Properties

 Label 1080.2.d.g Level $1080$ Weight $2$ Character orbit 1080.d Analytic conductor $8.624$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(109,1080)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1080.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$8.62384341830$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + 2243 x^{8} + 23019 x^{7} + 49749 x^{6} + 37798 x^{5} + 78784 x^{4} + 244612 x^{3} + \cdots + 45658$$ x^16 - 3*x^15 - 3*x^14 + 36*x^13 - 78*x^12 - 96*x^11 + 1194*x^10 + 1456*x^9 + 2243*x^8 + 23019*x^7 + 49749*x^6 + 37798*x^5 + 78784*x^4 + 244612*x^3 + 302532*x^2 + 157882*x + 45658 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{12} q^{2} + (\beta_{5} + \beta_{3} - 1) q^{4} - \beta_{10} q^{5} - \beta_{15} q^{7} + ( - \beta_{14} + \beta_{12} + \beta_{2}) q^{8}+O(q^{10})$$ q - b12 * q^2 + (b5 + b3 - 1) * q^4 - b10 * q^5 - b15 * q^7 + (-b14 + b12 + b2) * q^8 $$q - \beta_{12} q^{2} + (\beta_{5} + \beta_{3} - 1) q^{4} - \beta_{10} q^{5} - \beta_{15} q^{7} + ( - \beta_{14} + \beta_{12} + \beta_{2}) q^{8} + ( - \beta_{6} + \beta_{3}) q^{10} + (\beta_{13} + \beta_{9} - \beta_{6} + \beta_{4}) q^{11} + (\beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} - \beta_{3} - \beta_1) q^{13} + (\beta_{13} - \beta_{11}) q^{14} + (\beta_{10} - \beta_{9} - \beta_{4} - \beta_{3} - 2) q^{16} + (\beta_{14} - \beta_{10} + \beta_{9} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{17} + ( - \beta_{14} + 2 \beta_{12} + \beta_{4} - 2 \beta_{2}) q^{19} + ( - \beta_{14} + \beta_{4} - \beta_{2} + \beta_1) q^{20} + ( - \beta_{10} - \beta_{9} + \beta_{6} + \beta_1) q^{22} + ( - \beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{4} - \beta_{2}) q^{23} + (\beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_{2} - 2) q^{25} + ( - \beta_{15} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{4}) q^{26} + (\beta_{15} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7}) q^{28} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{4}) q^{29} + ( - \beta_{12} + \beta_{10} - \beta_{9} + 2 \beta_{3} - \beta_{2} - 1) q^{31} + (2 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{4} - \beta_{3}) q^{32} + (2 \beta_{14} - \beta_{12} - 2 \beta_{10} + 2 \beta_{9} - \beta_{5} + \beta_{3} - 2 \beta_{2} + 1) q^{34} + (\beta_{15} + \beta_{13} + \beta_{12} - \beta_{6} + \beta_{4} - 3 \beta_{3} + \beta_{2}) q^{35} + ( - \beta_{15} - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{4} - 2 \beta_1) q^{37} + (\beta_{14} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 5) q^{38} + (\beta_{14} - \beta_{13} - \beta_{11} + \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + 1) q^{40} + ( - \beta_{13} - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{41} + ( - \beta_{15} + \beta_{13} + \beta_{10} - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{43} + ( - \beta_{13} - \beta_{11} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{44} + ( - \beta_{14} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{3} - \beta_{2} + 2) q^{46} + ( - 2 \beta_{14} + \beta_{10} - \beta_{9} - 2 \beta_{5} - \beta_{3} + 1) q^{47} + (2 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + \beta_{3} + 2 \beta_{2} - 2) q^{49} + ( - \beta_{15} + 2 \beta_{12} - \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_1 - 1) q^{50} + (2 \beta_{15} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{6} + \beta_1) q^{52} + ( - \beta_{12} + \beta_{3} - \beta_{2} + 6) q^{53} + ( - \beta_{15} + 3 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 2) q^{55}+ \cdots + ( - \beta_{14} + 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{98}+O(q^{100})$$ q - b12 * q^2 + (b5 + b3 - 1) * q^4 - b10 * q^5 - b15 * q^7 + (-b14 + b12 + b2) * q^8 + (-b6 + b3) * q^10 + (b13 + b9 - b6 + b4) * q^11 + (b10 - b9 + b7 - b4 - b3 - b1) * q^13 + (b13 - b11) * q^14 + (b10 - b9 - b4 - b3 - 2) * q^16 + (b14 - b10 + b9 - 2*b5 + b4 - b3 + 1) * q^17 + (-b14 + 2*b12 + b4 - 2*b2) * q^19 + (-b14 + b4 - b2 + b1) * q^20 + (-b10 - b9 + b6 + b1) * q^22 + (-b14 + b12 + b10 - b9 - b4 - b2) * q^23 + (b12 - b11 + b10 - b9 + b8 - b4 + b2 - 2) * q^25 + (-b15 + b13 + b11 - b10 + b9 - b8 - b7 + 2*b4) * q^26 + (b15 - b10 - b9 - b8 + b7) * q^28 + (b11 + b9 - b8 + b4) * q^29 + (-b12 + b10 - b9 + 2*b3 - b2 - 1) * q^31 + (2*b12 + b10 - b9 + b4 - b3) * q^32 + (2*b14 - b12 - 2*b10 + 2*b9 - b5 + b3 - 2*b2 + 1) * q^34 + (b15 + b13 + b12 - b6 + b4 - 3*b3 + b2) * q^35 + (-b15 - b11 + b10 - b8 - b4 - 2*b1) * q^37 + (b14 - b10 + b9 - b5 - b4 - 2*b3 + b2 + 5) * q^38 + (b14 - b13 - b11 + b6 + b5 - 2*b4 + b2 + 1) * q^40 + (-b13 - b11 - b8 + b7 - b6 - b4 + b3 - b1) * q^41 + (-b15 + b13 + b10 - b7 + b6 - b3 - b1) * q^43 + (-b13 - b11 - b6 - b4 + b3 - b1) * q^44 + (-b14 + 2*b10 - 2*b9 - 2*b3 - b2 + 2) * q^46 + (-2*b14 + b10 - b9 - 2*b5 - b3 + 1) * q^47 + (2*b12 - 2*b10 + 2*b9 + b3 + 2*b2 - 2) * q^49 + (-b15 + 2*b12 - b8 + b7 - b5 - 2*b3 - b1 - 1) * q^50 + (2*b15 + b13 - b11 - b10 - b9 + 2*b8 - b6 + b1) * q^52 + (-b12 + b3 - b2 + 6) * q^53 + (-b15 + 3*b12 + b11 - b10 + b9 - b8 + b4 - b3 + 3*b2 + 2) * q^55 + (b15 - b13 + b11 - 2*b10 + 2*b9 - b8 - b7 - b6 + 2*b4 + 2*b3 + b1) * q^56 + (b15 + b8 - b7 + b6 + b1) * q^58 + (-b15 + b13 + b11 - 2*b10 - b8 - b6 + 2*b4) * q^59 + (-5*b14 + b12 + 2*b10 - 2*b9 + 2*b5 + b4 + b3 - b2 - 1) * q^61 + (-2*b14 + b12 - b10 + b9 + b5 + b4 - 2*b2 + 1) * q^62 + (2*b14 - 3*b10 + 3*b9 - 2*b5 + b4 - 3*b3 + 2*b2 + 2) * q^64 + (-b15 - 2*b14 - b13 + b9 - b7 - b6 + 2*b5 + 2*b4 + 2*b3 - b1 - 1) * q^65 + (-b15 - 2*b13 + b11 + b10 + b8 - 2*b6 - b4 - 2*b1) * q^67 + (-b14 - b12 - b5 + 3*b3 - 3*b2 + 5) * q^68 + (3*b14 - b13 + b11 - 2*b10 - b5 + b4 - b3 + 3*b2 + b1 - 1) * q^70 + (b15 + 2*b13 - b11 - b9 - b8 + b7 + 2*b6 - b3 + b1) * q^71 + (-b15 + b13 + b11 - b10 + b9 - b8 - b6 + 2*b4) * q^73 + (3*b15 + 3*b13 + b11 - b10 + b9 - b8 + b7 - b6 + 3*b4 - b3 + b1) * q^74 + (b14 - 5*b12 + 2*b10 - 2*b9 - b5 + b3 - b2 - 1) * q^76 + (2*b12 + 2*b10 - 2*b9 + b3 + 2*b2 - 3) * q^77 + (-b12 - 3*b10 + 3*b9 + 3*b3 - b2 + 2) * q^79 + (b15 - b12 + 2*b10 - b8 + b7 - b5 + 2*b2 - b1 - 1) * q^80 + (2*b15 + b13 + b11 - b10 + 3*b9 - 2*b8 + 3*b4 + b3 + 2*b1) * q^82 + (2*b12 - 3*b10 + 3*b9 - b3 + 2*b2 + 6) * q^83 + (-b15 + b14 + b12 + b11 + b10 + b8 - b7 - b4 - b3 - b2 - b1) * q^85 + (b15 + 2*b13 - 2*b9 + b8 + b7 - b6 - 2*b3 - b1) * q^86 + (b15 + b13 + b11 + 2*b9 - b8 + b7 - b6 + 2*b4 + b1) * q^88 + (b15 - b13 - 3*b10 + 2*b9 - b7 - b6 + 2*b4 + 3*b3 + 3*b1) * q^89 + (-6*b14 + 3*b10 - 3*b9 - 2*b5 - b3 + 1) * q^91 + (2*b14 - 2*b12 - b10 + b9 + b5 + b4 - 2*b3 + 2*b2 + 1) * q^92 + (b14 - b12 + b10 - b9 + 2*b5 - b4 - b3 - 3*b2 - 2) * q^94 + (b14 + 2*b13 + b12 - b10 + b9 - b7 + 2*b6 + 2*b5 + 2*b4 - b2 + b1 - 1) * q^95 + (-b15 - 2*b13 + b11 + 2*b10 + b9 - b8 + 2*b6 - b4) * q^97 + (-b14 + 2*b12 + 2*b10 - 2*b9 - 2*b5 - 2*b4 - b2 - 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 6 q^{4} - 6 q^{5} - 2 q^{8}+O(q^{10})$$ 16 * q - 2 * q^2 - 6 * q^4 - 6 * q^5 - 2 * q^8 $$16 q - 2 q^{2} - 6 q^{4} - 6 q^{5} - 2 q^{8} + 5 q^{10} - 30 q^{16} - q^{20} - 22 q^{25} + 18 q^{32} - 4 q^{34} - 2 q^{35} + 56 q^{38} + 19 q^{40} + 40 q^{46} - 44 q^{49} - 27 q^{50} + 96 q^{53} + 34 q^{55} + 2 q^{62} - 6 q^{64} + 72 q^{68} - 7 q^{70} - 12 q^{77} + 4 q^{79} - 9 q^{80} + 64 q^{83} + 20 q^{92} - 20 q^{94} - 36 q^{98}+O(q^{100})$$ 16 * q - 2 * q^2 - 6 * q^4 - 6 * q^5 - 2 * q^8 + 5 * q^10 - 30 * q^16 - q^20 - 22 * q^25 + 18 * q^32 - 4 * q^34 - 2 * q^35 + 56 * q^38 + 19 * q^40 + 40 * q^46 - 44 * q^49 - 27 * q^50 + 96 * q^53 + 34 * q^55 + 2 * q^62 - 6 * q^64 + 72 * q^68 - 7 * q^70 - 12 * q^77 + 4 * q^79 - 9 * q^80 + 64 * q^83 + 20 * q^92 - 20 * q^94 - 36 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + 2243 x^{8} + 23019 x^{7} + 49749 x^{6} + 37798 x^{5} + 78784 x^{4} + 244612 x^{3} + \cdots + 45658$$ :

 $$\beta_{1}$$ $$=$$ $$( - 29\!\cdots\!50 \nu^{15} + \cdots + 26\!\cdots\!36 ) / 53\!\cdots\!82$$ (-2968246925305068366016861850*v^15 + 100724542642732097318917507681*v^14 - 567607760522674625597150073539*v^13 + 1270521640866234227739985065478*v^12 + 690872865962071542233180933875*v^11 - 12862260501825953682216710285180*v^10 + 30709209012705539373498878120695*v^9 + 32334063594810278586330766195352*v^8 - 108042448863972106970412784976819*v^7 + 405163675978233988011284373334759*v^6 + 841460963211356679462474100466914*v^5 - 127696115268596669962320832755376*v^4 + 1317727487635383788120858856767730*v^3 + 4435294168143189478419454834972696*v^2 + 2732541523404331265441186456675686*v + 2689937510010371245965572803175936) / 537937288243246638195923625151482 $$\beta_{2}$$ $$=$$ $$( 91\!\cdots\!99 \nu^{15} + \cdots + 52\!\cdots\!72 ) / 29\!\cdots\!28$$ (912048055583909114267368116599*v^15 - 5698040312704688625458600718907*v^14 + 14115708744432730405169099899238*v^13 + 1280995371795872811510501294005*v^12 - 123794214856126853131814258112592*v^11 + 355136783502262874112895101270311*v^10 + 238764241238035190291337795854198*v^9 - 714160107263881642262295627250559*v^8 + 5106555474537776169119004536951975*v^7 + 8467394825371394503090403915954102*v^6 + 3777963322588830713383769205094970*v^5 + 23858007035730180850771394555342950*v^4 + 45344381428787029708635032124128754*v^3 + 45141980695973382302092265699617754*v^2 + 99826922284602257453687426322992042*v + 52568564805374419618599547102869772) / 29048613565135318462579875758180028 $$\beta_{3}$$ $$=$$ $$( - 46\!\cdots\!13 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 12\!\cdots\!72$$ (-46449794153507378462464313*v^15 + 247541873575584587339885975*v^14 - 391047289941977590220537926*v^13 - 1011247126758606846765662537*v^12 + 6391954850963295425202433488*v^11 - 9395358398162249739183319193*v^10 - 40685642718566191529383659706*v^9 + 38817024668411052751281577847*v^8 - 152801224095641326968018331549*v^7 - 778979448410751676709062383580*v^6 - 284369681153041160135598123346*v^5 - 271094238468154240554342214662*v^4 - 3153495124008224044780151788338*v^3 - 3634579657946262309219440042838*v^2 - 1928951608894628214963884861990*v - 1337216947772539479600848983644) / 1285393759243122194016543907172 $$\beta_{4}$$ $$=$$ $$( - 87\!\cdots\!94 \nu^{15} + \cdots - 16\!\cdots\!31 ) / 24\!\cdots\!69$$ (-87860439233952427110875383394*v^15 + 37263251707070194444744863223*v^14 + 1697786711553689228248940920153*v^13 - 6464862877355428644800549389922*v^12 + 4866844975149741053755465785922*v^11 + 42511857392227576452026209731523*v^10 - 185366960858756617517495506128185*v^9 - 246655569135727184170919021724532*v^8 + 112460540849969375003801107126426*v^7 - 3066101441992904461906948617949439*v^6 - 7002054938507546354794938174196250*v^5 - 2653892245875051729944810054208199*v^4 - 8766523120552899761705807462372940*v^3 - 34239618598105509035514735539030978*v^2 - 37790026492467100033699517719869512*v - 16990181512359178068636832973313631) / 2420717797094609871881656313181669 $$\beta_{5}$$ $$=$$ $$( 53\!\cdots\!17 \nu^{15} + \cdots - 38\!\cdots\!72 ) / 96\!\cdots\!76$$ (536496212669605662673476107317*v^15 - 3031628099783860500899046757931*v^14 + 5010504368748251349071998447510*v^13 + 12930280575804544144208147028949*v^12 - 84054762402289778006839537753592*v^11 + 130730311624230541313015806067437*v^10 + 488411285479145977958195793277498*v^9 - 720008839227572037312814014258415*v^8 + 1658000774363623058550026565275425*v^7 + 8850079069182169877354024787862828*v^6 - 817154759746499552597480606920214*v^5 - 4567354000268000403299377705361686*v^4 + 37060205580310396209554068038099546*v^3 + 27445211779793228070230591753899822*v^2 - 26638241791253248321152919660354946*v - 3845233369531395357024450336700672) / 9682871188378439487526625252726676 $$\beta_{6}$$ $$=$$ $$( 16\!\cdots\!42 \nu^{15} + \cdots + 65\!\cdots\!10 ) / 29\!\cdots\!28$$ (1678931869961246025751482656342*v^15 - 9922501609277149191081771501907*v^14 + 20188649462793866218084252073975*v^13 + 26062463748198874069772966237588*v^12 - 262639752234699500369196183809149*v^11 + 549709591732674664049541903736694*v^10 + 1073426688804900417923904013442399*v^9 - 2190048354033453433242699770361608*v^8 + 7910146154395458408804969175067633*v^7 + 23571699770791007139044388303758879*v^6 - 1582419709853013369276506762342836*v^5 + 20020407352584882924212303378445466*v^4 + 118752405678956579179929249305407242*v^3 + 61649576778676627296697081786586078*v^2 + 15120231805568494292014688489771966*v + 65880008362435834886181250593669910) / 29048613565135318462579875758180028 $$\beta_{7}$$ $$=$$ $$( 98\!\cdots\!73 \nu^{15} + \cdots - 71\!\cdots\!88 ) / 14\!\cdots\!14$$ (981733289547917239205031922273*v^15 - 2403344268270664082095633125341*v^14 - 11788544643920816249790473945714*v^13 + 77421827507796914931461143743850*v^12 - 149542567276496964297616324748777*v^11 - 236469348276488075538345678285800*v^10 + 2260578209286740445791742177475222*v^9 - 460678775942581977273304875462640*v^8 - 1054533793274871176390716695929090*v^7 + 32918211766122447150143301884213953*v^6 + 27237627010917239373486356055469288*v^5 - 25090437022170849116478328135778602*v^4 + 101455532695234330579648455701256366*v^3 + 212736973128501478417582565566352002*v^2 + 27119226155482261014122777726908648*v - 7120121894870179000886142728057188) / 14524306782567659231289937879090014 $$\beta_{8}$$ $$=$$ $$( - 12\!\cdots\!99 \nu^{15} + \cdots - 32\!\cdots\!52 ) / 14\!\cdots\!14$$ (-1293743257681942256442186325799*v^15 + 4968738705497027910097754403238*v^14 + 518950110945043540986636785293*v^13 - 50280854846991367442576118955505*v^12 + 142031045943419277209906176650229*v^11 + 46607007193792557113868933579847*v^10 - 1715007642909326614286140833046751*v^9 - 413064522998180380861220400335713*v^8 - 1294217798191495103187598002166976*v^7 - 29475107380374563927888171052871031*v^6 - 38059966625057139546185394795765896*v^5 + 2199535618314716984336541288078260*v^4 - 96306096851138238841515994060985652*v^3 - 252593482614583269343278949145479346*v^2 - 125262376724566867740849963423298916*v - 32851625501020684039007608623164452) / 14524306782567659231289937879090014 $$\beta_{9}$$ $$=$$ $$( 68\!\cdots\!85 \nu^{15} + \cdots + 75\!\cdots\!34 ) / 71\!\cdots\!76$$ (68460096829407729493072618685*v^15 - 341623495597449926990911857042*v^14 + 338022536738320361410346741407*v^13 + 2510640089244908582995590437747*v^12 - 11351830743787609314394630963035*v^11 + 12380345160350258667539354240149*v^10 + 77635620002219776839556714752383*v^9 - 84450951113311548939038816986753*v^8 + 198543627588457770517461457130486*v^7 + 1312356476908740198818910590863629*v^6 + 332199503775126359246903819246014*v^5 - 302412293843340936235784256784052*v^4 + 5257400432853981328558449299067812*v^3 + 6165283908794913503161657485950632*v^2 + 350327932206386288345113888784732*v + 758155685661014305806352162382534) / 717249717657662184261231500201976 $$\beta_{10}$$ $$=$$ $$( 70\!\cdots\!47 \nu^{15} + \cdots - 13\!\cdots\!70 ) / 71\!\cdots\!76$$ (70896363012619975352047365047*v^15 - 409016750792810739016049720256*v^14 + 740962837339389597865749684739*v^13 + 1488042873595497882371003908229*v^12 - 11303975568665029687638292569291*v^11 + 20683505336587692259796543586895*v^10 + 53390695339553739670087002742331*v^9 - 88287746232844903500229403060527*v^8 + 266219488261185805938979561060616*v^7 + 1042749769941272139410207444162757*v^6 + 33473401889470466941629285783106*v^5 - 27205067464592780445843442014512*v^4 + 4494433878046643427987644416961192*v^3 + 4580785124927317668191068692291748*v^2 - 624434644366499204342386836185704*v - 1385261056561774631404342637172970) / 717249717657662184261231500201976 $$\beta_{11}$$ $$=$$ $$( - 61\!\cdots\!37 \nu^{15} + \cdots - 20\!\cdots\!86 ) / 58\!\cdots\!56$$ (-6149563994619954688638779707937*v^15 + 24384290511076096526562020786398*v^14 - 4718552561761306915881069203927*v^13 - 220138357135127837547298790093483*v^12 + 711372172741298485663046882993911*v^11 - 143667817049424646174448098831829*v^10 - 7234739792925968930593802023929239*v^9 - 1470617694604805867906139806085295*v^8 - 13568216187504163801136339011810166*v^7 - 127916290173464130300754420540957025*v^6 - 173101196918431897285355745901015262*v^5 - 67851985676704121023772583567798508*v^4 - 401863543064934780565531559134870572*v^3 - 1036329442832847630470415460523284544*v^2 - 726884682171308571155783053543879340*v - 207794035139009378088396289841703886) / 58097227130270636925159751516360056 $$\beta_{12}$$ $$=$$ $$( - 88\!\cdots\!61 \nu^{15} + \cdots - 31\!\cdots\!10 ) / 72\!\cdots\!07$$ (-884749222076827840163383480061*v^15 + 3599043381514092590775304833862*v^14 - 941078557932148630775019122912*v^13 - 32565341693744781589549986685301*v^12 + 108171519253326925442652432374614*v^11 - 29523529413906777493617000674543*v^10 - 1068994750098133024151149897842617*v^9 - 14185174417871716231293317069707*v^8 - 1930516773969094645935373235463383*v^7 - 18741903802354980703715571404576057*v^6 - 22480618898714890313138256946084163*v^5 - 7985899677975973723172712922975579*v^4 - 64761128639918113258547739261608700*v^3 - 145171978655040313170872301388654991*v^2 - 105304314957815300573758882340184806*v - 31587100994435148956841949945540210) / 7262153391283829615644968939545007 $$\beta_{13}$$ $$=$$ $$( - 72\!\cdots\!65 \nu^{15} + \cdots - 66\!\cdots\!86 ) / 58\!\cdots\!56$$ (-7269103474487362172933174332865*v^15 + 31385661507909987353368135035934*v^14 - 17107023363453311887525186709639*v^13 - 252013692954719590152433256919851*v^12 + 926797171801198127075784520724839*v^11 - 503398277914670530340935274351237*v^10 - 8326297826207090073597894132202439*v^9 + 1094983459797783593718319523936489*v^8 - 16609911107011105456136005854111014*v^7 - 145808451479287995280982367655085153*v^6 - 155345327511197745613962098280501470*v^5 - 28194944284812483239401157917335316*v^4 - 481693103931873447446882666834166924*v^3 - 1036724982131751835039698759990491552*v^2 - 596064930865510949019247370622245756*v - 66185876186127129622090663942398286) / 58097227130270636925159751516360056 $$\beta_{14}$$ $$=$$ $$( - 47\!\cdots\!25 \nu^{15} + \cdots - 15\!\cdots\!04 ) / 29\!\cdots\!28$$ (-4777405049874405220290293508725*v^15 + 21869480600682083825586720360229*v^14 - 16930496194389552475606824693290*v^13 - 165894472843756913348070666740291*v^12 + 683600955386312395547326219767976*v^11 - 599047303927928447636178067892993*v^10 - 5266605292925521747930627375811942*v^9 + 2693891094309741866477833200482489*v^8 - 13703051666477086217585774710858421*v^7 - 92810415687486762484021665828620090*v^6 - 73467828060319886892184581181208762*v^5 - 30405655107395500282521243569159062*v^4 - 346112115238358927869461747300875766*v^3 - 582201197180551186444520919564636482*v^2 - 357672149290479265608983356352444654*v - 152095410191664098988657516871227004) / 29048613565135318462579875758180028 $$\beta_{15}$$ $$=$$ $$( 26\!\cdots\!78 \nu^{15} + \cdots + 69\!\cdots\!74 ) / 11\!\cdots\!96$$ (26752646231523755733786132878*v^15 - 128592701387822337905647833681*v^14 + 145337649633553291461789413149*v^13 + 752740054714243080564442352918*v^12 - 3602384666056073460301120040997*v^11 + 3984836439443536634231966000260*v^10 + 26000333648504466704692111919297*v^9 - 12362514354697864859638372770130*v^8 + 83329656222185135728278336221777*v^7 + 479683665037345623980106225703263*v^6 + 413736876395432004538516927161328*v^5 + 240551486807570209228379215086046*v^4 + 1798355690683910481445547905696838*v^3 + 3114340389660371141511675680175130*v^2 + 2109648831841915726153072408406390*v + 691319728804900074759618490740074) / 119541619609610364043538583366996
 $$\nu$$ $$=$$ $$( -\beta_{13} + \beta_{11} - \beta_{5} - \beta_{3} + 1 ) / 2$$ (-b13 + b11 - b5 - b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + \beta _1 + 4 ) / 2$$ (b15 - b14 - b13 + b12 + b11 - b8 - b7 - b6 - 4*b5 + 2*b4 - 4*b3 - 3*b2 + b1 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 8 \beta_{15} - 5 \beta_{14} - \beta_{13} + 9 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 5 \beta _1 - 8 ) / 2$$ (8*b15 - 5*b14 - b13 + 9*b12 + 4*b11 - 3*b10 - 4*b9 - 2*b8 - 3*b7 - 7*b6 + 2*b5 + 3*b4 - 2*b3 - 5*b2 + 5*b1 - 8) / 2 $$\nu^{4}$$ $$=$$ $$( 30 \beta_{15} - 39 \beta_{14} + 20 \beta_{13} + 39 \beta_{12} + 8 \beta_{11} + 7 \beta_{10} - 47 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 24 \beta_{6} + 10 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 33 \beta_{2} + 6 \beta_1 ) / 2$$ (30*b15 - 39*b14 + 20*b13 + 39*b12 + 8*b11 + 7*b10 - 47*b9 - 6*b8 + 2*b7 - 24*b6 + 10*b5 + 3*b4 - 3*b3 - 33*b2 + 6*b1) / 2 $$\nu^{5}$$ $$=$$ $$( 91 \beta_{15} - 166 \beta_{14} + 67 \beta_{13} + 134 \beta_{12} + 58 \beta_{11} + 6 \beta_{10} - 169 \beta_{9} - 57 \beta_{8} + 18 \beta_{7} - 23 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 46 \beta_{3} - 140 \beta_{2} - 19 \beta _1 + 24 ) / 2$$ (91*b15 - 166*b14 + 67*b13 + 134*b12 + 58*b11 + 6*b10 - 169*b9 - 57*b8 + 18*b7 - 23*b6 - 2*b5 + 5*b4 + 46*b3 - 140*b2 - 19*b1 + 24) / 2 $$\nu^{6}$$ $$=$$ $$( 256 \beta_{15} - 529 \beta_{14} + 262 \beta_{13} + 551 \beta_{12} + 142 \beta_{11} - 33 \beta_{10} - 391 \beta_{9} - 328 \beta_{8} + 84 \beta_{7} + 158 \beta_{6} - 272 \beta_{5} - 57 \beta_{4} + 79 \beta_{3} - 575 \beta_{2} - 202 \beta _1 - 132 ) / 2$$ (256*b15 - 529*b14 + 262*b13 + 551*b12 + 142*b11 - 33*b10 - 391*b9 - 328*b8 + 84*b7 + 158*b6 - 272*b5 - 57*b4 + 79*b3 - 575*b2 - 202*b1 - 132) / 2 $$\nu^{7}$$ $$=$$ $$( 892 \beta_{15} - 1132 \beta_{14} + 1176 \beta_{13} + 2266 \beta_{12} - 75 \beta_{11} - 583 \beta_{10} - 264 \beta_{9} - 1424 \beta_{8} + 397 \beta_{7} + 970 \beta_{6} - 990 \beta_{5} - 482 \beta_{4} + 797 \beta_{3} + \cdots - 2482 ) / 2$$ (892*b15 - 1132*b14 + 1176*b13 + 2266*b12 - 75*b11 - 583*b10 - 264*b9 - 1424*b8 + 397*b7 + 970*b6 - 990*b5 - 482*b4 + 797*b3 - 1158*b2 - 920*b1 - 2482) / 2 $$\nu^{8}$$ $$=$$ $$( 2010 \beta_{15} - 1747 \beta_{14} + 5656 \beta_{13} + 8167 \beta_{12} - 2868 \beta_{11} - 2173 \beta_{10} + 1117 \beta_{9} - 4946 \beta_{8} + 2502 \beta_{7} + 4468 \beta_{6} - 1412 \beta_{5} - 3313 \beta_{4} + \cdots - 15414 ) / 2$$ (2010*b15 - 1747*b14 + 5656*b13 + 8167*b12 - 2868*b11 - 2173*b10 + 1117*b9 - 4946*b8 + 2502*b7 + 4468*b6 - 1412*b5 - 3313*b4 + 5291*b3 + 963*b2 - 3834*b1 - 15414) / 2 $$\nu^{9}$$ $$=$$ $$( - 2158 \beta_{15} + 1246 \beta_{14} + 22952 \beta_{13} + 18538 \beta_{12} - 17171 \beta_{11} - 7597 \beta_{10} + 10402 \beta_{9} - 14804 \beta_{8} + 12011 \beta_{7} + 20178 \beta_{6} + 3214 \beta_{5} + \cdots - 66342 ) / 2$$ (-2158*b15 + 1246*b14 + 22952*b13 + 18538*b12 - 17171*b11 - 7597*b10 + 10402*b9 - 14804*b8 + 12011*b7 + 20178*b6 + 3214*b5 - 14544*b4 + 28289*b3 + 21992*b2 - 15896*b1 - 66342) / 2 $$\nu^{10}$$ $$=$$ $$( - 54552 \beta_{15} + 41459 \beta_{14} + 73850 \beta_{13} - 635 \beta_{12} - 78882 \beta_{11} - 23331 \beta_{10} + 74127 \beta_{9} - 34028 \beta_{8} + 43648 \beta_{7} + 91178 \beta_{6} + 28382 \beta_{5} + \cdots - 235586 ) / 2$$ (-54552*b15 + 41459*b14 + 73850*b13 - 635*b12 - 78882*b11 - 23331*b10 + 74127*b9 - 34028*b8 + 43648*b7 + 91178*b6 + 28382*b5 - 57421*b4 + 115063*b3 + 126405*b2 - 63832*b1 - 235586) / 2 $$\nu^{11}$$ $$=$$ $$( - 344812 \beta_{15} + 348494 \beta_{14} + 175628 \beta_{13} - 276596 \beta_{12} - 329057 \beta_{11} - 73911 \beta_{10} + 455018 \beta_{9} - 21320 \beta_{8} + 126395 \beta_{7} + 362424 \beta_{6} + \cdots - 740768 ) / 2$$ (-344812*b15 + 348494*b14 + 175628*b13 - 276596*b12 - 329057*b11 - 73911*b10 + 455018*b9 - 21320*b8 + 126395*b7 + 362424*b6 + 131804*b5 - 204174*b4 + 388181*b3 + 589764*b2 - 219270*b1 - 740768) / 2 $$\nu^{12}$$ $$=$$ $$( - 1622020 \beta_{15} + 2015805 \beta_{14} + 149988 \beta_{13} - 1865395 \beta_{12} - 1271436 \beta_{11} - 161197 \beta_{10} + 2217573 \beta_{9} + 388988 \beta_{8} + 293972 \beta_{7} + \cdots - 2002304 ) / 2$$ (-1622020*b15 + 2015805*b14 + 149988*b13 - 1865395*b12 - 1271436*b11 - 161197*b10 + 2217573*b9 + 388988*b8 + 293972*b7 + 1203284*b6 + 591254*b5 - 657473*b4 + 1096503*b3 + 2574947*b2 - 562428*b1 - 2002304) / 2 $$\nu^{13}$$ $$=$$ $$( - 6677652 \beta_{15} + 9219268 \beta_{14} - 1552498 \beta_{13} - 9380990 \beta_{12} - 4310753 \beta_{11} + 22809 \beta_{10} + 8854476 \beta_{9} + 3234118 \beta_{8} + 350405 \beta_{7} + \cdots - 3501172 ) / 2$$ (-6677652*b15 + 9219268*b14 - 1552498*b13 - 9380990*b12 - 4310753*b11 + 22809*b10 + 8854476*b9 + 3234118*b8 + 350405*b7 + 3070104*b6 + 2691456*b5 - 1585156*b4 + 2247253*b3 + 10446326*b2 - 602924*b1 - 3501172) / 2 $$\nu^{14}$$ $$=$$ $$( - 25180604 \beta_{15} + 35472515 \beta_{14} - 13780618 \beta_{13} - 41786267 \beta_{12} - 11693362 \beta_{11} + 2805451 \beta_{10} + 29945333 \beta_{9} + 18024544 \beta_{8} + \cdots + 6413714 ) / 2$$ (-25180604*b15 + 35472515*b14 - 13780618*b13 - 41786267*b12 - 11693362*b11 + 2805451*b10 + 29945333*b9 + 18024544*b8 - 2153472*b7 + 3832210*b6 + 11361702*b5 - 997443*b4 - 196079*b3 + 36966345*b2 + 4171884*b1 + 6413714) / 2 $$\nu^{15}$$ $$=$$ $$( - 84090122 \beta_{15} + 118817434 \beta_{14} - 78383728 \beta_{13} - 168880386 \beta_{12} - 17956033 \beta_{11} + 20252189 \beta_{10} + 85214078 \beta_{9} + \cdots + 108552702 ) / 2$$ (-84090122*b15 + 118817434*b14 - 78383728*b13 - 168880386*b12 - 17956033*b11 + 20252189*b10 + 85214078*b9 + 83927062*b8 - 23246011*b7 - 19135792*b6 + 41411014*b5 + 17964946*b4 - 37824487*b3 + 105206836*b2 + 39087836*b1 + 108552702) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.774725 − 2.71943i −1.57997 + 0.888702i −1.57997 − 0.888702i 0.774725 + 2.71943i −0.315931 − 0.438405i 1.15135 − 1.37875i 1.15135 + 1.37875i −0.315931 + 0.438405i −2.05868 − 0.306986i 3.74844 + 1.37690i 3.74844 − 1.37690i −2.05868 + 0.306986i 1.37578 + 2.65312i −1.59571 − 0.665253i −1.59571 + 0.665253i 1.37578 − 2.65312i
−1.18432 0.772900i 0 0.805250 + 1.83073i −1.90735 1.16705i 0 3.04658i 0.461295 2.79056i 0 1.35692 + 2.85636i
109.2 −1.18432 0.772900i 0 0.805250 + 1.83073i −1.90735 + 1.16705i 0 3.04658i 0.461295 2.79056i 0 3.16094 + 0.0920331i
109.3 −1.18432 + 0.772900i 0 0.805250 1.83073i −1.90735 1.16705i 0 3.04658i 0.461295 + 2.79056i 0 3.16094 0.0920331i
109.4 −1.18432 + 0.772900i 0 0.805250 1.83073i −1.90735 + 1.16705i 0 3.04658i 0.461295 + 2.79056i 0 1.35692 2.85636i
109.5 −0.763079 1.19068i 0 −0.835421 + 1.81716i 1.27498 1.83696i 0 1.23231i 2.80114 0.391920i 0 −3.16014 0.116351i
109.6 −0.763079 1.19068i 0 −0.835421 + 1.81716i 1.27498 + 1.83696i 0 1.23231i 2.80114 0.391920i 0 1.21431 2.91984i
109.7 −0.763079 + 1.19068i 0 −0.835421 1.81716i 1.27498 1.83696i 0 1.23231i 2.80114 + 0.391920i 0 1.21431 + 2.91984i
109.8 −0.763079 + 1.19068i 0 −0.835421 1.81716i 1.27498 + 1.83696i 0 1.23231i 2.80114 + 0.391920i 0 −3.16014 + 0.116351i
109.9 0.393855 1.35826i 0 −1.68976 1.06992i −1.33104 1.79676i 0 4.27541i −2.11875 + 1.87374i 0 −2.96471 + 1.10024i
109.10 0.393855 1.35826i 0 −1.68976 1.06992i −1.33104 + 1.79676i 0 4.27541i −2.11875 + 1.87374i 0 1.91623 + 2.51556i
109.11 0.393855 + 1.35826i 0 −1.68976 + 1.06992i −1.33104 1.79676i 0 4.27541i −2.11875 1.87374i 0 1.91623 2.51556i
109.12 0.393855 + 1.35826i 0 −1.68976 + 1.06992i −1.33104 + 1.79676i 0 4.27541i −2.11875 1.87374i 0 −2.96471 1.10024i
109.13 1.05355 0.943417i 0 0.219928 1.98787i 0.463409 2.18752i 0 3.14971i −1.64369 2.30180i 0 −1.57552 2.74185i
109.14 1.05355 0.943417i 0 0.219928 1.98787i 0.463409 + 2.18752i 0 3.14971i −1.64369 2.30180i 0 2.55197 + 1.86747i
109.15 1.05355 + 0.943417i 0 0.219928 + 1.98787i 0.463409 2.18752i 0 3.14971i −1.64369 + 2.30180i 0 2.55197 1.86747i
109.16 1.05355 + 0.943417i 0 0.219928 + 1.98787i 0.463409 + 2.18752i 0 3.14971i −1.64369 + 2.30180i 0 −1.57552 + 2.74185i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.g 16
3.b odd 2 1 1080.2.d.h yes 16
4.b odd 2 1 4320.2.d.g 16
5.b even 2 1 1080.2.d.h yes 16
8.b even 2 1 1080.2.d.h yes 16
8.d odd 2 1 4320.2.d.h 16
12.b even 2 1 4320.2.d.h 16
15.d odd 2 1 inner 1080.2.d.g 16
20.d odd 2 1 4320.2.d.h 16
24.f even 2 1 4320.2.d.g 16
24.h odd 2 1 inner 1080.2.d.g 16
40.e odd 2 1 4320.2.d.g 16
40.f even 2 1 inner 1080.2.d.g 16
60.h even 2 1 4320.2.d.g 16
120.i odd 2 1 1080.2.d.h yes 16
120.m even 2 1 4320.2.d.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.g 16 1.a even 1 1 trivial
1080.2.d.g 16 15.d odd 2 1 inner
1080.2.d.g 16 24.h odd 2 1 inner
1080.2.d.g 16 40.f even 2 1 inner
1080.2.d.h yes 16 3.b odd 2 1
1080.2.d.h yes 16 5.b even 2 1
1080.2.d.h yes 16 8.b even 2 1
1080.2.d.h yes 16 120.i odd 2 1
4320.2.d.g 16 4.b odd 2 1
4320.2.d.g 16 24.f even 2 1
4320.2.d.g 16 40.e odd 2 1
4320.2.d.g 16 60.h even 2 1
4320.2.d.h 16 8.d odd 2 1
4320.2.d.h 16 12.b even 2 1
4320.2.d.h 16 20.d odd 2 1
4320.2.d.h 16 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}^{8} + 39T_{7}^{6} + 500T_{7}^{4} + 2356T_{7}^{2} + 2556$$ T7^8 + 39*T7^6 + 500*T7^4 + 2356*T7^2 + 2556 $$T_{11}^{8} + 43T_{11}^{6} + 556T_{11}^{4} + 2032T_{11}^{2} + 284$$ T11^8 + 43*T11^6 + 556*T11^4 + 2032*T11^2 + 284 $$T_{13}^{8} - 88T_{13}^{6} + 2492T_{13}^{4} - 27228T_{13}^{2} + 101104$$ T13^8 - 88*T13^6 + 2492*T13^4 - 27228*T13^2 + 101104 $$T_{53}^{4} - 24T_{53}^{3} + 206T_{53}^{2} - 754T_{53} + 999$$ T53^4 - 24*T53^3 + 206*T53^2 - 754*T53 + 999

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + T^{7} + 2 T^{6} + 2 T^{5} + 6 T^{4} + \cdots + 16)^{2}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 3 T^{7} + 10 T^{6} + 25 T^{5} + \cdots + 625)^{2}$$
$7$ $$(T^{8} + 39 T^{6} + 500 T^{4} + 2356 T^{2} + \cdots + 2556)^{2}$$
$11$ $$(T^{8} + 43 T^{6} + 556 T^{4} + 2032 T^{2} + \cdots + 284)^{2}$$
$13$ $$(T^{8} - 88 T^{6} + 2492 T^{4} + \cdots + 101104)^{2}$$
$17$ $$(T^{8} + 77 T^{6} + 1487 T^{4} + 8435 T^{2} + \cdots + 356)^{2}$$
$19$ $$(T^{8} + 81 T^{6} + 1643 T^{4} + \cdots + 28836)^{2}$$
$23$ $$(T^{8} + 73 T^{6} + 391 T^{4} + 667 T^{2} + \cdots + 356)^{2}$$
$29$ $$(T^{8} + 80 T^{6} + 2040 T^{4} + \cdots + 72704)^{2}$$
$31$ $$(T^{4} - 44 T^{2} + 22 T + 257)^{4}$$
$37$ $$(T^{8} - 296 T^{6} + 28608 T^{4} + \cdots + 1617664)^{2}$$
$41$ $$(T^{8} - 264 T^{6} + 20528 T^{4} + \cdots + 909936)^{2}$$
$43$ $$(T^{8} - 216 T^{6} + 14856 T^{4} + \cdots + 1617664)^{2}$$
$47$ $$(T^{8} + 140 T^{6} + 5360 T^{4} + \cdots + 22784)^{2}$$
$53$ $$(T^{4} - 24 T^{3} + 206 T^{2} - 754 T + 999)^{4}$$
$59$ $$(T^{8} + 208 T^{6} + 14380 T^{4} + \cdots + 3954416)^{2}$$
$61$ $$(T^{8} + 389 T^{6} + 51139 T^{4} + \cdots + 44612496)^{2}$$
$67$ $$(T^{8} - 468 T^{6} + 71472 T^{4} + \cdots + 6470656)^{2}$$
$71$ $$(T^{8} - 452 T^{6} + 64068 T^{4} + \cdots + 73704816)^{2}$$
$73$ $$(T^{8} + 135 T^{6} + 5424 T^{4} + \cdots + 207036)^{2}$$
$79$ $$(T^{4} - T^{3} - 165 T^{2} + 267 T + 3364)^{4}$$
$83$ $$(T^{4} - 16 T^{3} - 34 T^{2} + 164 T + 213)^{4}$$
$89$ $$(T^{8} - 464 T^{6} + 63960 T^{4} + \cdots + 32757696)^{2}$$
$97$ $$(T^{8} + 399 T^{6} + 50052 T^{4} + \cdots + 39301056)^{2}$$