# Properties

 Label 380.2.u.a Level $380$ Weight $2$ Character orbit 380.u Analytic conductor $3.034$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(61,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("380.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.u (of order $$9$$, degree $$6$$, 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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + \cdots + 9$$ x^18 - 3*x^17 + 3*x^15 + 36*x^14 + 72*x^13 - 134*x^12 - 741*x^11 + 486*x^10 + 5700*x^9 + 11637*x^8 + 13878*x^7 + 13978*x^6 + 10005*x^5 + 5949*x^4 + 2649*x^3 + 819*x^2 + 135*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_{3}) q^{3} - \beta_{4} q^{5} + ( - \beta_{17} + \beta_{15} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{17} - \beta_{13} - \beta_{11} + \cdots - 1) q^{9}+O(q^{10})$$ q + (-b7 + b3) * q^3 - b4 * q^5 + (-b17 + b15 - b13 - b12 - 2*b11 - b10 + b4) * q^7 + (-b17 - b13 - b11 - b6 + b5 - b4 + b3 + b1 - 1) * q^9 $$q + ( - \beta_{7} + \beta_{3}) q^{3} - \beta_{4} q^{5} + ( - \beta_{17} + \beta_{15} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{17} + \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q + (-b7 + b3) * q^3 - b4 * q^5 + (-b17 + b15 - b13 - b12 - 2*b11 - b10 + b4) * q^7 + (-b17 - b13 - b11 - b6 + b5 - b4 + b3 + b1 - 1) * q^9 + (-b17 + b14 - 2*b13 + b12 - b10 + b9 - b7 - b6 - b4 + 2*b3 + 2*b1 - 2) * q^11 + (b15 - b13 - b10 + b9 + b8 + 2*b6) * q^13 - b1 * q^15 + (b16 + b13 + b12 + b10 + b9 - b8 - 2*b7 + b5 + b4 - b1 + 1) * q^17 + (-b15 - b14 - 3*b12 - b11 - b10 + b8 - b6 - b3 - b2) * q^19 + (b12 + b11 - b7 + 2*b6 - b5 - b4 + 2*b2 - b1 + 1) * q^21 + (b14 + b13 + b12 + b8 + b4 + b1 - 1) * q^23 + b9 * q^25 + (-b17 - b16 + b14 - b13 - b12 + 3*b11 - b10 + 2*b9 - 3*b4 + b2 - 1) * q^27 + (b17 + b16 - b15 - 2*b14 + b13 - 3*b12 - b9 + b8 + b7 + b6 - b5 - b4 - 2*b3 - b2 - b1 - 1) * q^29 + (-b17 - b16 + b15 + b13 - 2*b12 - 3*b11 - b10 - b9 + b8 - b5 + b2) * q^31 + (-2*b17 - 2*b16 + b13 - 3*b12 - 2*b11 - b10 - 2*b9 + b8 - 2*b6 - 2*b5 - 4*b4 + b3 + 3*b2 + 3*b1 + 1) * q^33 + (-b17 - b13 - b12 - b9 - b8 + 1) * q^35 + (b17 + b15 + b14 + 2*b12 - b11 + b10 + 2*b9 - 2*b8 + b5 + b4 + b3 - b1 + 4) * q^37 + (-2*b12 + 3*b11 - 3*b9 + 2*b8 + b7 + b6 - 2*b5 - 2*b4 - b1 - 4) * q^39 + (2*b17 - 2*b14 + b13 - b12 + b11 + 2*b9 + 2*b8 + 3*b7 + 2*b6 - 3*b3 - 2*b2) * q^41 + (b17 + b16 - 4*b13 + 5*b12 + b11 + 3*b9 - b8 + b6 + b5 + 4*b4 - b3 + b2 + b1 - 2) * q^43 + (b16 + b9 - b8 - b6 + b5 + 2*b4 - b3 - b2) * q^45 + (-b17 + b15 + b14 + 3*b12 + b11 + 2*b9 - b8 + 2*b7 + 2*b6 + 2*b5 - 2*b2 - 2*b1 + 1) * q^47 + (b17 + b16 - b14 - 2*b13 + 3*b12 - 4*b11 + b10 - b9 - 2*b6 + 2*b4 + 2*b3 + b2 + 2*b1 - 2) * q^49 + (b16 + b14 - b13 + b12 + 2*b10 - 3*b9 - b8 - 3*b7 - 2*b6 + 3*b5 + 2*b4 + 2*b3 - 2*b2 - 2) * q^51 + (b16 - b15 - b14 + b13 - b11 - b8 - 2*b7 - b5 + b4 + b3 - b2 + 2) * q^53 + (b17 + 2*b13 + b12 + b10 + b9 - b8 - b6 + 2*b5 + b4 - b2 - b1 + 1) * q^55 + (b17 - b15 - b14 + 3*b13 + b12 + b11 + b10 - b8 - 2*b7 - b6 - b5 - b3 + b2 - 2*b1 + 5) * q^57 + (b17 - b16 - 3*b15 + 2*b13 + 2*b12 + 2*b11 - 2*b9 - b5 - 2*b4 + b1 + 2) * q^59 + (2*b17 + 4*b13 - b12 + b11 + 2*b10 + 2*b7 + b5 + 2*b4 - b3 + b2 + 1) * q^61 + (b16 - b15 + b14 - 2*b13 + b12 + b11 + b10 - 6*b9 - 2*b8 + b7 - 2*b6 - b5 + 3*b4 + 2*b3 - 2*b2 - 3) * q^63 + (-b17 - b16 + b15 + b14 - b13 + b12 - b11 - b4 + 2*b2) * q^65 + (-b17 - b16 + 2*b15 + 3*b14 - 4*b13 + 2*b12 - 2*b11 - b9 - 2*b8 - 2*b7 - b6 - b5 - b4 + b3 + 2*b2 + b1 - 4) * q^67 + (-b16 + b13 + b8 + 3*b7 - b6 - b4 - b3 + 3*b1) * q^69 + (b17 + b16 + b13 + b11 - b10 + b9 - 2*b8 + b6 + b5 + 5*b4 + b3 - 3*b2 - 3*b1 + 1) * q^71 + (3*b17 - b16 - b14 + 2*b13 - 3*b12 + 5*b11 + b10 - b9 + 3*b8 + 2*b7 + 2*b6 - b5 - b4 - 2*b3 - b2 - b1 + 2) * q^73 - b5 * q^75 + (b14 + 2*b11 + b10 - 2*b9 - 3*b8 - b5 - b4 + 2*b3 - 2*b1 + 2) * q^77 + (b17 - 2*b16 + 2*b14 + b13 + b12 + b11 + 2*b10 - b9 + b8 - b7 - b6 + 3*b5 - 2*b4 + b3 - 2*b2 + 3*b1 - 1) * q^79 + (-2*b15 - 2*b14 + 2*b13 - 2*b12 - b9 - 3*b6 - 3*b5 + 3*b4 - 2*b3 - b2 - b1 + 1) * q^81 + (b17 - b16 - b15 + 7*b13 - 2*b12 - b11 + b10 - b9 + b8 + b7 - 2*b6 + 4*b5 - 2*b3 - 4*b2 + b1) * q^83 + (b16 - b14 + b13 - b12 + b11 + 2*b6 - b5 - b3 - 2*b1 + 1) * q^85 + (2*b17 + b16 - b15 - 2*b14 + 4*b13 - b12 + b11 + b10 + 3*b7 + 3*b6 + b4 - 6*b3 - b2 - 6*b1 + 3) * q^87 + (-2*b16 - b15 - 2*b14 - 3*b13 - 2*b12 + 2*b11 - 2*b10 - 2*b9 + b8 + 2*b7 + 2*b6 - 2*b5) * q^89 + (-b17 - b16 + b15 - 5*b13 + 4*b12 - b10 + 3*b7 + 2*b5 - 4*b4 - 2*b3 + b2 + 4*b1 - 1) * q^91 + (b15 - 2*b13 + b12 - b11 - b9 + 3*b7 + 2*b6 - b5 + b4 + 2*b2 - b1 + 1) * q^93 + (-b16 + b15 + b14 - b13 - b11 - b10 + b7 + b6 - b2 + 1) * q^95 + (3*b15 - 2*b13 + 3*b12 + 2*b11 + b9 + 3*b7 + b6 - b5 - 2*b4 + b2 + 3) * q^97 + (-b17 + b14 + 3*b13 - 2*b11 - b10 + b8 - 2*b5 + 4*b4 + 2*b3 + 2*b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 3 q^{3} - 9 q^{9}+O(q^{10})$$ 18 * q + 3 * q^3 - 9 * q^9 $$18 q + 3 q^{3} - 9 q^{9} + 9 q^{13} - 3 q^{15} + 12 q^{17} + 18 q^{19} + 15 q^{21} - 9 q^{23} - 33 q^{29} - 18 q^{31} + 9 q^{33} + 12 q^{35} + 60 q^{37} - 84 q^{39} - 18 q^{41} + 18 q^{43} + 6 q^{45} - 15 q^{47} - 15 q^{49} - 9 q^{51} + 24 q^{53} + 3 q^{55} + 66 q^{57} + 48 q^{59} - 18 q^{61} - 54 q^{63} + 3 q^{65} - 36 q^{67} - 9 q^{69} + 18 q^{73} - 6 q^{75} + 9 q^{79} - 9 q^{81} - 30 q^{83} - 12 q^{85} - 9 q^{87} + 6 q^{89} + 30 q^{91} + 21 q^{95} + 21 q^{97} - 21 q^{99}+O(q^{100})$$ 18 * q + 3 * q^3 - 9 * q^9 + 9 * q^13 - 3 * q^15 + 12 * q^17 + 18 * q^19 + 15 * q^21 - 9 * q^23 - 33 * q^29 - 18 * q^31 + 9 * q^33 + 12 * q^35 + 60 * q^37 - 84 * q^39 - 18 * q^41 + 18 * q^43 + 6 * q^45 - 15 * q^47 - 15 * q^49 - 9 * q^51 + 24 * q^53 + 3 * q^55 + 66 * q^57 + 48 * q^59 - 18 * q^61 - 54 * q^63 + 3 * q^65 - 36 * q^67 - 9 * q^69 + 18 * q^73 - 6 * q^75 + 9 * q^79 - 9 * q^81 - 30 * q^83 - 12 * q^85 - 9 * q^87 + 6 * q^89 + 30 * q^91 + 21 * q^95 + 21 * q^97 - 21 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + \cdots + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 23\!\cdots\!01 \nu^{17} + \cdots - 93\!\cdots\!61 ) / 12\!\cdots\!51$$ (-231667570354872196288782201*v^17 + 389488635224348211306582021*v^16 + 754451987251829617693613317*v^15 - 35259195860322802131384636*v^14 - 9794519403131638766351044047*v^13 - 28110054430217622150128255619*v^12 + 3652170209608927941590351160*v^11 + 207492395843744952615724230528*v^10 + 146700786197641007166642360739*v^9 - 1372603081518822273455517698208*v^8 - 4645145281589102147013300825927*v^7 - 7597964985987669865304021738517*v^6 - 8371284501364384832505708957972*v^5 - 6893402601950139853407587360982*v^4 - 4594195196132758725916285640297*v^3 - 2079805066868622893127635890107*v^2 - 724820953738053752312865986643*v - 93468141871554998285173833861) / 120643294798248855786063810651 $$\beta_{3}$$ $$=$$ $$( 80\!\cdots\!23 \nu^{17} + \cdots + 42\!\cdots\!45 ) / 12\!\cdots\!51$$ (807839106278777849009544823*v^17 - 2986838739776381567581894698*v^16 + 1867134315360308741890730413*v^15 + 1872146384745245363395544859*v^14 + 27491556962166055159584220869*v^13 + 38231611828007321887478658966*v^12 - 142210181709551486691443966756*v^11 - 511297681362483201488696840775*v^10 + 786397483721716848008874249031*v^9 + 4202942863098206765227775666769*v^8 + 6283146946918122114431660747100*v^7 + 5638806059763457689981530357466*v^6 + 5476091241485392246427824624081*v^5 + 2585843437669590078289979596122*v^4 + 1555597775787327760469102806360*v^3 + 448599006470879866260842212611*v^2 + 174118689851659309586610929895*v + 42473161608363356652161376045) / 120643294798248855786063810651 $$\beta_{4}$$ $$=$$ $$( - 50\!\cdots\!40 \nu^{17} + \cdots - 11\!\cdots\!00 ) / 36\!\cdots\!53$$ (-5056680892738733138987859440*v^17 + 9681353074242183377921426397*v^16 + 19262761143237546207843032094*v^15 - 23905807425362658095768011503*v^14 - 197551343744754797559938310306*v^13 - 553196771225424350242177704678*v^12 + 382497897260625740583745348186*v^11 + 4671603490342689808772463505917*v^10 + 1204973689785979136472301560222*v^9 - 33526262422953324128398272683790*v^8 - 88517720476447807117081195674810*v^7 - 118182272846585061463821669844143*v^6 - 116186278391742293875602560844578*v^5 - 93113696997188752021658139742551*v^4 - 52734366035525201525416390588620*v^3 - 25697125146828966347819863988076*v^2 - 7632714788422950749050149857136*v - 1125829928136219537092382779400) / 361929884394746567358191431953 $$\beta_{5}$$ $$=$$ $$( 18\!\cdots\!41 \nu^{17} + \cdots - 15\!\cdots\!20 ) / 12\!\cdots\!51$$ (1829563201324672013014050641*v^17 - 6420920381079182069281010698*v^16 + 2911921582382152892934811061*v^15 + 5170277202053468185458456822*v^14 + 63038582316078521411683941666*v^13 + 98365780788788166680209272258*v^12 - 308200982941096184260819886959*v^11 - 1220840201219001147340133749354*v^10 + 1567727111447515078722491291930*v^9 + 9891041642549056526226491790510*v^8 + 16001885139052307653649385511941*v^7 + 15167997624346760686276753864086*v^6 + 14173868221779242322028202015117*v^5 + 7550723801540826027192538260020*v^4 + 4100659153988020754213674777172*v^3 + 1163764379089976102739697658592*v^2 + 147726002538830187776340585000*v - 15170042678216199416963578320) / 120643294798248855786063810651 $$\beta_{6}$$ $$=$$ $$( - 30\!\cdots\!74 \nu^{17} + \cdots - 17\!\cdots\!96 ) / 16\!\cdots\!69$$ (-30103266206878174*v^17 + 89982817832438707*v^16 + 6253022931308790*v^15 - 108503893341585330*v^14 - 1078183524913991538*v^13 - 2161022396547272742*v^12 + 4191654100641764381*v^11 + 22642405411909503703*v^10 - 15290748700288928310*v^9 - 175348207542851549889*v^8 - 347746927977772126974*v^7 - 392225643626669668809*v^6 - 378292079701243787365*v^5 - 263844419857316268116*v^4 - 145783314394989192354*v^3 - 62431169231720649660*v^2 - 15738989858455126452*v - 1709886947115762396) / 1682094761273482869 $$\beta_{7}$$ $$=$$ $$( 35\!\cdots\!66 \nu^{17} + \cdots + 13\!\cdots\!58 ) / 12\!\cdots\!51$$ (3536302434880648031879629866*v^17 - 11606077193716845543735118000*v^16 + 3280809811429364816273280074*v^15 + 9758629102280018401481148780*v^14 + 124141909103749983691694315628*v^13 + 219988240913339033861968672866*v^12 - 535279020919672186464908408868*v^11 - 2465836184704645536770366989816*v^10 + 2415947337928304110549428763523*v^9 + 19448384398452141897836541341535*v^8 + 35617030686888582634964968403676*v^7 + 39213883648824552384755713403175*v^6 + 38966703955178806804966846380378*v^5 + 24959225289339265167818515585577*v^4 + 14238217546395200812155448068131*v^3 + 5636976593543755718081028754596*v^2 + 1401825661972236631277972365740*v + 136842401503388517133286809758) / 120643294798248855786063810651 $$\beta_{8}$$ $$=$$ $$( 47\!\cdots\!25 \nu^{17} + \cdots + 29\!\cdots\!10 ) / 12\!\cdots\!51$$ (4708369673869593399298388725*v^17 - 17164182987196920070734585102*v^16 + 9847849734011595224211342946*v^15 + 12025944304201831968979068261*v^14 + 160067077246513429012050656694*v^13 + 232057455274442155329868273131*v^12 - 823917634880641106998089274391*v^11 - 3024696734804781109543805746167*v^10 + 4447277475580313475751806858292*v^9 + 24809030795358045390030609259893*v^8 + 37760730451739470522652875178577*v^7 + 34180682744244520276659443944434*v^6 + 32591589108106884786549048930397*v^5 + 15887024892114604236697882455699*v^4 + 9231516801203950741097039548813*v^3 + 2639446136111125377872051718177*v^2 + 318677330702219001240573098697*v + 298892682009739571794194805110) / 120643294798248855786063810651 $$\beta_{9}$$ $$=$$ $$( 14\!\cdots\!15 \nu^{17} + \cdots + 13\!\cdots\!40 ) / 36\!\cdots\!53$$ (14157720536121118884053792015*v^17 - 44896678927199690199190010514*v^16 + 8960516219329144702745684094*v^15 + 36871758662282430426489184806*v^14 + 504061500146124543735749877963*v^13 + 936881207714222394173120362473*v^12 - 2011829387324251896125644106908*v^11 - 10064240372137094633009527982847*v^10 + 8414545224642313382116233441615*v^9 + 78339814604725227095079991738407*v^8 + 152144565289546840158050650678248*v^7 + 177631404759534521529603543342870*v^6 + 180980199474610626691359313713272*v^5 + 125219720239435617695674715237832*v^4 + 76466749156375766006366069908869*v^3 + 32837008372822860642451186628655*v^2 + 10249376099670556767257529022452*v + 1388936202821373120587429132340) / 361929884394746567358191431953 $$\beta_{10}$$ $$=$$ $$( 24\!\cdots\!75 \nu^{17} + \cdots + 29\!\cdots\!70 ) / 36\!\cdots\!53$$ (24134694293349454004581350575*v^17 - 78386602493529797622915003228*v^16 + 25086343727955317484603153960*v^15 + 47518852501807108475746686825*v^14 + 861605260065477276330788235882*v^13 + 1543046061815684023931851468491*v^12 - 3418708473022950048432022891090*v^11 - 16692064037065800505751984971101*v^10 + 14965728076743555008150337642471*v^9 + 129867328650235822830051858940539*v^8 + 252768523143786780400681337681589*v^7 + 304099319275526362418945412995493*v^6 + 317615938056742590676112901126572*v^5 + 218215822550642285736880260380340*v^4 + 139909011227748225994330579946706*v^3 + 58649800977992398636342483692897*v^2 + 20211643620277816813286237788500*v + 2976411464728962890751342479970) / 361929884394746567358191431953 $$\beta_{11}$$ $$=$$ $$( 26\!\cdots\!47 \nu^{17} + \cdots + 90\!\cdots\!16 ) / 36\!\cdots\!53$$ (26099366397779599622736751847*v^17 - 84481791508377431496118754067*v^16 + 20431227048910590841762778157*v^15 + 71825690407947829042486662309*v^14 + 923960581126324213455753542118*v^13 + 1660655075482500692302941175845*v^12 - 3876742602959483715937737331129*v^11 - 18404071041302567983840702404270*v^10 + 16969289860509123716517635337288*v^9 + 144503309491594095634931938734327*v^8 + 269927392598757564410741552777385*v^7 + 300265915606461054162352583119062*v^6 + 296519055677159951871871990509557*v^5 + 193488702640354012761879469309968*v^4 + 111932751489917940513860559874797*v^3 + 43052658537355820960239774390296*v^2 + 11644672741905695103419399116596*v + 905773594869594128801841784416) / 361929884394746567358191431953 $$\beta_{12}$$ $$=$$ $$( 31\!\cdots\!71 \nu^{17} + \cdots + 56\!\cdots\!50 ) / 36\!\cdots\!53$$ (31456413298341720160375144571*v^17 - 102554629880830771029735688842*v^16 + 25857715361821391928459669906*v^15 + 90128213406817992257977784730*v^14 + 1108771430587697606659248392793*v^13 + 1974910701055852065950680124835*v^12 - 4760429269233785637413739414214*v^11 - 22129995736440852539517588800775*v^10 + 21151432373020563003787330708629*v^9 + 174412906237928043126516660511224*v^8 + 320321956946740792108459260348429*v^7 + 348550452534475010824086333386610*v^6 + 338972512317037280317401221676311*v^5 + 214249576908828666528936256163964*v^4 + 120014057156825867965486127084514*v^3 + 45280179515483597549774722183266*v^2 + 10197669730123241255886683777694*v + 563494878914400256576560209550) / 361929884394746567358191431953 $$\beta_{13}$$ $$=$$ $$( 18\!\cdots\!44 \nu^{17} + \cdots + 82\!\cdots\!19 ) / 16\!\cdots\!69$$ (189987438568418044*v^17 - 600065581912132306*v^16 + 89982817832438707*v^15 + 576215338636562922*v^14 + 6731043895121464254*v^13 + 12600912052012107630*v^12 - 27619339164715290638*v^11 - 136589037878556006223*v^10 + 114976300556160673087*v^9 + 1067637651139693922490*v^8 + 2035535615077829228139*v^7 + 2288898744474733487658*v^6 + 2263418772682677750223*v^5 + 1522532243175778742855*v^4 + 866390852186202675640*v^3 + 357493410372750206202*v^2 + 93168542955813728376*v + 8227219587007826619) / 1682094761273482869 $$\beta_{14}$$ $$=$$ $$( - 80\!\cdots\!05 \nu^{17} + \cdots - 19\!\cdots\!59 ) / 36\!\cdots\!53$$ (-80630175602154817518359481205*v^17 + 264117948234902985757056530625*v^16 - 73427545097228993276337226650*v^15 - 218076138028049051639858845488*v^14 - 2849075754042316911861432159000*v^13 - 5017846030687465904709934153830*v^12 + 12171478211277403323055932235784*v^11 + 56402896243569861607931076400485*v^10 - 54576341753744721356352025652700*v^9 - 444418014929759976563162613059253*v^8 - 817156588370805399614519825215218*v^7 - 895360592880579434138336386034775*v^6 - 878308700007851697281432277434101*v^5 - 561308015758463249213577972615246*v^4 - 325398253173498984455527720324788*v^3 - 122345133605207142999154480162032*v^2 - 32863260029673004729491231994482*v - 1980070461003814924427007343659) / 361929884394746567358191431953 $$\beta_{15}$$ $$=$$ $$( 97\!\cdots\!98 \nu^{17} + \cdots + 64\!\cdots\!72 ) / 36\!\cdots\!53$$ (97840735245757496952441748198*v^17 - 294838584364082053076958198396*v^16 - 182970566122048167634003701*v^15 + 306092528716769569399603915284*v^14 + 3519367475587308301094228173509*v^13 + 6979110664921566068828115634815*v^12 - 13351076616056099151555081409463*v^11 - 72609249362629318412633570647449*v^10 + 49144802098094464272591283364502*v^9 + 560171429626759246241523693312414*v^8 + 1128649389830737329608374078668663*v^7 + 1318007320667633481354959995501425*v^6 + 1303742089520979617712468607412464*v^5 + 913764294252665316427285108666593*v^4 + 526267581000308760834583628307498*v^3 + 222913619760447244997467117590867*v^2 + 63050469609168302079740373881163*v + 6425272946973976591535625553272) / 361929884394746567358191431953 $$\beta_{16}$$ $$=$$ $$( 12\!\cdots\!10 \nu^{17} + \cdots + 65\!\cdots\!98 ) / 36\!\cdots\!53$$ (121178432938992740065612370510*v^17 - 378227016597730483612935398547*v^16 + 39718996396657289945372076846*v^15 + 380894308509631369046156015151*v^14 + 4304029744822788899245792812699*v^13 + 8187478749883765804969541869569*v^12 - 17440571908579674653224044024403*v^11 - 87987542134716366628403891634327*v^10 + 70620033263800197889765411078722*v^9 + 686135470459053135811244625151257*v^8 + 1321321628951129536458685493429655*v^7 + 1489037204604029789694857919456141*v^6 + 1462823034972996670835002102997819*v^5 + 988468767332924450417299021621032*v^4 + 557047160714328843359937077750859*v^3 + 232398341906193359626124929418850*v^2 + 59808537765923187156305464395603*v + 6521006495666350439000470393398) / 361929884394746567358191431953 $$\beta_{17}$$ $$=$$ $$( - 60\!\cdots\!28 \nu^{17} + \cdots - 24\!\cdots\!27 ) / 12\!\cdots\!51$$ (-60831464903533644776576914128*v^17 + 197403744180344768155502710427*v^16 - 49194581531733157875959643138*v^15 - 168104863256813191644791236626*v^14 - 2147941192052956377218016522642*v^13 - 3857469405821145774070148093334*v^12 + 9067838064355178520284742138651*v^11 + 42792806003222162513695015142045*v^10 - 39934222918559384687818942547154*v^9 - 336308406802845254051004895789524*v^8 - 625863257684411583571716684788523*v^7 - 695836164695687258747899685673585*v^6 - 689440353013060506895522947455952*v^5 - 449499336262842668834143983463585*v^4 - 260024167903081543413774359717745*v^3 - 102636871859771567128804017534321*v^2 - 26695315854869903246030107012287*v - 2421304566653719784443358422527) / 120643294798248855786063810651
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{14} + \beta_{13} - \beta_{12} - 3\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2$$ -b14 + b13 - b12 - 3*b11 - b9 + b7 + b6 - b3 - b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 4 \beta_{9} + \cdots + 1$$ b16 - b15 - b14 + b13 - 3*b12 - 2*b11 + b10 - 4*b9 - b8 + 7*b5 - b4 - 7*b2 + 1 $$\nu^{4}$$ $$=$$ $$- 7 \beta_{17} - 8 \beta_{13} - 21 \beta_{12} - 8 \beta_{11} - 21 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + \cdots - 9$$ -7*b17 - 8*b13 - 21*b12 - 8*b11 - 21*b9 + 4*b7 - 7*b6 + 12*b5 - 8*b4 + 8*b3 - 4*b2 + 7*b1 - 9 $$\nu^{5}$$ $$=$$ $$- 12 \beta_{17} + 4 \beta_{16} + 7 \beta_{15} - 46 \beta_{13} - 48 \beta_{12} - 6 \beta_{11} + \cdots - 48$$ -12*b17 + 4*b16 + 7*b15 - 46*b13 - 48*b12 - 6*b11 - 8*b10 - 27*b9 + 8*b8 + 55*b7 - 4*b6 + 5*b5 - 2*b4 - 4*b2 - b1 - 48 $$\nu^{6}$$ $$=$$ $$- 5 \beta_{17} + 55 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} - 189 \beta_{13} - 81 \beta_{12} + \cdots - 185$$ -5*b17 + 55*b16 + 4*b15 + 5*b14 - 189*b13 - 81*b12 + 67*b11 - b10 - 14*b9 + 119*b7 - 59*b6 + 69*b4 - 60*b3 - 46*b2 - 60*b1 - 185 $$\nu^{7}$$ $$=$$ $$119 \beta_{16} + 59 \beta_{15} + 119 \beta_{14} - 550 \beta_{13} + 119 \beta_{12} + 199 \beta_{11} + \cdots - 411$$ 119*b16 + 59*b15 + 119*b14 - 550*b13 + 119*b12 + 199*b11 + 46*b10 + 93*b9 - 60*b8 + 94*b7 - 469*b6 - 94*b5 + 411*b4 - 23*b3 + 23*b2 - 411 $$\nu^{8}$$ $$=$$ $$94 \beta_{17} + 94 \beta_{16} + 469 \beta_{15} + 469 \beta_{14} - 1234 \beta_{13} + 1328 \beta_{12} + \cdots - 634$$ 94*b17 + 94*b16 + 469*b15 + 469*b14 - 1234*b13 + 1328*b12 + 94*b11 + 71*b10 + 694*b9 - 23*b8 - 1138*b6 - 1138*b5 + 1750*b4 + 304*b3 + 457*b2 + 457*b1 - 634 $$\nu^{9}$$ $$=$$ $$1138 \beta_{17} + 1138 \beta_{15} + 681 \beta_{14} + 5515 \beta_{12} - 727 \beta_{11} + 681 \beta_{10} + \cdots + 423$$ 1138*b17 + 1138*b15 + 681*b14 + 5515*b12 - 727*b11 + 681*b10 + 1865*b9 + 304*b8 - 1248*b7 - 1248*b6 - 4189*b5 + 4834*b4 + 906*b3 + 342*b1 + 423 $$\nu^{10}$$ $$=$$ $$4189 \beta_{17} - 1248 \beta_{16} + 1248 \beta_{15} + 906 \beta_{14} + 11563 \beta_{13} + 16578 \beta_{12} + \cdots + 5942$$ 4189*b17 - 1248*b16 + 1248*b15 + 906*b14 + 11563*b13 + 16578*b12 - 2659*b11 + 4189*b10 + 906*b8 - 10842*b7 - 7246*b5 + 7374*b4 + 7246*b3 - 3596*b2 - 2023*b1 + 5942 $$\nu^{11}$$ $$=$$ $$7246 \beta_{17} - 10842 \beta_{16} + 2023 \beta_{14} + 54370 \beta_{13} + 31546 \beta_{12} + \cdots + 15578$$ 7246*b17 - 10842*b16 + 2023*b14 + 54370*b13 + 31546*b12 - 4748*b11 + 10842*b10 - 17601*b9 + 7246*b8 - 38476*b7 + 14438*b6 - 10177*b5 - 10842*b4 + 38476*b3 - 4261*b2 - 10177*b1 + 15578 $$\nu^{12}$$ $$=$$ $$10177 \beta_{17} - 38476 \beta_{16} - 14438 \beta_{15} - 4261 \beta_{14} + 174807 \beta_{13} + \cdots + 4261$$ 10177*b17 - 38476*b16 - 14438*b15 - 4261*b14 + 174807*b13 + 13855*b12 + 28293*b11 + 14438*b10 - 67935*b9 + 38476*b8 - 74249*b7 + 103688*b6 - 13456*b5 - 124527*b4 + 103688*b3 + 13456*b2 - 74249*b1 + 4261 $$\nu^{13}$$ $$=$$ $$13456 \beta_{17} - 74249 \beta_{16} - 103688 \beta_{15} - 29439 \beta_{14} + 407806 \beta_{13} + \cdots - 138370$$ 13456*b17 - 74249*b16 - 103688*b15 - 29439*b14 + 407806*b13 - 154758*b12 + 304118*b11 - 125319*b9 + 103688*b8 - 107318*b7 + 359866*b6 + 48563*b5 - 532720*b4 + 155881*b3 + 107318*b2 - 359866*b1 - 138370 $$\nu^{14}$$ $$=$$ $$- 48563 \beta_{17} - 107318 \beta_{16} - 359866 \beta_{15} + 490832 \beta_{13} - 673789 \beta_{12} + \cdots - 673789$$ -48563*b17 - 107318*b16 - 359866*b15 + 490832*b13 - 673789*b12 + 1399960*b11 - 155881*b10 + 179529*b9 + 155881*b8 - 88369*b7 + 745254*b6 + 251020*b5 - 1555841*b4 + 745254*b2 - 996274*b1 - 673789 $$\nu^{15}$$ $$=$$ $$- 251020 \beta_{17} - 88369 \beta_{16} - 745254 \beta_{15} + 251020 \beta_{14} - 789064 \beta_{13} + \cdots - 1534318$$ -251020*b17 - 88369*b16 - 745254*b15 + 251020*b14 - 789064*b13 - 1363369*b12 + 4210017*b11 - 996274*b10 + 2846648*b9 + 526665*b7 + 1092384*b6 - 2935017*b4 - 1619049*b3 + 3407251*b2 - 1619049*b1 - 1534318 $$\nu^{16}$$ $$=$$ $$526665 \beta_{16} - 1092384 \beta_{15} + 526665 \beta_{14} - 5580452 \beta_{13} + 526665 \beta_{12} + \cdots + 1219807$$ 526665*b16 - 1092384*b15 + 526665*b14 - 5580452*b13 + 526665*b12 + 8419308*b11 - 3407251*b10 + 14020626*b9 - 1619049*b8 + 2221838*b7 + 560842*b6 - 2221838*b5 - 1219807*b4 - 9609816*b3 + 9609816*b2 + 1219807 $$\nu^{17}$$ $$=$$ $$2221838 \beta_{17} + 2221838 \beta_{16} - 560842 \beta_{15} - 560842 \beta_{14} - 12855684 \beta_{13} + \cdots + 28353865$$ 2221838*b17 + 2221838*b16 - 560842*b15 - 560842*b14 - 12855684*b13 + 15077522*b12 + 2221838*b11 - 7387978*b10 + 43431387*b9 - 9609816*b8 - 5546275*b6 - 5546275*b5 + 15212498*b4 - 32531806*b3 + 16431857*b2 + 16431857*b1 + 28353865

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$\beta_{12}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 −1.37827 − 1.15651i −0.344606 − 0.289159i 2.39653 + 2.01093i −1.37827 + 1.15651i −0.344606 + 0.289159i 2.39653 − 2.01093i 0.128174 − 0.726910i −0.124000 + 0.703239i −0.443867 + 2.51729i 2.91971 − 1.06269i −0.168248 + 0.0612373i −1.48541 + 0.540646i 0.128174 + 0.726910i −0.124000 − 0.703239i −0.443867 − 2.51729i 2.91971 + 1.06269i −0.168248 − 0.0612373i −1.48541 − 0.540646i
0 −0.312429 1.77187i 0 −0.766044 + 0.642788i 0 −0.336777 + 0.583316i 0 −0.222848 + 0.0811099i 0
61.2 0 −0.0781159 0.443017i 0 −0.766044 + 0.642788i 0 −1.06981 + 1.85297i 0 2.62892 0.956847i 0
61.3 0 0.543249 + 3.08092i 0 −0.766044 + 0.642788i 0 0.0481505 0.0833990i 0 −6.37785 + 2.32135i 0
81.1 0 −0.312429 + 1.77187i 0 −0.766044 0.642788i 0 −0.336777 0.583316i 0 −0.222848 0.0811099i 0
81.2 0 −0.0781159 + 0.443017i 0 −0.766044 0.642788i 0 −1.06981 1.85297i 0 2.62892 + 0.956847i 0
81.3 0 0.543249 3.08092i 0 −0.766044 0.642788i 0 0.0481505 + 0.0833990i 0 −6.37785 2.32135i 0
101.1 0 −0.693610 0.252453i 0 −0.173648 0.984808i 0 −2.32856 + 4.03318i 0 −1.88077 1.57815i 0
101.2 0 0.671023 + 0.244232i 0 −0.173648 0.984808i 0 1.15961 2.00851i 0 −1.90751 1.60059i 0
101.3 0 2.40197 + 0.874246i 0 −0.173648 0.984808i 0 −0.118043 + 0.204456i 0 2.70703 + 2.27147i 0
161.1 0 −2.38017 + 1.99720i 0 0.939693 + 0.342020i 0 2.42255 4.19598i 0 1.15545 6.55290i 0
161.2 0 0.137157 0.115088i 0 0.939693 + 0.342020i 0 −1.55670 + 2.69628i 0 −0.515378 + 2.92285i 0
161.3 0 1.21092 1.01608i 0 0.939693 + 0.342020i 0 1.77958 3.08232i 0 −0.0870410 + 0.493634i 0
301.1 0 −0.693610 + 0.252453i 0 −0.173648 + 0.984808i 0 −2.32856 4.03318i 0 −1.88077 + 1.57815i 0
301.2 0 0.671023 0.244232i 0 −0.173648 + 0.984808i 0 1.15961 + 2.00851i 0 −1.90751 + 1.60059i 0
301.3 0 2.40197 0.874246i 0 −0.173648 + 0.984808i 0 −0.118043 0.204456i 0 2.70703 2.27147i 0
321.1 0 −2.38017 1.99720i 0 0.939693 0.342020i 0 2.42255 + 4.19598i 0 1.15545 + 6.55290i 0
321.2 0 0.137157 + 0.115088i 0 0.939693 0.342020i 0 −1.55670 2.69628i 0 −0.515378 2.92285i 0
321.3 0 1.21092 + 1.01608i 0 0.939693 0.342020i 0 1.77958 + 3.08232i 0 −0.0870410 0.493634i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.u.a 18
19.e even 9 1 inner 380.2.u.a 18
19.e even 9 1 7220.2.a.v 9
19.f odd 18 1 7220.2.a.x 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.a 18 1.a even 1 1 trivial
380.2.u.a 18 19.e even 9 1 inner
7220.2.a.v 9 19.e even 9 1
7220.2.a.x 9 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} - 3 T_{3}^{17} + 9 T_{3}^{16} - 24 T_{3}^{15} + 54 T_{3}^{14} - 297 T_{3}^{13} + 1135 T_{3}^{12} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$T^{18} - 3 T^{17} + \cdots + 9$$
$5$ $$(T^{6} - T^{3} + 1)^{3}$$
$7$ $$T^{18} + 39 T^{16} + \cdots + 361$$
$11$ $$T^{18} + 60 T^{16} + \cdots + 531441$$
$13$ $$T^{18} + \cdots + 199402641$$
$17$ $$T^{18} - 12 T^{17} + \cdots + 6561$$
$19$ $$T^{18} + \cdots + 322687697779$$
$23$ $$T^{18} + 9 T^{17} + \cdots + 998001$$
$29$ $$T^{18} + \cdots + 387420489$$
$31$ $$T^{18} + 18 T^{17} + \cdots + 13082689$$
$37$ $$(T^{9} - 30 T^{8} + \cdots - 1864711)^{2}$$
$41$ $$T^{18} + \cdots + 15485980059729$$
$43$ $$T^{18} + \cdots + 213364449$$
$47$ $$T^{18} + \cdots + 130045445640009$$
$53$ $$T^{18} + \cdots + 1556065809$$
$59$ $$T^{18} + \cdots + 191850201$$
$61$ $$T^{18} + \cdots + 1917895384161$$
$67$ $$T^{18} + \cdots + 8000542361529$$
$71$ $$T^{18} + \cdots + 77975557962129$$
$73$ $$T^{18} + \cdots + 18281242781649$$
$79$ $$T^{18} + \cdots + 400527093527929$$
$83$ $$T^{18} + \cdots + 2004181007481$$
$89$ $$T^{18} + \cdots + 51059270662929$$
$97$ $$T^{18} + \cdots + 3020598812169$$