# Properties

 Label 1110.2.e.c Level $1110$ Weight $2$ Character orbit 1110.e Analytic conductor $8.863$ Analytic rank $0$ Dimension $16$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(739,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1110.739");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.e (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: $$8.86339462436$$ 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: $$x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64$$ x^16 - 2*x^15 + 2*x^14 + 8*x^13 + 138*x^12 - 220*x^11 + 196*x^10 + 744*x^9 + 4241*x^8 - 3018*x^7 + 658*x^6 - 1584*x^5 + 16372*x^4 - 18840*x^3 + 10952*x^2 - 1184*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{7} q^{3} + q^{4} - \beta_{11} q^{5} - \beta_{7} q^{6} - \beta_{14} q^{7} - q^{8} - q^{9}+O(q^{10})$$ q - q^2 + b7 * q^3 + q^4 - b11 * q^5 - b7 * q^6 - b14 * q^7 - q^8 - q^9 $$q - q^{2} + \beta_{7} q^{3} + q^{4} - \beta_{11} q^{5} - \beta_{7} q^{6} - \beta_{14} q^{7} - q^{8} - q^{9} + \beta_{11} q^{10} + \beta_{2} q^{11} + \beta_{7} q^{12} - \beta_{8} q^{13} + \beta_{14} q^{14} + \beta_{6} q^{15} + q^{16} + (\beta_{9} + \beta_{6} + \beta_{5} - 3) q^{17} + q^{18} + (\beta_{13} - \beta_{7} - \beta_{3} - \beta_1) q^{19} - \beta_{11} q^{20} + \beta_{10} q^{21} - \beta_{2} q^{22} + ( - \beta_{10} - \beta_{9} - \beta_{8} - 1) q^{23} - \beta_{7} q^{24} + (\beta_{15} - \beta_{10} + \cdots + \beta_1) q^{25}+ \cdots - \beta_{2} q^{99}+O(q^{100})$$ q - q^2 + b7 * q^3 + q^4 - b11 * q^5 - b7 * q^6 - b14 * q^7 - q^8 - q^9 + b11 * q^10 + b2 * q^11 + b7 * q^12 - b8 * q^13 + b14 * q^14 + b6 * q^15 + q^16 + (b9 + b6 + b5 - 3) * q^17 + q^18 + (b13 - b7 - b3 - b1) * q^19 - b11 * q^20 + b10 * q^21 - b2 * q^22 + (-b10 - b9 - b8 - 1) * q^23 - b7 * q^24 + (b15 - b10 - b9 - b8 - b6 - b5 - b4 + b1) * q^25 + b8 * q^26 - b7 * q^27 - b14 * q^28 + (-b15 + b14 - b6 + b5 - b4 + b1) * q^29 - b6 * q^30 + (-b15 + b14 - b12 + b11 - 2*b7 - b6 + b5 - b4 + b1) * q^31 - q^32 + b13 * q^33 + (-b9 - b6 - b5 + 3) * q^34 + (-b15 + b14 + b11 - b10 + b9 - b8 - b7 - b3 + b2 + b1) * q^35 - q^36 + (b13 + b12 + b11 - b9 + b7 - b4 - 1) * q^37 + (-b13 + b7 + b3 + b1) * q^38 - b1 * q^39 + b11 * q^40 + (-b12 - b11 - b10 - 2*b9 + b8) * q^41 - b10 * q^42 + (-b12 - b11 - b10 + 2*b9 + b8 + b6 + b5) * q^43 + b2 * q^44 + b11 * q^45 + (b10 + b9 + b8 + 1) * q^46 + (-b15 + b14 - b13 - b12 + b11 - b4 + 2*b1) * q^47 + b7 * q^48 + (b15 - b12 - b11 - 2*b10 + b9 - 2*b8 - b6 - b5 - b4) * q^49 + (-b15 + b10 + b9 + b8 + b6 + b5 + b4 - b1) * q^50 + (-b12 + b11 - 3*b7 - b3) * q^51 - b8 * q^52 + (-b15 + 2*b14 - 2*b12 + 2*b11 - 3*b7 - b4 - b3 + 2*b1) * q^53 + b7 * q^54 + (b15 - b13 - b10 - 2*b9 - b8 - b7 - b6 - 2*b5 + b2 - b1 + 2) * q^55 + b14 * q^56 + (-b9 + b8 - b2 + 1) * q^57 + (b15 - b14 + b6 - b5 + b4 - b1) * q^58 + (b12 - b11 + 2*b6 - 2*b5 + 2*b3 + 2*b1) * q^59 + b6 * q^60 + (b15 - 2*b14 + b13 - 2*b7 + b4 - b1) * q^61 + (b15 - b14 + b12 - b11 + 2*b7 + b6 - b5 + b4 - b1) * q^62 + b14 * q^63 + q^64 + (b14 + b13 - b8 - b7 - b6 + 2*b5 - b4) * q^65 - b13 * q^66 + (b15 + 2*b6 - 2*b5 + b4) * q^67 + (b9 + b6 + b5 - 3) * q^68 + (-b14 - b7 + b3 - b1) * q^69 + (b15 - b14 - b11 + b10 - b9 + b8 + b7 + b3 - b2 - b1) * q^70 + (-b15 + 2*b12 + 2*b11 + 2*b9 + b6 + b5 + b4 + 2*b2 - 2) * q^71 + q^72 + (-2*b14 + 2*b13 + 2*b12 - 2*b11 - b6 + b5 + 2*b3 - 3*b1) * q^73 + (-b13 - b12 - b11 + b9 - b7 + b4 + 1) * q^74 + (b15 - b14 + b12 - b11 - b8 + b4 + b3 - b1) * q^75 + (b13 - b7 - b3 - b1) * q^76 + (-b15 - 2*b12 + 2*b11 - 3*b7 - b4 - 3*b3) * q^77 + b1 * q^78 + (-b15 - 2*b14 + b6 - b5 - b4) * q^79 - b11 * q^80 + q^81 + (b12 + b11 + b10 + 2*b9 - b8) * q^82 + (-b15 - b14 - b12 + b11 - b7 - b4 - 3*b3 + b1) * q^83 + b10 * q^84 + (b12 + 2*b11 - b9 - 3*b7 + b4 + b3 - b2) * q^85 + (b12 + b11 + b10 - 2*b9 - b8 - b6 - b5) * q^86 + (b15 - b12 - b11 - b10 - b8 - b4) * q^87 - b2 * q^88 + (-b15 + 2*b13 - b12 + b11 - b6 + b5 - b4 - 4*b3 + b1) * q^89 - b11 * q^90 + (b14 - b12 + b11 - 3*b7 - b6 + b5 + b3 + b1) * q^91 + (-b10 - b9 - b8 - 1) * q^92 + (b15 - b12 - b11 - b10 - b8 - b6 - b5 - b4 + 2) * q^93 + (b15 - b14 + b13 + b12 - b11 + b4 - 2*b1) * q^94 + (b15 - b14 - 2*b11 - 2*b10 + 2*b7 - b6 + 3*b3 - b1) * q^95 - b7 * q^96 + (3*b15 - 2*b12 - 2*b11 - 2*b10 - 2*b9 - 3*b8 - b6 - b5 - 3*b4 - b2 + 2) * q^97 + (-b15 + b12 + b11 + 2*b10 - b9 + 2*b8 + b6 + b5 + b4) * q^98 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{2} + 16 q^{4} - 2 q^{5} - 16 q^{8} - 16 q^{9}+O(q^{10})$$ 16 * q - 16 * q^2 + 16 * q^4 - 2 * q^5 - 16 * q^8 - 16 * q^9 $$16 q - 16 q^{2} + 16 q^{4} - 2 q^{5} - 16 q^{8} - 16 q^{9} + 2 q^{10} + 2 q^{11} + 6 q^{15} + 16 q^{16} - 38 q^{17} + 16 q^{18} - 2 q^{20} + 6 q^{21} - 2 q^{22} - 20 q^{23} - 4 q^{25} - 6 q^{30} - 16 q^{32} + 38 q^{34} - 10 q^{35} - 16 q^{36} - 4 q^{37} + 2 q^{40} - 6 q^{41} - 6 q^{42} - 2 q^{43} + 2 q^{44} + 2 q^{45} + 20 q^{46} - 18 q^{49} + 4 q^{50} + 20 q^{55} + 16 q^{57} + 6 q^{60} + 16 q^{64} + 12 q^{65} - 38 q^{68} + 10 q^{70} - 24 q^{71} + 16 q^{72} + 4 q^{74} - 2 q^{80} + 16 q^{81} + 6 q^{82} + 6 q^{84} + 2 q^{86} + 2 q^{87} - 2 q^{88} - 2 q^{90} - 20 q^{92} + 22 q^{93} - 16 q^{95} + 38 q^{97} + 18 q^{98} - 2 q^{99}+O(q^{100})$$ 16 * q - 16 * q^2 + 16 * q^4 - 2 * q^5 - 16 * q^8 - 16 * q^9 + 2 * q^10 + 2 * q^11 + 6 * q^15 + 16 * q^16 - 38 * q^17 + 16 * q^18 - 2 * q^20 + 6 * q^21 - 2 * q^22 - 20 * q^23 - 4 * q^25 - 6 * q^30 - 16 * q^32 + 38 * q^34 - 10 * q^35 - 16 * q^36 - 4 * q^37 + 2 * q^40 - 6 * q^41 - 6 * q^42 - 2 * q^43 + 2 * q^44 + 2 * q^45 + 20 * q^46 - 18 * q^49 + 4 * q^50 + 20 * q^55 + 16 * q^57 + 6 * q^60 + 16 * q^64 + 12 * q^65 - 38 * q^68 + 10 * q^70 - 24 * q^71 + 16 * q^72 + 4 * q^74 - 2 * q^80 + 16 * q^81 + 6 * q^82 + 6 * q^84 + 2 * q^86 + 2 * q^87 - 2 * q^88 - 2 * q^90 - 20 * q^92 + 22 * q^93 - 16 * q^95 + 38 * q^97 + 18 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( 35\!\cdots\!56 \nu^{15} + \cdots - 19\!\cdots\!68 ) / 87\!\cdots\!36$$ (3598583486206356*v^15 + 12442080899330691*v^14 - 14569191096743604*v^13 + 39468243127992567*v^12 + 700061195842025382*v^11 + 1963789626693085265*v^10 - 974073933365141220*v^9 + 3679942840789531641*v^8 + 34385434998275232414*v^7 + 74659043196870143072*v^6 + 39504937564477019472*v^5 - 6917313879099567612*v^4 + 37944345263941329576*v^3 + 57688770756908863728*v^2 + 32004315856384857120*v - 1913670067113595968) / 8776577717871405136 $$\beta_{2}$$ $$=$$ $$( - 50\!\cdots\!44 \nu^{15} + \cdots + 23\!\cdots\!60 ) / 87\!\cdots\!36$$ (-5019160714525244*v^15 + 7302267297319127*v^14 - 7397401618349153*v^13 - 27789629224556003*v^12 - 739353168840258249*v^11 + 714784098392565685*v^10 - 629714596255782179*v^9 - 2097418788500457013*v^8 - 25695585297125062851*v^7 + 2402052437683258296*v^6 + 7074323891414175984*v^5 + 51547587830387511516*v^4 - 78239049472925927388*v^3 + 48290456870435422360*v^2 - 5254999736306159424*v + 23585035147057026960) / 8776577717871405136 $$\beta_{3}$$ $$=$$ $$( 95\!\cdots\!46 \nu^{15} + \cdots - 38\!\cdots\!40 ) / 87\!\cdots\!36$$ (9567165448408746*v^15 - 2762615888851235*v^14 + 7897444240156323*v^13 + 80748754982251899*v^12 + 1460496131630607439*v^11 + 369699396297893537*v^10 + 1338037921924729557*v^9 + 7563565552882221785*v^8 + 53127278044494191983*v^7 + 60306332089804664822*v^6 + 56448524114568934548*v^5 - 10785012944336174024*v^4 + 82276993584777983684*v^3 + 20812138400247811624*v^2 + 64382242804295173056*v - 3846979372012104640) / 8776577717871405136 $$\beta_{4}$$ $$=$$ $$( - 10\!\cdots\!49 \nu^{15} + \cdots + 38\!\cdots\!40 ) / 87\!\cdots\!36$$ (-10747369620592049*v^15 + 49507716150097919*v^14 - 59292368394327099*v^13 - 69436016039014749*v^12 - 1234651710034151587*v^11 + 6372699016733347429*v^10 - 5776516973434541039*v^9 - 7026367954906604219*v^8 - 23166341077983495856*v^7 + 163792457990670368680*v^6 - 16498400553027426082*v^5 - 39842557895772371100*v^4 - 290703731837693963136*v^3 + 568979970447300755648*v^2 - 350212003392840468312*v + 38099452666799307040) / 8776577717871405136 $$\beta_{5}$$ $$=$$ $$( - 23\!\cdots\!03 \nu^{15} + \cdots + 39\!\cdots\!12 ) / 17\!\cdots\!72$$ (-23656976378661003*v^15 + 30447616523487557*v^14 - 5668307481205641*v^13 - 227788358439582645*v^12 - 3371744582497375019*v^11 + 2928984459775409985*v^10 + 183920614188561673*v^9 - 20557827614532110923*v^8 - 109211225452771384886*v^7 + 4858276358302257694*v^6 + 68517725209056647836*v^5 + 104967433823144605564*v^4 - 263375550046521833464*v^3 + 174184766214316358360*v^2 - 1536500573724590784*v + 39218675595124970912) / 17553155435742810272 $$\beta_{6}$$ $$=$$ $$( - 27\!\cdots\!09 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 17\!\cdots\!72$$ (-27270263108412109*v^15 + 35845002295782009*v^14 - 10266165292669687*v^13 - 258385448345775489*v^12 - 3888009054007888081*v^11 + 3464120130945415681*v^10 - 265227763229672233*v^9 - 23400222459835818047*v^8 - 126253736340414947654*v^7 + 7307757322320005194*v^6 + 67175719164418371636*v^5 + 111309066839156150716*v^4 - 328339387849681228408*v^3 + 212479679502370074056*v^2 - 40789217922827096384*v + 41557711249901783200) / 17553155435742810272 $$\beta_{7}$$ $$=$$ $$( 18\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 87\!\cdots\!36$$ (18273716052943846*v^15 - 34740788741012139*v^14 + 33848739219740466*v^13 + 148488657329282791*v^12 + 2537071360259347170*v^11 - 3762085295892389589*v^10 + 3314080510791990968*v^9 + 13820218932099338377*v^8 + 78920027203186704448*v^7 - 46628819603962745844*v^6 + 10799364680828176918*v^5 - 28274563205543913964*v^4 + 296006462710790874136*v^3 - 311794891535882361168*v^2 + 180986281567814143544*v - 10786298850005666000) / 8776577717871405136 $$\beta_{8}$$ $$=$$ $$( - 21\!\cdots\!85 \nu^{15} + \cdots + 35\!\cdots\!52 ) / 87\!\cdots\!36$$ (-21379513563644685*v^15 + 30517922200804757*v^14 - 24808754347643015*v^13 - 155365536772579389*v^12 - 3105505244059654869*v^11 + 2986426268519524431*v^10 - 2060018014627625601*v^9 - 13421212405044163915*v^8 - 105151622543957426978*v^7 + 9942740781114420512*v^6 + 28298939092911453964*v^5 + 127950227372178531508*v^4 - 324346718373826828408*v^3 + 200025933426027654000*v^2 - 21765048210218762368*v + 3543341730699935552) / 8776577717871405136 $$\beta_{9}$$ $$=$$ $$( 24\!\cdots\!59 \nu^{15} + \cdots - 52\!\cdots\!04 ) / 87\!\cdots\!36$$ (24216523190739559*v^15 - 32826769491708205*v^14 + 15191250088530019*v^13 + 218066683956470309*v^12 + 3466704866406129737*v^11 - 3190075392666935965*v^10 + 946193107947504485*v^9 + 19727658351803595975*v^8 + 113783901684109617420*v^7 - 8456590044398138202*v^6 - 47181359656513821644*v^5 - 95486908052697491976*v^4 + 321463007301980811984*v^3 - 203241633505665399864*v^2 + 22172893866218873632*v - 5271315577848317504) / 8776577717871405136 $$\beta_{10}$$ $$=$$ $$( - 12\!\cdots\!34 \nu^{15} + \cdots - 13\!\cdots\!72 ) / 43\!\cdots\!68$$ (-12817278110967634*v^15 + 17727172949013492*v^14 - 9114568564984668*v^13 - 118516004781748970*v^12 - 1827553739206846274*v^11 + 1731816050083731475*v^10 - 682650127547049224*v^9 - 10968243851454832160*v^8 - 59671526276950665476*v^7 + 5472559110178869761*v^6 + 17389998241110704844*v^5 + 28357264878402748114*v^4 - 183727634589725420008*v^3 + 113728649431467197100*v^2 - 12379865386235900624*v - 13297933955641522072) / 4388288858935702568 $$\beta_{11}$$ $$=$$ $$( 83\!\cdots\!77 \nu^{15} + \cdots - 46\!\cdots\!88 ) / 17\!\cdots\!72$$ (83459165975960877*v^15 - 142854828745674593*v^14 + 150156848474466771*v^13 + 677067493024494429*v^12 + 11703383233180742765*v^11 - 14685343779497331501*v^10 + 15464887800268384161*v^9 + 63102508808192934739*v^8 + 370192197216499213802*v^7 - 116570414566402812598*v^6 + 120150201939706955400*v^5 - 120684227016313567036*v^4 + 1228622370375316235368*v^3 - 1224969327699563325368*v^2 + 789420777992913348560*v - 46015836365558307488) / 17553155435742810272 $$\beta_{12}$$ $$=$$ $$( - 87\!\cdots\!83 \nu^{15} + \cdots + 48\!\cdots\!76 ) / 17\!\cdots\!72$$ (-87072452705711983*v^15 + 148252214517969045*v^14 - 154754706285930817*v^13 - 707664582930687273*v^12 - 12219647704691255827*v^11 + 15220479450667337197*v^10 - 15914036177686618067*v^9 - 65944903653496641863*v^8 - 387234708104142776570*v^7 + 119019895530420560098*v^6 - 121492207984345231600*v^5 + 127025860032325112188*v^4 - 1293586208178475630312*v^3 + 1263264240987617041064*v^2 - 793567184470530233616*v + 48354872020335119776) / 17553155435742810272 $$\beta_{13}$$ $$=$$ $$( 48\!\cdots\!94 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 87\!\cdots\!36$$ (48118279587099894*v^15 - 84633788878151129*v^14 + 81011792960581768*v^13 + 395068305903552419*v^12 + 6753590184184493108*v^11 - 8963801050572878679*v^10 + 8071383029047362522*v^9 + 36768398939132960325*v^8 + 214738955638592641550*v^7 - 94977690341340840884*v^6 + 39676510248119000838*v^5 - 75392677385581876676*v^4 + 777438211441026403136*v^3 - 805058818164724298664*v^2 + 477800959874748124440*v - 28477425678718074000) / 8776577717871405136 $$\beta_{14}$$ $$=$$ $$( - 58\!\cdots\!38 \nu^{15} + \cdots + 35\!\cdots\!12 ) / 87\!\cdots\!36$$ (-58599146537079738*v^15 + 123266479279552438*v^14 - 115989458543362729*v^13 - 473747488729663366*v^12 - 8038705530926452605*v^11 + 13879856120878845218*v^10 - 10956815022075974549*v^9 - 44060072494787372566*v^8 - 244375119257663357073*v^7 + 216915316012027345288*v^6 + 1175795409551302370*v^5 + 93341958315418191324*v^4 - 999232944627119782372*v^3 + 1203959456301324504768*v^2 - 601092945511793476232*v + 35816544221695449712) / 8776577717871405136 $$\beta_{15}$$ $$=$$ $$( - 59\!\cdots\!79 \nu^{15} + \cdots + 63\!\cdots\!12 ) / 87\!\cdots\!36$$ (-59923129802572879*v^15 + 117113881990875321*v^14 - 95951644204439093*v^13 - 495669233213731053*v^12 - 8288330089041619629*v^11 + 12958762198920303113*v^10 - 8373350468913758013*v^9 - 45494715681625362075*v^8 - 255837304581511347840*v^7 + 182824512099824497730*v^6 + 67554990970372022286*v^5 + 154569987779827399044*v^4 - 971789885387320911184*v^3 + 995490369264377828984*v^2 - 396696295674212250488*v + 6396540916876267712) / 8776577717871405136
 $$\nu$$ $$=$$ $$( \beta_{12} + \beta_{11} - \beta_{6} + \beta_{5} ) / 2$$ (b12 + b11 - b6 + b5) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} + 5\beta_{7} + \beta_{4} + \beta_{3} - \beta_1$$ b15 - b14 + b12 - b11 + 5*b7 + b4 + b3 - b1 $$\nu^{3}$$ $$=$$ $$( - 2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - 7 \beta_{6} + \cdots - 2 ) / 2$$ (-2*b14 - 2*b13 - 7*b12 - 7*b11 + 2*b10 + 2*b7 - 7*b6 + 7*b5 + 2*b4 + 2*b2 - 2) / 2 $$\nu^{4}$$ $$=$$ $$- 10 \beta_{15} - 2 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} + 10 \beta_{6} + \cdots - 41$$ -10*b15 - 2*b12 - 2*b11 + 11*b10 + 11*b9 + 11*b8 + 10*b6 + 10*b5 + 10*b4 - 2*b2 - 41 $$\nu^{5}$$ $$=$$ $$( - 22 \beta_{15} + 22 \beta_{14} + 26 \beta_{13} - 61 \beta_{12} - 61 \beta_{11} + 22 \beta_{10} + \cdots - 34 ) / 2$$ (-22*b15 + 22*b14 + 26*b13 - 61*b12 - 61*b11 + 22*b10 - 2*b8 - 34*b7 + 61*b6 - 61*b5 + 26*b2 - 2*b1 - 34) / 2 $$\nu^{6}$$ $$=$$ $$- 97 \beta_{15} + 109 \beta_{14} - 26 \beta_{13} - 99 \beta_{12} + 99 \beta_{11} - 379 \beta_{7} + \cdots + 111 \beta_1$$ -97*b15 + 109*b14 - 26*b13 - 99*b12 + 99*b11 - 379*b7 + 28*b6 - 28*b5 - 97*b4 - 113*b3 + 111*b1 $$\nu^{7}$$ $$=$$ $$( 222 \beta_{14} + 278 \beta_{13} + 569 \beta_{12} + 577 \beta_{11} - 222 \beta_{10} - 8 \beta_{9} + \cdots + 394 ) / 2$$ (222*b14 + 278*b13 + 569*b12 + 577*b11 - 222*b10 - 8*b9 + 28*b8 - 394*b7 + 569*b6 - 577*b5 - 218*b4 - 8*b3 - 278*b2 - 28*b1 + 394) / 2 $$\nu^{8}$$ $$=$$ $$942 \beta_{15} + 306 \beta_{12} + 306 \beta_{11} - 1073 \beta_{10} - 1141 \beta_{9} - 1097 \beta_{8} + \cdots + 3643$$ 942*b15 + 306*b12 + 306*b11 - 1073*b10 - 1141*b9 - 1097*b8 - 978*b6 - 978*b5 - 942*b4 + 270*b2 + 3643 $$\nu^{9}$$ $$=$$ $$( 2122 \beta_{15} - 2218 \beta_{14} - 2822 \beta_{13} + 5615 \beta_{12} + 5439 \beta_{11} - 2218 \beta_{10} + \cdots + 4126 ) / 2$$ (2122*b15 - 2218*b14 - 2822*b13 + 5615*b12 + 5439*b11 - 2218*b10 - 192*b9 + 302*b8 + 4126*b7 - 5615*b6 + 5439*b5 + 192*b3 - 2822*b2 + 302*b1 + 4126) / 2 $$\nu^{10}$$ $$=$$ $$9163 \beta_{15} - 10567 \beta_{14} + 2630 \beta_{13} + 9657 \beta_{12} - 9657 \beta_{11} + \cdots - 10765 \beta_1$$ 9163*b15 - 10567*b14 + 2630*b13 + 9657*b12 - 9657*b11 + 35497*b7 - 3124*b6 + 3124*b5 + 9163*b4 + 11459*b3 - 10765*b1 $$\nu^{11}$$ $$=$$ $$( - 22122 \beta_{14} - 28178 \beta_{13} - 52459 \beta_{12} - 55187 \beta_{11} + 22122 \beta_{10} + \cdots - 41854 ) / 2$$ (-22122*b14 - 28178*b13 - 52459*b12 - 55187*b11 + 22122*b10 + 3128*b9 - 3012*b8 + 41854*b7 - 52459*b6 + 55187*b5 + 20574*b4 + 3128*b3 + 28178*b2 + 3012*b1 - 41854) / 2 $$\nu^{12}$$ $$=$$ $$- 89202 \beta_{15} - 31214 \beta_{12} - 31214 \beta_{11} + 104123 \beta_{10} + 114855 \beta_{9} + \cdots - 347593$$ -89202*b15 - 31214*b12 - 31214*b11 + 104123*b10 + 114855*b9 + 105371*b8 + 95358*b6 + 95358*b5 + 89202*b4 - 25050*b2 - 347593 $$\nu^{13}$$ $$=$$ $$( - 199358 \beta_{15} + 220574 \beta_{14} + 279810 \beta_{13} - 544365 \beta_{12} - 507629 \beta_{11} + \cdots - 420186 ) / 2$$ (-199358*b15 + 220574*b14 + 279810*b13 - 544365*b12 - 507629*b11 + 220574*b10 + 43376*b9 - 29162*b8 - 420186*b7 + 544365*b6 - 507629*b5 - 43376*b3 + 279810*b2 - 29162*b1 - 420186) / 2 $$\nu^{14}$$ $$=$$ $$- 868761 \beta_{15} + 1026381 \beta_{14} - 236434 \beta_{13} - 941739 \beta_{12} + 941739 \beta_{11} + \cdots + 1030535 \beta_1$$ -868761*b15 + 1026381*b14 - 236434*b13 - 941739*b12 + 941739*b11 - 3410179*b7 + 309772*b6 - 309772*b5 - 868761*b4 - 1150209*b3 + 1030535*b1 $$\nu^{15}$$ $$=$$ $$( 2199438 \beta_{14} + 2773286 \beta_{13} + 4918065 \beta_{12} + 5377337 \beta_{11} - 2199438 \beta_{10} + \cdots + 4204746 ) / 2$$ (2199438*b14 + 2773286*b13 + 4918065*b12 + 5377337*b11 - 2199438*b10 - 551944*b9 + 278572*b8 - 4204746*b7 + 4918065*b6 - 5377337*b5 - 1932538*b4 - 551944*b3 - 2773286*b2 - 278572*b1 + 4204746) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 2.23405 − 2.23405i 2.18974 − 2.18974i 0.718347 − 0.718347i 0.623809 − 0.623809i 0.0603067 − 0.0603067i −1.28817 + 1.28817i −1.32920 + 1.32920i −2.20888 + 2.20888i 2.23405 + 2.23405i 2.18974 + 2.18974i 0.718347 + 0.718347i 0.623809 + 0.623809i 0.0603067 + 0.0603067i −1.28817 − 1.28817i −1.32920 − 1.32920i −2.20888 − 2.20888i
−1.00000 1.00000i 1.00000 −2.23405 0.0950871i 1.00000i 3.20752i −1.00000 −1.00000 2.23405 + 0.0950871i
739.2 −1.00000 1.00000i 1.00000 −2.18974 + 0.452819i 1.00000i 0.0776078i −1.00000 −1.00000 2.18974 0.452819i
739.3 −1.00000 1.00000i 1.00000 −0.718347 2.11754i 1.00000i 2.74990i −1.00000 −1.00000 0.718347 + 2.11754i
739.4 −1.00000 1.00000i 1.00000 −0.623809 + 2.14729i 1.00000i 5.25373i −1.00000 −1.00000 0.623809 2.14729i
739.5 −1.00000 1.00000i 1.00000 −0.0603067 + 2.23525i 1.00000i 3.18244i −1.00000 −1.00000 0.0603067 2.23525i
739.6 −1.00000 1.00000i 1.00000 1.28817 + 1.82773i 1.00000i 1.25548i −1.00000 −1.00000 −1.28817 1.82773i
739.7 −1.00000 1.00000i 1.00000 1.32920 1.79812i 1.00000i 1.87266i −1.00000 −1.00000 −1.32920 + 1.79812i
739.8 −1.00000 1.00000i 1.00000 2.20888 + 0.347650i 1.00000i 2.08112i −1.00000 −1.00000 −2.20888 0.347650i
739.9 −1.00000 1.00000i 1.00000 −2.23405 + 0.0950871i 1.00000i 3.20752i −1.00000 −1.00000 2.23405 0.0950871i
739.10 −1.00000 1.00000i 1.00000 −2.18974 0.452819i 1.00000i 0.0776078i −1.00000 −1.00000 2.18974 + 0.452819i
739.11 −1.00000 1.00000i 1.00000 −0.718347 + 2.11754i 1.00000i 2.74990i −1.00000 −1.00000 0.718347 2.11754i
739.12 −1.00000 1.00000i 1.00000 −0.623809 2.14729i 1.00000i 5.25373i −1.00000 −1.00000 0.623809 + 2.14729i
739.13 −1.00000 1.00000i 1.00000 −0.0603067 2.23525i 1.00000i 3.18244i −1.00000 −1.00000 0.0603067 + 2.23525i
739.14 −1.00000 1.00000i 1.00000 1.28817 1.82773i 1.00000i 1.25548i −1.00000 −1.00000 −1.28817 + 1.82773i
739.15 −1.00000 1.00000i 1.00000 1.32920 + 1.79812i 1.00000i 1.87266i −1.00000 −1.00000 −1.32920 1.79812i
739.16 −1.00000 1.00000i 1.00000 2.20888 0.347650i 1.00000i 2.08112i −1.00000 −1.00000 −2.20888 + 0.347650i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 739.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.e.c 16
3.b odd 2 1 3330.2.e.f 16
5.b even 2 1 1110.2.e.d yes 16
15.d odd 2 1 3330.2.e.e 16
37.b even 2 1 1110.2.e.d yes 16
111.d odd 2 1 3330.2.e.e 16
185.d even 2 1 inner 1110.2.e.c 16
555.b odd 2 1 3330.2.e.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.c 16 1.a even 1 1 trivial
1110.2.e.c 16 185.d even 2 1 inner
1110.2.e.d yes 16 5.b even 2 1
1110.2.e.d yes 16 37.b even 2 1
3330.2.e.e 16 15.d odd 2 1
3330.2.e.e 16 111.d odd 2 1
3330.2.e.f 16 3.b odd 2 1
3330.2.e.f 16 555.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{16} + 65 T_{7}^{14} + 1582 T_{7}^{12} + 19194 T_{7}^{10} + 126197 T_{7}^{8} + 448469 T_{7}^{6} + \cdots + 3136$$ T7^16 + 65*T7^14 + 1582*T7^12 + 19194*T7^10 + 126197*T7^8 + 448469*T7^6 + 791452*T7^4 + 525424*T7^2 + 3136 $$T_{13}^{8} - 34T_{13}^{6} + 285T_{13}^{4} + 16T_{13}^{3} - 600T_{13}^{2} + 80T_{13} + 64$$ T13^8 - 34*T13^6 + 285*T13^4 + 16*T13^3 - 600*T13^2 + 80*T13 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{16}$$
$3$ $$(T^{2} + 1)^{8}$$
$5$ $$T^{16} + 2 T^{15} + \cdots + 390625$$
$7$ $$T^{16} + 65 T^{14} + \cdots + 3136$$
$11$ $$(T^{8} - T^{7} - 50 T^{6} + \cdots + 716)^{2}$$
$13$ $$(T^{8} - 34 T^{6} + \cdots + 64)^{2}$$
$17$ $$(T^{8} + 19 T^{7} + \cdots - 4504)^{2}$$
$19$ $$T^{16} + 156 T^{14} + \cdots + 86415616$$
$23$ $$(T^{8} + 10 T^{7} + \cdots + 31168)^{2}$$
$29$ $$T^{16} + 193 T^{14} + \cdots + 25563136$$
$31$ $$T^{16} + \cdots + 1349533696$$
$37$ $$T^{16} + \cdots + 3512479453921$$
$41$ $$(T^{8} + 3 T^{7} + \cdots - 1327648)^{2}$$
$43$ $$(T^{8} + T^{7} + \cdots + 1418624)^{2}$$
$47$ $$T^{16} + \cdots + 1124663296$$
$53$ $$T^{16} + \cdots + 24049806400$$
$59$ $$T^{16} + \cdots + 50462269542400$$
$61$ $$T^{16} + \cdots + 229356703744$$
$67$ $$T^{16} + \cdots + 505031950336$$
$71$ $$(T^{8} + 12 T^{7} + \cdots - 6032896)^{2}$$
$73$ $$T^{16} + \cdots + 67\!\cdots\!04$$
$79$ $$T^{16} + \cdots + 19573129216$$
$83$ $$T^{16} + \cdots + 502941945856$$
$89$ $$T^{16} + \cdots + 60\!\cdots\!24$$
$97$ $$(T^{8} - 19 T^{7} + \cdots + 26655952)^{2}$$