# Properties

 Label 380.2.u.b Level $380$ Weight $2$ Character orbit 380.u Analytic conductor $3.034$ Analytic rank $0$ Dimension $18$ 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: $$x^{18} - 3 x^{17} + 21 x^{16} - 30 x^{15} + 192 x^{14} - 207 x^{13} + 1178 x^{12} - 705 x^{11} + \cdots + 5329$$ x^18 - 3*x^17 + 21*x^16 - 30*x^15 + 192*x^14 - 207*x^13 + 1178*x^12 - 705*x^11 + 4413*x^10 - 2224*x^9 + 11430*x^8 - 4101*x^7 + 19237*x^6 - 7125*x^5 + 21573*x^4 - 5266*x^3 + 13851*x^2 - 3285*x + 5329 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.

 $$f(q)$$ $$=$$ $$q + \beta_{13} q^{3} + ( - \beta_{11} + \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{16} - \beta_{11} + \beta_{8} + \cdots + 1) q^{9}+O(q^{10})$$ q + b13 * q^3 + (-b11 + b2) * q^5 + (-b17 + b9 - b8 - b4 - b1) * q^7 + (-b16 - b11 + b8 - b6 + b2 + 1) * q^9 $$q + \beta_{13} q^{3} + ( - \beta_{11} + \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{17} + \beta_{16} - \beta_{15} + \cdots - 2) q^{99}+O(q^{100})$$ q + b13 * q^3 + (-b11 + b2) * q^5 + (-b17 + b9 - b8 - b4 - b1) * q^7 + (-b16 - b11 + b8 - b6 + b2 + 1) * q^9 + (-b17 + b16 - b15 + b13 + b11 - b8 - b7 + b6 - b4 - b2 - 1) * q^11 + (-b17 - b16 - b13 + 2*b12 - b8 - 2*b6 - b2 - b1 + 1) * q^13 - b7 * q^15 + (-b16 - b15 - b13 + b12 + b10 + b9 - b7 + b5 + b3 - 2*b2 - b1 + 1) * q^17 + (-b17 - b15 - b12 + b10 - b8 + b6 - b5 - b4 - 2*b3 + b2 + b1 - 2) * q^19 + (-2*b17 + b13 + b12 - b10 - b7 - 2*b6 - 2*b4 + b3 + b2 - b1 - 1) * q^21 + (b16 + b15 + 2*b13 + 2*b11 - b10 - b9 + b8 - b5 - b4 + 2*b3 - b2 - b1 + 1) * q^23 + (-b12 + b6) * q^25 + (2*b17 - b16 + 2*b15 - b14 + b13 - b12 - 3*b11 - b10 + 2*b8 - 2*b6 + 3*b4 + 2*b3 + 2*b2 + 3) * q^27 + (-b17 + b16 - 3*b15 + b14 + 3*b11 + 2*b10 - b8 - b7 + 4*b6 - b5 - 2*b4 - 2*b3 - b2 + 2*b1 - 3) * q^29 + (2*b17 + b15 - b14 + 2*b12 + b11 + b10 - 2*b9 + b8 - b6 - 2*b2 + b1) * q^31 + (-b17 + b16 - 2*b15 + 2*b12 + 2*b11 + b9 - b8 - b6 + b5 - b4 - 2*b2 - 2) * q^33 + (-b16 - b13 - b11 + b9) * q^35 + (-b17 - b16 + b15 - b14 - b13 + 2*b12 - 2*b11 - b10 - b9 - b8 - b7 - 2*b6 + b5 + 2*b3 - 2*b1 - 1) * q^37 + (2*b16 + b14 + b13 - b12 + b11 - b10 - b7 + 3*b6 - 2*b5 - 2*b3 + 2*b2 + 2*b1 - 2) * q^39 + (-2*b16 - b13 + b12 - 3*b11 + 2*b9 + 2*b7 - 2*b6 + b4 - 2*b1 + 2) * q^41 + (-b17 + b16 - 3*b15 - b14 + 2*b12 + 3*b11 + 3*b10 + b9 - b8 + 2*b5 - 2*b4 - b3 - 3*b2 + b1 - 2) * q^43 + (b17 + b15 - b12 - b11 - b9 + b6 + b4 + b2) * q^45 + (2*b16 + b15 + 3*b14 + b13 + b11 - b10 - 2*b8 - 3*b7 + 3*b6 - 2*b4 + b2 - b1 - 1) * q^47 + (-b16 + 2*b15 - b14 - b13 - 2*b12 - 3*b11 - b10 - 2*b9 + 2*b8 + 2*b7 - b6 + 2*b4 + b2 + 2) * q^49 + (b17 + b16 + 2*b14 - 3*b12 + 4*b11 + b9 + 3*b6 - b5 + 4*b4 - b2 + 1) * q^51 + (b17 + b16 + 2*b15 - 5*b12 + 2*b11 - b10 - 2*b9 + 2*b8 + 3*b7 - 2*b5 + b4 + b2 + b1 + 3) * q^53 + (-b17 + b12 + b9 - b7 - b6 + b5 - b4 + b3 - b2 - b1) * q^55 + (b17 + b16 + b15 - b14 - 3*b12 - 5*b11 - 2*b10 - b7 - b5 + b4 + 3*b2 - b1 - 1) * q^57 + (b17 - b16 - b15 + b14 - 2*b13 + 2*b10 + b9 + 2*b7 - b6 + b5 - b4 - 2*b3 + 3*b1 - 1) * q^59 + (-2*b16 - 2*b13 - b12 - b11 + b10 - 2*b8 + 4*b7 + 2*b5 + 2*b4 - 2*b3 + 2*b2 + b1 + 1) * q^61 + (b13 + b12 - 2*b10 + b9 - b8 - b6 - b5 - 2*b3 - b2 + b1 + 1) * q^63 + (b17 - b13 - b11 + b9 + b7 - b6 + 2*b4 + b2 + 2) * q^65 + (3*b17 - 2*b16 + 2*b15 - b14 - 2*b13 - 6*b11 + 2*b10 + 2*b8 + b7 + 3*b5 + 4*b4 + 2*b2 + 2*b1 + 6) * q^67 + (b17 - b15 - 2*b14 + 3*b13 + 4*b11 - b10 - b9 + 2*b8 + 3*b7 - 4*b6 - 2*b4 + b1) * q^69 + (2*b17 + b15 - b14 + 2*b12 - 2*b11 - 2*b10 + 2*b8 + 3*b7 - b6 - b5 - b4 + 2*b3 + 2*b2 + b1 - 2) * q^71 + (-b16 + 2*b15 + 2*b14 - 2*b13 + b11 - b9 + b8 + 3*b6 + 2*b5 - 2*b3 - 3) * q^73 + (b3 - b1) * q^75 + (-b17 - 2*b16 + b15 - b14 - b13 - b10 + b9 + b8 - b7 + b6 + 2*b5 + 3*b3 + b2 - 3*b1) * q^77 + (3*b16 + b14 + 2*b13 + 2*b12 + 2*b11 - 3*b9 + b7 + 2*b4 - b3 - b1) * q^79 + (2*b17 + b14 - b12 + 3*b11 + 2*b10 + 2*b8 - b7 - b6 + 2*b4 - 3*b2 - b1 + 1) * q^81 + (2*b17 + b15 - b14 + b13 - 2*b12 - 5*b11 - 2*b9 + b8 + b7 + 5*b6 - 2*b5 + 2*b4 + 2*b2 + 2*b1) * q^83 + (b17 - b16 - b14 - b11 - b10 + b8 + b7 - 2*b6 + b5 + b3 + b2 - b1 + 1) * q^85 + (2*b17 + 3*b16 - b15 - 2*b14 - 4*b12 - b11 - 2*b10 + 3*b9 - b8 + 2*b7 + 3*b6 + 2*b4 + b3 - 3*b2 + 2) * q^87 + (2*b17 + 2*b16 + b14 + 4*b11 - 4*b9 + b8 - b5 + 4*b4 - b2 + 1) * q^89 + (-2*b17 + 2*b16 + b15 - 2*b13 - b12 + b10 - b9 - b7 - 2*b4 - 2*b3 - 2*b2 + b1 - 2) * q^91 + (4*b17 + 2*b14 - 8*b12 - 3*b9 + 2*b7 + 5*b6 - 3*b5 + 5*b4 - 2*b3 + 4*b2 + 4*b1 - 3) * q^93 + (-b14 - b13 + b11 + 2*b10 + b9 - b8 + b6 + b5 - b4 - 2*b2 - 1) * q^95 + (2*b17 + b14 - 4*b13 - 2*b12 + 4*b10 + b9 - b7 - b6 + b5 - b4 + b3 + 3*b2 - 3) * q^97 + (-b17 + b16 - b15 + b13 + 4*b12 + 2*b11 - 2*b10 + b9 + b7 - 4*b4 + b3 - 4*b2 + b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 3 q^{3} + 15 q^{9}+O(q^{10})$$ 18 * q + 3 * q^3 + 15 * q^9 $$18 q + 3 q^{3} + 15 q^{9} + 9 q^{13} + 3 q^{15} + 12 q^{17} - 18 q^{19} - 9 q^{21} + 21 q^{23} + 18 q^{27} - 9 q^{29} + 6 q^{31} - 21 q^{33} - 6 q^{35} - 36 q^{37} - 12 q^{39} + 6 q^{41} - 12 q^{43} - 6 q^{45} + 21 q^{47} - 3 q^{49} - 9 q^{51} + 36 q^{53} - 3 q^{55} - 24 q^{57} - 6 q^{61} + 36 q^{63} + 15 q^{65} + 60 q^{67} + 27 q^{69} - 36 q^{71} - 60 q^{73} - 6 q^{75} - 36 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{85} + 21 q^{87} + 6 q^{89} - 30 q^{91} - 48 q^{93} - 21 q^{95} - 57 q^{97} + 3 q^{99}+O(q^{100})$$ 18 * q + 3 * q^3 + 15 * q^9 + 9 * q^13 + 3 * q^15 + 12 * q^17 - 18 * q^19 - 9 * q^21 + 21 * q^23 + 18 * q^27 - 9 * q^29 + 6 * q^31 - 21 * q^33 - 6 * q^35 - 36 * q^37 - 12 * q^39 + 6 * q^41 - 12 * q^43 - 6 * q^45 + 21 * q^47 - 3 * q^49 - 9 * q^51 + 36 * q^53 - 3 * q^55 - 24 * q^57 - 6 * q^61 + 36 * q^63 + 15 * q^65 + 60 * q^67 + 27 * q^69 - 36 * q^71 - 60 * q^73 - 6 * q^75 - 36 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^83 + 12 * q^85 + 21 * q^87 + 6 * q^89 - 30 * q^91 - 48 * q^93 - 21 * q^95 - 57 * q^97 + 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 3 x^{17} + 21 x^{16} - 30 x^{15} + 192 x^{14} - 207 x^{13} + 1178 x^{12} - 705 x^{11} + \cdots + 5329$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 72\!\cdots\!11 \nu^{17} + \cdots + 83\!\cdots\!10 ) / 32\!\cdots\!31$$ (72588030392443584911*v^17 + 938502298439910332299*v^16 - 1082950131172633771102*v^15 + 20706548927390622659350*v^14 - 5462451793914015963121*v^13 + 199327286464235880130583*v^12 + 4799544629438998301443*v^11 + 1251727781167403855252088*v^10 + 512712857132822002380135*v^9 + 4902051035191064292959244*v^8 + 2529576063289792535006307*v^7 + 11996669827340915478927898*v^6 + 6947441225516565162602538*v^5 + 19073878896864219357530980*v^4 + 9578655669900112712881923*v^3 + 14949615736949600842337642*v^2 + 5598357377266876545624887*v + 8370431853812903182862410) / 3264877556804777523627231 $$\beta_{3}$$ $$=$$ $$( 17\!\cdots\!95 \nu^{17} + \cdots - 97\!\cdots\!37 ) / 14\!\cdots\!49$$ (1762594869473805695*v^17 - 6696296929870308215*v^16 + 40285356401548879955*v^15 - 77743397152460978677*v^14 + 356489391421994590502*v^13 - 575066345264455808427*v^12 + 2165706466003388024758*v^11 - 2441610904589764855613*v^10 + 7566467576851410325763*v^9 - 7985060560687636406792*v^8 + 19608849791167040068362*v^7 - 16298186811256142049697*v^6 + 30354611698953191690155*v^5 - 24948897977118508977655*v^4 + 39733165983985501193325*v^3 - 18524274308803076317816*v^2 + 26368786322069071762341*v - 9736596397829991568437) / 14908116697738710153549 $$\beta_{4}$$ $$=$$ $$( - 13\!\cdots\!69 \nu^{17} + \cdots - 14\!\cdots\!28 ) / 10\!\cdots\!77$$ (-133378032846986185869*v^17 + 271464673069370741872*v^16 - 2312109013906177403554*v^15 + 1060509968096517339355*v^14 - 19933314314491696243427*v^13 + 1585527225520535368237*v^12 - 115139479489444452938511*v^11 - 64065058861122064769689*v^10 - 410359662918697203780148*v^9 - 255719388058455676408043*v^8 - 941601494510854646786854*v^7 - 884462722049703576741657*v^6 - 1376025580655774887934072*v^5 - 1265568169988806419064690*v^4 - 1056094750278381832383122*v^3 - 2198152395858712332326571*v^2 - 495147108420981089270951*v - 1486774563608692618071228) / 1088292518934925841209077 $$\beta_{5}$$ $$=$$ $$( - 44\!\cdots\!86 \nu^{17} + \cdots + 33\!\cdots\!19 ) / 32\!\cdots\!31$$ (-442212880196994897886*v^17 + 4220750067020519154780*v^16 - 14437983897868950123567*v^15 + 66145959855158017756544*v^14 - 107894544664080142228551*v^13 + 598580415938491326282842*v^12 - 553488823223162106831878*v^11 + 3460216855488662633343342*v^10 - 695363948229899544772580*v^9 + 13941886809484379352479194*v^8 + 294677207456648036155671*v^7 + 34937119859460274189243232*v^6 + 5447530220821467549484696*v^5 + 61427698601007504842856678*v^4 + 5901952825752027816812748*v^3 + 56015456368122710036969567*v^2 - 64039772940553280496555*v + 33291202955060128186575019) / 3264877556804777523627231 $$\beta_{6}$$ $$=$$ $$( - 46\!\cdots\!86 \nu^{17} + \cdots - 55\!\cdots\!65 ) / 32\!\cdots\!31$$ (-468946316097082340686*v^17 - 635646176949590442294*v^16 - 5504441685633559934810*v^15 - 21460967704782793169764*v^14 - 69029813922024177357640*v^13 - 211247290644224446788494*v^12 - 458199462385021029754642*v^11 - 1463404013007698785435279*v^10 - 2474059130662381100174370*v^9 - 5389378686003254044165898*v^8 - 6132287675328064754947078*v^7 - 12886312833113341841748094*v^6 - 11915026050746750062916263*v^5 - 18712725857698781995916975*v^4 - 8098723638552391009500750*v^3 - 14335010326725246950184969*v^2 - 7191008624601249328004758*v - 5566245871183997282606965) / 3264877556804777523627231 $$\beta_{7}$$ $$=$$ $$( 75\!\cdots\!09 \nu^{17} + \cdots - 14\!\cdots\!54 ) / 44\!\cdots\!47$$ (7554887320900910209*v^17 - 30360069462158512762*v^16 + 158482999719350396117*v^15 - 355174428132210955470*v^14 + 1349461034229607490736*v^13 - 3181110026967528827256*v^12 + 7790092331848739198413*v^11 - 16203736789672018658813*v^10 + 23662922765747523882271*v^9 - 70506307704077449177073*v^8 + 49410951240254333342651*v^7 - 162898210717527811205449*v^6 + 54168259871253657915924*v^5 - 293687102107459508255833*v^4 + 21773109633302028686600*v^3 - 206500140195702739705254*v^2 - 2778575893352198940811*v - 147383621555844699314254) / 44724350093216130460647 $$\beta_{8}$$ $$=$$ $$( 30\!\cdots\!92 \nu^{17} + \cdots - 39\!\cdots\!07 ) / 17\!\cdots\!49$$ (30420868161822609292*v^17 - 220581128263053631354*v^16 + 815880790156100794207*v^15 - 2836479865744693880466*v^14 + 5030500149080102077476*v^13 - 22214967103148448587065*v^12 + 22364214584156972713412*v^11 - 108945742182957440511799*v^10 - 8363356283010984571806*v^9 - 371869004795968406012846*v^8 - 117495906477202012861816*v^7 - 729370067773788505970229*v^6 - 584532532677773225720225*v^5 - 985083650993286325864395*v^4 - 639737263082241296154020*v^3 - 412553789222239296676538*v^2 - 829670656455500889919494*v - 399420308295416703390007) / 171835660884461974927749 $$\beta_{9}$$ $$=$$ $$( - 66\!\cdots\!53 \nu^{17} + \cdots - 60\!\cdots\!01 ) / 32\!\cdots\!31$$ (-665890582381961174853*v^17 + 3777578311259166566891*v^16 - 16643571145826057881732*v^15 + 49181111021974779761862*v^14 - 124035952153400344318727*v^13 + 390592404341681319881812*v^12 - 615743041621814239908305*v^11 + 1904690994123788670714558*v^10 - 912196648108191924426393*v^9 + 6695919935466374389619292*v^8 + 615993581116716236329671*v^7 + 13746369533249610704352795*v^6 + 8852032983181270293141370*v^5 + 18703535604640744638252800*v^4 + 15659036519938035730927727*v^3 + 8024963837534662320290671*v^2 + 15035368648422414355509793*v - 609300281963012419687901) / 3264877556804777523627231 $$\beta_{10}$$ $$=$$ $$( 15\!\cdots\!84 \nu^{17} + \cdots - 52\!\cdots\!03 ) / 44\!\cdots\!47$$ (15839265611195083384*v^17 - 35716421498821219921*v^16 + 313482052591286715160*v^15 - 265744570263879236521*v^14 + 2936342585691393180920*v^13 - 1105604865381637598955*v^12 + 17847977295809268254991*v^11 + 2635368205629704961204*v^10 + 69362833079230942819196*v^9 + 23286231176769347390049*v^8 + 168415799177812693433533*v^7 + 76041990203522300269447*v^6 + 268370802923429861644135*v^5 + 109763206715670236378040*v^4 + 210025538424605599458616*v^3 + 62916994086316992479762*v^2 + 117929911419891511771165*v - 5298926218648381698503) / 44724350093216130460647 $$\beta_{11}$$ $$=$$ $$( - 14\!\cdots\!82 \nu^{17} + \cdots + 65\!\cdots\!91 ) / 32\!\cdots\!31$$ (-1400146641318054942582*v^17 + 3492136588151732671137*v^16 - 26568220880531348437261*v^15 + 27629306222725454190635*v^14 - 240217963843168571173978*v^13 + 184145421383724062098719*v^12 - 1426716842895226895827096*v^11 + 432338968929381371517185*v^10 - 5171420537081301358345328*v^9 + 1909159667197969976140115*v^8 - 11751276989958837934651441*v^7 + 3846858547549398629761418*v^6 - 16326714626870072998276409*v^5 + 10560618139618382884391422*v^4 - 10449405688650489957499441*v^3 + 7788443355198799388803202*v^2 - 2245039560559940106174628*v + 6517636338449475747513191) / 3264877556804777523627231 $$\beta_{12}$$ $$=$$ $$( 15\!\cdots\!12 \nu^{17} + \cdots - 12\!\cdots\!98 ) / 32\!\cdots\!31$$ (1550007403845995732112*v^17 - 6141000562353058215431*v^16 + 34677301362433508162322*v^15 - 70460320323562556437163*v^14 + 292681567053395412870266*v^13 - 530660055173680261033952*v^12 + 1687906987739295335611714*v^11 - 2318089645342610865273720*v^10 + 5252710849800365072989855*v^9 - 8152138397257090434662867*v^8 + 11797392881223663827207733*v^7 - 17559042598061338684279169*v^6 + 15032017277418712605501086*v^5 - 29143488141691696236740273*v^4 + 14017006507935088152294695*v^3 - 23377383612714447987417437*v^2 + 5699009116044025059836798*v - 12401344881411719274427598) / 3264877556804777523627231 $$\beta_{13}$$ $$=$$ $$( 25\!\cdots\!17 \nu^{17} + \cdots - 10\!\cdots\!89 ) / 44\!\cdots\!47$$ (25542051033146208817*v^17 - 74550100774818553578*v^16 + 510401135013426360185*v^15 - 657664998892618658463*v^14 + 4384081398966917989355*v^13 - 4155685695209607825455*v^12 + 25447489805396218433116*v^11 - 11164996055401460837002*v^10 + 85866483258592370437857*v^9 - 34965811567155721912154*v^8 + 194376659842140702882401*v^7 - 69271794894633178253478*v^6 + 260362949221686828514071*v^5 - 160866352250136740592125*v^4 + 204336892643538173963186*v^3 - 171992479452538334965836*v^2 + 91653820711402946550188*v - 107509631034866392506989) / 44724350093216130460647 $$\beta_{14}$$ $$=$$ $$( 27\!\cdots\!24 \nu^{17} + \cdots - 34\!\cdots\!78 ) / 44\!\cdots\!47$$ (27979248290970376224*v^17 - 59499054142536564652*v^16 + 486703523119113197128*v^15 - 287779161213912767864*v^14 + 4223550384607130017952*v^13 - 1290675314758109144842*v^12 + 24575495422686874460533*v^11 + 5542452571588448366124*v^10 + 88113908123331029723994*v^9 + 10578510716964569875426*v^8 + 202869338019554472889060*v^7 + 39642544766947631166297*v^6 + 302109155614938269497325*v^5 - 27641852583698168850936*v^4 + 230198378456061404879965*v^3 + 9529221922473449687164*v^2 + 97352527665245380435075*v - 34233081075087010870078) / 44724350093216130460647 $$\beta_{15}$$ $$=$$ $$( - 25\!\cdots\!40 \nu^{17} + \cdots + 11\!\cdots\!27 ) / 32\!\cdots\!31$$ (-2550941223326384809840*v^17 + 8826878006643638457053*v^16 - 56169435087370468063288*v^15 + 100188300582856770505647*v^14 - 507855620355252029781105*v^13 + 750566400723259883346436*v^12 - 3051343071834764577987106*v^11 + 3173532795701403211840478*v^10 - 10713521113977364074515900*v^9 + 11091070271126731611639173*v^8 - 25164006060599141579895085*v^7 + 24306382813693891534523805*v^6 - 35003773949798138602865505*v^5 + 39769326410921339818640502*v^4 - 27867509745943421112425649*v^3 + 32189520617944740676756419*v^2 - 5012797064760956379244395*v + 11502835051092886711310927) / 3264877556804777523627231 $$\beta_{16}$$ $$=$$ $$( 33\!\cdots\!30 \nu^{17} + \cdots + 16\!\cdots\!24 ) / 32\!\cdots\!31$$ (3364751683713185370930*v^17 - 4214474302339995999244*v^16 + 53622242483980463103732*v^15 + 17649350883661346753301*v^14 + 488395140907377253891972*v^13 + 357088412648994171003008*v^12 + 2895173804487840476514771*v^11 + 3896629729151288176238480*v^10 + 11501056751411689838883868*v^9 + 14788172029734766643964660*v^8 + 26947741431034589219072736*v^7 + 39181978475515950523178048*v^6 + 42318469248434988943640040*v^5 + 54339356177712512835976492*v^4 + 28097672354037368183697804*v^3 + 48820935320643241061486184*v^2 + 11896384343989276778076864*v + 16854691098376367338219624) / 3264877556804777523627231 $$\beta_{17}$$ $$=$$ $$( 37\!\cdots\!50 \nu^{17} + \cdots - 50\!\cdots\!89 ) / 32\!\cdots\!31$$ (3723341884943530253250*v^17 - 10542276620679393908085*v^16 + 76522088723180841541064*v^15 - 101852316851702897095581*v^14 + 703939492506857286926473*v^13 - 694022986259641983546278*v^12 + 4269286893453183971479028*v^11 - 2220533664304976558636564*v^10 + 15878031151098002941386517*v^9 - 7199905253360984239021335*v^8 + 38741762109061061478268129*v^7 - 13546982911907407054707321*v^6 + 61327501689093768153983321*v^5 - 24448789004256535950789568*v^4 + 56263255567131185922744747*v^3 - 16826987686738072174224811*v^2 + 30172274393951331312628363*v - 5015773200357011551469389) / 3264877556804777523627231
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{17} - \beta_{15} + \beta_{12} + \beta_{11} - \beta_{8} - 4\beta_{4} - 4$$ -b17 - b15 + b12 + b11 - b8 - 4*b4 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{16} - 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + \cdots - 3$$ b16 - 2*b12 + 2*b11 + b10 - 2*b9 - 2*b8 + b7 + 3*b6 - b5 + 4*b3 + b2 - 4*b1 - 3 $$\nu^{4}$$ $$=$$ $$9 \beta_{17} + 9 \beta_{15} - \beta_{14} - 11 \beta_{12} - 10 \beta_{11} + \beta_{10} - 9 \beta_{9} + \cdots - 2 \beta_1$$ 9*b17 + 9*b15 - b14 - 11*b12 - 10*b11 + b10 - 9*b9 + 10*b6 - 2*b5 + 24*b4 + 11*b2 - 2*b1 $$\nu^{5}$$ $$=$$ $$21 \beta_{17} - 12 \beta_{16} + 23 \beta_{15} - 8 \beta_{14} + 10 \beta_{13} - 16 \beta_{12} - 37 \beta_{11} + \cdots + 39$$ 21*b17 - 12*b16 + 23*b15 - 8*b14 + 10*b13 - 16*b12 - 37*b11 - 8*b10 - 2*b9 + 23*b8 - 2*b7 - 21*b6 + 39*b4 - 22*b3 + 21*b2 + 39 $$\nu^{6}$$ $$=$$ $$- 4 \beta_{17} - 31 \beta_{16} + 4 \beta_{15} + 14 \beta_{14} + 14 \beta_{13} + 28 \beta_{12} + \cdots + 181$$ -4*b17 - 31*b16 + 4*b15 + 14*b14 + 14*b13 + 28*b12 - 28*b11 - 14*b10 + 76*b9 + 76*b8 - 14*b7 - 113*b6 + 31*b5 - 27*b3 - 85*b2 + 27*b1 + 181 $$\nu^{7}$$ $$=$$ $$- 228 \beta_{17} - 197 \beta_{15} + 81 \beta_{14} - 48 \beta_{13} + 381 \beta_{12} + 176 \beta_{11} + \cdots + 150 \beta_1$$ -228*b17 - 197*b15 + 81*b14 - 48*b13 + 381*b12 + 176*b11 - 33*b10 + 228*b9 - 31*b8 - 48*b7 - 176*b6 + 129*b5 - 407*b4 - 381*b2 + 150*b1 $$\nu^{8}$$ $$=$$ $$- 656 \beta_{17} + 373 \beta_{16} - 737 \beta_{15} - 8 \beta_{14} - 145 \beta_{13} + 719 \beta_{12} + \cdots - 1527$$ -656*b17 + 373*b16 - 737*b15 - 8*b14 - 145*b13 + 719*b12 + 1145*b11 - 8*b10 + 81*b9 - 737*b8 + 153*b7 + 426*b6 - 1527*b4 + 278*b3 - 426*b2 - 1527 $$\nu^{9}$$ $$=$$ $$381 \beta_{17} + 1352 \beta_{16} - 381 \beta_{15} - 408 \beta_{14} - 408 \beta_{13} - 2032 \beta_{12} + \cdots - 3994$$ 381*b17 + 1352*b16 - 381*b15 - 408*b14 - 408*b13 - 2032*b12 + 2032*b11 + 638*b10 - 1816*b9 - 1816*b8 + 638*b7 + 3774*b6 - 1352*b5 + 1154*b3 + 1742*b2 - 1154*b1 - 3994 $$\nu^{10}$$ $$=$$ $$6927 \beta_{17} + 5805 \beta_{15} - 1361 \beta_{14} - 216 \beta_{13} - 11470 \beta_{12} + \cdots - 2642 \beta_1$$ 6927*b17 + 5805*b15 - 1361*b14 - 216*b13 - 11470*b12 - 6236*b11 + 1577*b10 - 6927*b9 + 1122*b8 - 216*b7 + 6236*b6 - 4108*b5 + 13605*b4 + 11470*b2 - 2642*b1 $$\nu^{11}$$ $$=$$ $$16735 \beta_{17} - 13950 \beta_{16} + 21059 \beta_{15} - 571 \beta_{14} + 5114 \beta_{13} - 16638 \beta_{12} + \cdots + 38395$$ 16735*b17 - 13950*b16 + 21059*b15 - 571*b14 + 5114*b13 - 16638*b12 - 37041*b11 - 571*b10 - 4324*b9 + 21059*b8 - 4543*b7 - 20403*b6 + 38395*b4 - 9497*b3 + 20403*b2 + 38395 $$\nu^{12}$$ $$=$$ $$- 13379 \beta_{17} - 43305 \beta_{16} + 13379 \beta_{15} + 15982 \beta_{14} + 15982 \beta_{13} + \cdots + 124772$$ -13379*b17 - 43305*b16 + 13379*b15 + 15982*b14 + 15982*b13 + 58327*b12 - 58327*b11 - 12314*b10 + 52405*b9 + 52405*b8 - 12314*b7 - 113851*b6 + 43305*b5 - 24445*b3 - 55524*b2 + 24445*b1 + 124772 $$\nu^{13}$$ $$=$$ $$- 201921 \beta_{17} - 154948 \beta_{15} + 42145 \beta_{14} + 5922 \beta_{13} + 362624 \beta_{12} + \cdots + 81467 \beta_1$$ -201921*b17 - 154948*b15 + 42145*b14 + 5922*b13 + 362624*b12 + 157097*b11 - 48067*b10 + 201921*b9 - 46973*b8 + 5922*b7 - 157097*b6 + 142118*b5 - 366733*b4 - 362624*b2 + 81467*b1 $$\nu^{14}$$ $$=$$ $$- 480481 \beta_{17} + 445001 \beta_{16} - 628521 \beta_{15} - 50579 \beta_{14} - 110124 \beta_{13} + \cdots - 1163961$$ -480481*b17 + 445001*b16 - 628521*b15 - 50579*b14 - 110124*b13 + 504968*b12 + 1122683*b11 - 50579*b10 + 148040*b9 - 628521*b8 + 160703*b7 + 617715*b6 - 1163961*b4 + 224615*b3 - 617715*b2 - 1163961 $$\nu^{15}$$ $$=$$ $$495580 \beta_{17} + 1432844 \beta_{16} - 495580 \beta_{15} - 494162 \beta_{14} - 494162 \beta_{13} + \cdots - 3499851$$ 495580*b17 + 1432844*b16 - 495580*b15 - 494162*b14 - 494162*b13 - 2065062*b12 + 2065062*b11 + 356928*b10 - 1443741*b9 - 1443741*b8 + 356928*b7 + 3545531*b6 - 1432844*b5 + 718960*b3 + 1480469*b2 - 718960*b1 - 3499851 $$\nu^{16}$$ $$=$$ $$6029028 \beta_{17} + 4458950 \beta_{15} - 984889 \beta_{14} - 621321 \beta_{13} - 11020913 \beta_{12} + \cdots - 2067007 \beta_1$$ 6029028*b17 + 4458950*b15 - 984889*b14 - 621321*b13 - 11020913*b12 - 4666329*b11 + 1606210*b10 - 6029028*b9 + 1570078*b8 - 621321*b7 + 4666329*b6 - 4499141*b5 + 10977737*b4 + 11020913*b2 - 2067007*b1 $$\nu^{17}$$ $$=$$ $$13539874 \beta_{17} - 14325778 \beta_{16} + 18660336 \beta_{15} + 1895634 \beta_{14} + 3096251 \beta_{13} + \cdots + 33445352$$ 13539874*b17 - 14325778*b16 + 18660336*b15 + 1895634*b14 + 3096251*b13 - 13979765*b12 - 34628456*b11 + 1895634*b10 - 5120462*b9 + 18660336*b8 - 4991885*b7 - 20648691*b6 + 33445352*b4 - 6478596*b3 + 20648691*b2 + 33445352

## 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
 0.793225 − 1.37391i −0.478554 + 0.828880i −0.754364 + 1.30660i 0.793225 + 1.37391i −0.478554 − 0.828880i −0.754364 − 1.30660i −0.838693 + 1.45266i 0.546970 − 0.947380i 1.55777 − 2.69813i 1.37427 − 2.38031i 0.443231 − 0.767698i −1.14386 + 1.98122i −0.838693 − 1.45266i 0.546970 + 0.947380i 1.55777 + 2.69813i 1.37427 + 2.38031i 0.443231 + 0.767698i −1.14386 − 1.98122i
0 −0.275484 1.56235i 0 0.766044 0.642788i 0 0.778817 1.34895i 0 0.454036 0.165255i 0
61.2 0 0.166200 + 0.942568i 0 0.766044 0.642788i 0 −2.28066 + 3.95023i 0 1.95827 0.712751i 0
61.3 0 0.261988 + 1.48581i 0 0.766044 0.642788i 0 1.67550 2.90204i 0 0.680094 0.247534i 0
81.1 0 −0.275484 + 1.56235i 0 0.766044 + 0.642788i 0 0.778817 + 1.34895i 0 0.454036 + 0.165255i 0
81.2 0 0.166200 0.942568i 0 0.766044 + 0.642788i 0 −2.28066 3.95023i 0 1.95827 + 0.712751i 0
81.3 0 0.261988 1.48581i 0 0.766044 + 0.642788i 0 1.67550 + 2.90204i 0 0.680094 + 0.247534i 0
101.1 0 −1.57623 0.573700i 0 0.173648 + 0.984808i 0 0.230636 0.399473i 0 −0.142773 0.119801i 0
101.2 0 1.02797 + 0.374150i 0 0.173648 + 0.984808i 0 −1.81409 + 3.14209i 0 −1.38140 1.15914i 0
101.3 0 2.92764 + 1.06558i 0 0.173648 + 0.984808i 0 0.643760 1.11502i 0 5.13752 + 4.31089i 0
161.1 0 −2.10551 + 1.76673i 0 −0.939693 0.342020i 0 −1.23778 + 2.14391i 0 0.790885 4.48533i 0
161.2 0 −0.679069 + 0.569806i 0 −0.939693 0.342020i 0 0.460431 0.797489i 0 −0.384489 + 2.18055i 0
161.3 0 1.75249 1.47051i 0 −0.939693 0.342020i 0 1.54340 2.67324i 0 0.387867 2.19970i 0
301.1 0 −1.57623 + 0.573700i 0 0.173648 0.984808i 0 0.230636 + 0.399473i 0 −0.142773 + 0.119801i 0
301.2 0 1.02797 0.374150i 0 0.173648 0.984808i 0 −1.81409 3.14209i 0 −1.38140 + 1.15914i 0
301.3 0 2.92764 1.06558i 0 0.173648 0.984808i 0 0.643760 + 1.11502i 0 5.13752 4.31089i 0
321.1 0 −2.10551 1.76673i 0 −0.939693 + 0.342020i 0 −1.23778 2.14391i 0 0.790885 + 4.48533i 0
321.2 0 −0.679069 0.569806i 0 −0.939693 + 0.342020i 0 0.460431 + 0.797489i 0 −0.384489 2.18055i 0
321.3 0 1.75249 + 1.47051i 0 −0.939693 + 0.342020i 0 1.54340 + 2.67324i 0 0.387867 + 2.19970i 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.b 18
19.e even 9 1 inner 380.2.u.b 18
19.e even 9 1 7220.2.a.w 9
19.f odd 18 1 7220.2.a.y 9

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} - 3 T_{3}^{17} - 3 T_{3}^{16} + 6 T_{3}^{15} + 60 T_{3}^{14} - 111 T_{3}^{13} + 113 T_{3}^{12} + \cdots + 5329$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$T^{18} - 3 T^{17} + \cdots + 5329$$
$5$ $$(T^{6} + T^{3} + 1)^{3}$$
$7$ $$T^{18} + 33 T^{16} + \cdots + 130321$$
$11$ $$T^{18} + 36 T^{16} + \cdots + 12321$$
$13$ $$T^{18} - 9 T^{17} + \cdots + 942841$$
$17$ $$T^{18} - 12 T^{17} + \cdots + 9162729$$
$19$ $$T^{18} + \cdots + 322687697779$$
$23$ $$T^{18} - 21 T^{17} + \cdots + 76055841$$
$29$ $$T^{18} + \cdots + 12637107826641$$
$31$ $$T^{18} + \cdots + 1031244769$$
$37$ $$(T^{9} + 18 T^{8} + \cdots - 587421)^{2}$$
$41$ $$T^{18} + \cdots + 4797190682001$$
$43$ $$T^{18} + \cdots + 16131286081$$
$47$ $$T^{18} + \cdots + 62131046121$$
$53$ $$T^{18} + \cdots + 55278122769$$
$59$ $$T^{18} + \cdots + 25345414942929$$
$61$ $$T^{18} + \cdots + 189464681156689$$
$67$ $$T^{18} + \cdots + 20\!\cdots\!81$$
$71$ $$T^{18} + \cdots + 176384961$$
$73$ $$T^{18} + \cdots + 95\!\cdots\!81$$
$79$ $$T^{18} + \cdots + 385979893469641$$
$83$ $$T^{18} + \cdots + 71913539889$$
$89$ $$T^{18} + \cdots + 156305980013121$$
$97$ $$T^{18} + \cdots + 45168117642729$$