$$$24\!\cdots\!54$$$$\nu^{10} +$$$$12\!\cdots\!72$$$$\nu^{9} -$$$$14\!\cdots\!30$$$$\nu^{8} +$$$$43\!\cdots\!16$$$$\nu^{7} +$$$$31\!\cdots\!00$$$$\nu^{6} -$$$$10\!\cdots\!68$$$$\nu^{5} +$$$$21\!\cdots\!60$$$$\nu^{4} +$$$$28\!\cdots\!20$$$$\nu^{3} -$$$$17\!\cdots\!76$$$$\nu^{2} -$$$$54\!\cdots\!28$$$$\nu +$$$$63\!\cdots\!76$$$$)/$$$$35\!\cdots\!44$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!43$$$$\nu^{15} +$$$$69\!\cdots\!67$$$$\nu^{14} +$$$$65\!\cdots\!76$$$$\nu^{13} -$$$$32\!\cdots\!98$$$$\nu^{12} -$$$$69\!\cdots\!53$$$$\nu^{11} +$$$$48\!\cdots\!01$$$$\nu^{10} +$$$$26\!\cdots\!44$$$$\nu^{9} -$$$$29\!\cdots\!90$$$$\nu^{8} +$$$$67\!\cdots\!96$$$$\nu^{7} +$$$$64\!\cdots\!32$$$$\nu^{6} -$$$$20\!\cdots\!36$$$$\nu^{5} +$$$$43\!\cdots\!80$$$$\nu^{4} +$$$$55\!\cdots\!36$$$$\nu^{3} -$$$$20\!\cdots\!08$$$$\nu^{2} -$$$$11\!\cdots\!24$$$$\nu +$$$$11\!\cdots\!96$$$$)/$$$$70\!\cdots\!88$$ $$\beta_{15}$$ $$=$$ $$($$$$28\!\cdots\!32$$$$\nu^{15} -$$$$52\!\cdots\!45$$$$\nu^{14} -$$$$83\!\cdots\!02$$$$\nu^{13} +$$$$31\!\cdots\!26$$$$\nu^{12} +$$$$88\!\cdots\!52$$$$\nu^{11} -$$$$51\!\cdots\!43$$$$\nu^{10} -$$$$36\!\cdots\!42$$$$\nu^{9} +$$$$32\!\cdots\!50$$$$\nu^{8} +$$$$77\!\cdots\!76$$$$\nu^{7} -$$$$71\!\cdots\!60$$$$\nu^{6} +$$$$21\!\cdots\!00$$$$\nu^{5} -$$$$71\!\cdots\!32$$$$\nu^{4} -$$$$51\!\cdots\!44$$$$\nu^{3} +$$$$29\!\cdots\!60$$$$\nu^{2} +$$$$11\!\cdots\!44$$$$\nu -$$$$13\!\cdots\!52$$$$)/$$$$70\!\cdots\!88$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} + 3 \beta_{13} - 6 \beta_{12} + 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 12 \beta_{2} - 9 \beta_{1} + 9$$$$)/54$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{15} + 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 6 \beta_{10} + 32 \beta_{9} - 60 \beta_{8} - 15 \beta_{7} + 15 \beta_{6} - 58 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 66 \beta_{2} + 171 \beta_{1} + 2097$$$$)/54$$ $$\nu^{3}$$ $$=$$ $$($$$$262 \beta_{15} + 153 \beta_{14} + 207 \beta_{13} - 360 \beta_{12} - 126 \beta_{11} + 63 \beta_{10} - 8 \beta_{9} + 594 \beta_{8} - 333 \beta_{7} + 279 \beta_{6} - 259 \beta_{5} - 258 \beta_{4} + 63 \beta_{3} - 954 \beta_{2} - 693 \beta_{1} - 5841$$$$)/54$$ $$\nu^{4}$$ $$=$$ $$($$$$-1512 \beta_{15} + 612 \beta_{14} - 747 \beta_{13} + 1386 \beta_{12} + 2016 \beta_{11} - 972 \beta_{10} + 3240 \beta_{9} - 4824 \beta_{8} - 27 \beta_{7} + 1647 \beta_{6} - 4732 \beta_{5} + 1496 \beta_{4} - 972 \beta_{3} + 7290 \beta_{2} + 29763 \beta_{1} + 156969$$$$)/54$$ $$\nu^{5}$$ $$=$$ $$($$$$32630 \beta_{15} + 16515 \beta_{14} + 19083 \beta_{13} - 27726 \beta_{12} - 19350 \beta_{11} + 9045 \beta_{10} - 15340 \beta_{9} + 54780 \beta_{8} - 30477 \beta_{7} + 13197 \beta_{6} - 6157 \beta_{5} - 30902 \beta_{4} + 10485 \beta_{3} - 82146 \beta_{2} - 179703 \beta_{1} - 816273$$$$)/54$$ $$\nu^{6}$$ $$=$$ $$($$$$-247460 \beta_{15} + 30918 \beta_{14} - 112923 \beta_{13} + 164556 \beta_{12} + 254220 \beta_{11} - 112062 \beta_{10} + 359104 \beta_{9} - 403758 \beta_{8} + 144729 \beta_{7} + 186021 \beta_{6} - 406954 \beta_{5} + 232452 \beta_{4} - 116382 \beta_{3} + 768654 \beta_{2} + 4634181 \beta_{1} + 13960017$$$$)/54$$ $$\nu^{7}$$ $$=$$ $$($$$$3810126 \beta_{15} + 1493289 \beta_{14} + 1989039 \beta_{13} - 2371200 \beta_{12} - 2477664 \beta_{11} + 1037421 \beta_{10} - 2684670 \beta_{9} + 4879122 \beta_{8} - 3297981 \beta_{7} - 135645 \beta_{6} + 663401 \beta_{5} - 3370162 \beta_{4} + 1398789 \beta_{3} - 7714254 \beta_{2} - 35715861 \beta_{1} - 96370029$$$$)/54$$ $$\nu^{8}$$ $$=$$ $$($$$$-33101968 \beta_{15} + 310632 \beta_{14} - 15078675 \beta_{13} + 17453526 \beta_{12} + 29246880 \beta_{11} - 11119128 \beta_{10} + 41540144 \beta_{9} - 34834524 \beta_{8} + 27802893 \beta_{7} + 23997219 \beta_{6} - 36285976 \beta_{5} + 28678192 \beta_{4} - 13139160 \beta_{3} + 79332690 \beta_{2} + 646343919 \beta_{1} + 1341758745$$$$)/54$$ $$\nu^{9}$$ $$=$$ $$($$$$433006054 \beta_{15} + 126418959 \beta_{14} + 217504551 \beta_{13} - 213140382 \beta_{12} - 290064474 \beta_{11} + 101356605 \beta_{10} - 374035880 \beta_{9} + 420319500 \beta_{8} - 394012569 \beta_{7} - 152949975 \beta_{6} + 149655719 \beta_{5} - 348832710 \beta_{4} + 165791997 \beta_{3} - 762484458 \beta_{2} - 5729178195 \beta_{1} - 10853159733$$$$)/54$$ $$\nu^{10}$$ $$=$$ $$($$$$-4062381420 \beta_{15} - 138233742 \beta_{14} - 1892854743 \beta_{13} + 1767898236 \beta_{12} + 3216969492 \beta_{11} - 969690954 \beta_{10} + 4874326800 \beta_{9} - 2977985478 \beta_{8} + 4022831661 \beta_{7} + 3184620969 \beta_{6} - 3297732478 \beta_{5} + 3160458860 \beta_{4} - 1462291914 \beta_{3} + 8082357414 \beta_{2} + 84322941993 \beta_{1} + 133822224693$$$$)/54$$ $$\nu^{11}$$ $$=$$ $$($$$$48499514030 \beta_{15} + 10112577321 \beta_{14} + 24136734963 \beta_{13} - 19515342264 \beta_{12} - 32215401768 \beta_{11} + 8414179125 \beta_{10} - 47933990566 \beta_{9} + 34614474882 \beta_{8} - 48169974153 \beta_{7} - 28885566849 \beta_{6} + 20000068937 \beta_{5} - 34702271426 \beta_{4} + 18381347853 \beta_{3} - 76725341526 \beta_{2} - 810563558121 \beta_{1} - 1187441814825$$$$)/54$$ $$\nu^{12}$$ $$=$$ $$($$$$-475514212088 \beta_{15} - 19107195348 \beta_{14} - 227182483671 \beta_{13} + 172645902462 \beta_{12} + 343680414960 \beta_{11} - 70338926292 \beta_{10} + 572400682216 \beta_{9} - 240825302796 \beta_{8} + 521917357929 \beta_{7} + 415038592551 \beta_{6} - 302093990164 \beta_{5} + 323331111720 \beta_{4} - 160686785652 \beta_{3} + 812336246250 \beta_{2} + 10530888691203 \beta_{1} + 13532011977573$$$$)/54$$ $$\nu^{13}$$ $$=$$ $$($$$$5371594977942 \beta_{15} + 739350340911 \beta_{14} + 2677131100059 \beta_{13} - 1777856438046 \beta_{12} - 3450978812034 \beta_{11} + 531828979461 \beta_{10} - 5875603977024 \beta_{9} + 2631892112868 \beta_{8} - 5828570580597 \beta_{7} - 4237713608403 \beta_{6} + 2243982590087 \beta_{5} - 3318605490406 \beta_{4} + 1957956794469 \beta_{3} - 7689064460082 \beta_{2} - 105979054369167 \beta_{1} - 126521367686001$$$$)/54$$ $$\nu^{14}$$ $$=$$ $$($$$$-53952898642708 \beta_{15} - 1366278539058 \beta_{14} - 26310946432011 \beta_{13} + 16173600573612 \beta_{12} + 35853220709676 \beta_{11} - 3169094144310 \beta_{10} + 66778246261040 \beta_{9} - 17055274287342 \beta_{8} + 63894013845801 \beta_{7} + 52531623420501 \beta_{6} - 27514722147010 \beta_{5} + 31070385811348 \beta_{4} - 17346556422198 \beta_{3} + 80162137539438 \beta_{2} + 1274209961062293 \beta_{1} + 1366755753041361$$$$)/54$$ $$\nu^{15}$$ $$=$$ $$($$$$588210720499726 \beta_{15} + 44260971175041 \beta_{14} + 294499803340167 \beta_{13} - 157561454266608 \beta_{12} - 359035155404016 \beta_{11} + 9304639275477 \beta_{10} - 698920301264366 \beta_{9} + 168165221674818 \beta_{8} - 691024482816405 \beta_{7} - 559281785494917 \beta_{6} + 227591471220929 \beta_{5} - 302504235040530 \beta_{4} + 202943484516045 \beta_{3} - 756406138662510 \beta_{2} - 13139670825969885 \beta_{1} - 13121441550661221$$$$)/54$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 4.65022 − 0.707107i −10.4548 + 0.707107i −1.35249 + 0.707107i 8.15703 − 0.707107i 5.70754 + 0.707107i 1.09700 − 0.707107i −8.00527 − 0.707107i 2.20073 + 0.707107i 4.65022 + 0.707107i −10.4548 − 0.707107i −1.35249 − 0.707107i 8.15703 + 0.707107i 5.70754 − 0.707107i 1.09700 + 0.707107i −8.00527 + 0.707107i 2.20073 − 0.707107i
0 0 0 −10.1259 17.5386i 0 12.0269 14.0838i 0 0 0
17.2 0 0 0 −4.36813 7.56582i 0 14.4904 + 11.5338i 0 0 0
17.3 0 0 0 −4.27106 7.39769i 0 −12.5787 13.5933i 0 0 0
17.4 0 0 0 −3.41226 5.91021i 0 −14.9386 + 10.9471i 0 0 0
17.5 0 0 0 3.41226 + 5.91021i 0 −14.9386 + 10.9471i 0 0 0
17.6 0 0 0 4.27106 + 7.39769i 0 −12.5787 13.5933i 0 0 0
17.7 0 0 0 4.36813 + 7.56582i 0 14.4904 + 11.5338i 0 0 0
17.8 0 0 0 10.1259 + 17.5386i 0 12.0269 14.0838i 0 0 0
89.1 0 0 0 −10.1259 + 17.5386i 0 12.0269 + 14.0838i 0 0 0
89.2 0 0 0 −4.36813 + 7.56582i 0 14.4904 11.5338i 0 0 0
89.3 0 0 0 −4.27106 + 7.39769i 0 −12.5787 + 13.5933i 0 0 0
89.4 0 0 0 −3.41226 + 5.91021i 0 −14.9386 10.9471i 0 0 0
89.5 0 0 0 3.41226 5.91021i 0 −14.9386 10.9471i 0 0 0
89.6 0 0 0 4.27106 7.39769i 0 −12.5787 + 13.5933i 0 0 0
89.7 0 0 0 4.36813 7.56582i 0 14.4904 11.5338i 0 0 0
89.8 0 0 0 10.1259 17.5386i 0 12.0269 + 14.0838i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.8 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a 16 1.a even 1 1 trivial
252.4.t.a 16 3.b odd 2 1 inner
252.4.t.a 16 7.d odd 6 1 inner
252.4.t.a 16 21.g even 6 1 inner
1008.4.bt.b 16 4.b odd 2 1
1008.4.bt.b 16 12.b even 2 1
1008.4.bt.b 16 28.f even 6 1
1008.4.bt.b 16 84.j odd 6 1
1764.4.f.a 16 7.c even 3 1
1764.4.f.a 16 7.d odd 6 1
1764.4.f.a 16 21.g even 6 1
1764.4.f.a 16 21.h odd 6 1
1764.4.t.b 16 7.b odd 2 1
1764.4.t.b 16 7.c even 3 1
1764.4.t.b 16 21.c even 2 1
1764.4.t.b 16 21.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$11316405686931216 + 573927418423440 T^{2} + 19229963845212 T^{4} + 372024076068 T^{6} + 5245846173 T^{8} + 45478638 T^{10} + 274383 T^{12} + 606 T^{14} + T^{16}$$
$7$ $$( 13841287201 + 80707214 T - 11529602 T^{2} + 192080 T^{3} + 221431 T^{4} + 560 T^{5} - 98 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$11$ $$70\!\cdots\!96$$$$-$$$$12\!\cdots\!72$$$$T^{2} + 1703868846409635804 T^{4} - 6791181573369132 T^{6} + 19862336359773 T^{8} - 22036449654 T^{10} + 17903511 T^{12} - 4806 T^{14} + T^{16}$$
$13$ $$( 22573860516 + 426062052 T^{2} + 2267613 T^{4} + 2826 T^{6} + T^{8} )^{2}$$
$17$ $$49\!\cdots\!76$$$$+$$$$99\!\cdots\!40$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{4} + 9962008776094977792 T^{6} + 5337923687148816 T^{8} + 1682121675600 T^{10} + 365954796 T^{12} + 21396 T^{14} + T^{16}$$
$19$ $$( 46677071171844 - 4897686621816 T + 67363750602 T^{2} + 10905712884 T^{3} + 216000891 T^{4} - 547668 T^{5} - 14781 T^{6} + 36 T^{7} + T^{8} )^{2}$$
$23$ $$49\!\cdots\!36$$$$-$$$$23\!\cdots\!88$$$$T^{2} +$$$$75\!\cdots\!72$$$$T^{4} -$$$$12\!\cdots\!36$$$$T^{6} + 1477439351941048848 T^{8} - 86217178855824 T^{10} + 3608759628 T^{12} - 71460 T^{14} + T^{16}$$
$29$ $$( 416548977121475136 + 86086124811216 T^{2} + 5723596161 T^{4} + 133110 T^{6} + T^{8} )^{2}$$
$31$ $$( 17938763430477561 - 4104504382503798 T + 301548356400690 T^{2} + 2630657447964 T^{3} + 3618762939 T^{4} - 30388068 T^{5} - 44070 T^{6} + 354 T^{7} + T^{8} )^{2}$$
$37$ $$( 179850423092764516 - 5973918932125064 T + 152069932014238 T^{2} - 1572126338876 T^{3} + 12909581851 T^{4} - 24018986 T^{5} + 110761 T^{6} - 38 T^{7} + T^{8} )^{2}$$
$41$ $$( 3423608728179335424 - 403281542442816 T^{2} + 16046933220 T^{4} - 240060 T^{6} + T^{8} )^{2}$$
$43$ $$( -6254809742 + 68765576 T - 146451 T^{2} - 352 T^{3} + T^{4} )^{4}$$
$47$ $$47\!\cdots\!36$$$$+$$$$30\!\cdots\!00$$$$T^{2} +$$$$13\!\cdots\!36$$$$T^{4} +$$$$29\!\cdots\!56$$$$T^{6} +$$$$45\!\cdots\!80$$$$T^{8} + 34589931830684928 T^{10} + 190320805800 T^{12} + 522912 T^{14} + T^{16}$$
$53$ $$39\!\cdots\!76$$$$-$$$$75\!\cdots\!56$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{4} -$$$$60\!\cdots\!24$$$$T^{6} +$$$$24\!\cdots\!21$$$$T^{8} - 3936297096958518 T^{10} + 44910196227 T^{12} - 252054 T^{14} + T^{16}$$
$59$ $$56\!\cdots\!76$$$$+$$$$11\!\cdots\!12$$$$T^{2} +$$$$22\!\cdots\!84$$$$T^{4} +$$$$63\!\cdots\!48$$$$T^{6} +$$$$14\!\cdots\!69$$$$T^{8} + 85770996108094134 T^{10} + 362612345931 T^{12} + 705414 T^{14} + T^{16}$$
$61$ $$( 10077652565550144 - 274392800500032 T - 3939444913968 T^{2} + 175070170800 T^{3} + 4946257404 T^{4} + 52264800 T^{5} + 286002 T^{6} + 816 T^{7} + T^{8} )^{2}$$
$67$ $$($$$$90\!\cdots\!56$$$$- 5710030600549908304 T + 46396767109353724 T^{2} + 19594109854844 T^{3} + 234348933949 T^{4} + 115322636 T^{5} + 929389 T^{6} + 764 T^{7} + T^{8} )^{2}$$
$71$ $$($$$$17\!\cdots\!44$$$$+ 42297868577973888 T^{2} + 337035071448 T^{4} + 1006848 T^{6} + T^{8} )^{2}$$
$73$ $$( 11798164103488007184 + 579941181193088880 T + 11266873327061460 T^{2} + 86734229600700 T^{3} + 191349429597 T^{4} - 693501750 T^{5} + 93795 T^{6} + 1350 T^{7} + T^{8} )^{2}$$
$79$ $$( 1530560661733206601 - 214369146481110838 T + 30810683064091618 T^{2} + 109679037609848 T^{3} + 434253280891 T^{4} + 230876408 T^{5} + 668698 T^{6} + 182 T^{7} + T^{8} )^{2}$$
$83$ $$($$$$10\!\cdots\!84$$$$- 79735840708215204 T^{2} + 734790325365 T^{4} - 1589970 T^{6} + T^{8} )^{2}$$
$89$ $$10\!\cdots\!96$$$$+$$$$17\!\cdots\!44$$$$T^{2} +$$$$20\!\cdots\!24$$$$T^{4} +$$$$10\!\cdots\!24$$$$T^{6} +$$$$40\!\cdots\!72$$$$T^{8} + 6291396333799327872 T^{10} + 7072435128816 T^{12} + 3076488 T^{14} + T^{16}$$
$97$ $$($$$$47\!\cdots\!36$$$$+ 3276863917344744120 T^{2} + 6899032044873 T^{4} + 5135454 T^{6} + T^{8} )^{2}$$