# Properties

 Label 440.2.y.c Level $440$ Weight $2$ Character orbit 440.y Analytic conductor $3.513$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(81,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 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("440.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$440 = 2^{3} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 440.y (of order $$5$$, degree $$4$$, 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: $$3.51341768894$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1$$ x^12 - x^11 + 5*x^10 + 4*x^9 + 28*x^8 - 81*x^7 + 335*x^6 - 235*x^5 + 782*x^4 - 302*x^3 + 711*x^2 - 43*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{8} + \beta_{3}) q^{3} - \beta_{6} q^{5} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{9} + 2 \beta_{7} + \beta_{4} + \cdots - 2) q^{9}+O(q^{10})$$ q + (b8 + b3) * q^3 - b6 * q^5 + (b11 - 2*b7 + b6 + b5 + b2 + 1) * q^7 + (-b9 + 2*b7 + b4 - b1 - 2) * q^9 $$q + (\beta_{8} + \beta_{3}) q^{3} - \beta_{6} q^{5} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + 2 \beta_{10} - 6 \beta_{8} + \cdots + 7) q^{99}+O(q^{100})$$ q + (b8 + b3) * q^3 - b6 * q^5 + (b11 - 2*b7 + b6 + b5 + b2 + 1) * q^7 + (-b9 + 2*b7 + b4 - b1 - 2) * q^9 + (-b10 - b8 - b5 + 1) * q^11 + (2*b8 - b7 - b6 + b5 + b3 + 1) * q^13 + (b8 - b7 + b5 - b4 + b3 + b2 + 1) * q^15 + (b11 + b10 - b9 - b6 + b3 + b2 + 1) * q^17 + (-b10 - b8 + 2*b7 + 2*b4 - b3 - 2*b2 - 2) * q^19 + (-b10 - b9 - 2*b8 - 2*b7 + 3*b2 - b1) * q^21 + (-2*b10 - 2*b9 + 3*b8 + 2*b7 + b5 - b4 - 2*b1 - 3) * q^23 - b8 * q^25 + (-b11 + 2*b10 + b9 - 3*b8 - 3*b6 - b5 - b3 - b2 + 3*b1 + 2) * q^27 + (-b11 - b9 + b8 + 2*b7 - b5 - b4 + b3 - b2 + 1) * q^29 + (-2*b11 - 2*b10 + 2*b8 + b7 - b6 + b5 - 2*b4 + b3 - 1) * q^31 + (-b11 - b9 + 5*b7 - 2*b6 - 2*b5 - b3 - b2 - b1 - 2) * q^33 + (b9 - 2*b8 - 2*b7 + b6 - b5 - b3 + b1 + 2) * q^35 + (-b9 - b8 + b6 - 2*b5 + b4 - b3 - 2*b2) * q^37 + (b11 - b9 + 6*b8 - 2*b6 + 2*b5 + 3*b3 + 3*b2 - b1 - 3) * q^39 + (2*b11 + b7 + b4 + b3 - b2 - 2*b1 - 1) * q^41 + (-b8 - b5 + b4 + b2 + b1 - 2) * q^43 + (-b10 - b9 + b8 + b7 + b2 - b1 - 1) * q^45 + (b11 + b10 - 3*b8 - 3*b7 + 4*b6 + b4 - 3*b3 - b2 - b1 - 1) * q^47 + (-b10 - b8 + 2*b6 - 4*b5 - 3*b3 - 3*b2 - b1 - 2) * q^49 + (-b11 + 2*b8 + 3*b7 - 3*b6 + 3*b5 - 2*b4 + 2*b3 + 3*b2 - 1) * q^51 + (2*b11 + 2*b10 + b9 - 4*b8 - 4*b7 + 5*b6 + b5 + 3*b4 + b3 + b1 + 4) * q^53 + (b9 + b8 + b7 - b6 + b3) * q^55 + (2*b11 + 2*b10 + 3*b8 - 6*b7 - 2*b6 + b5 - b4 + b3 + 6) * q^57 + (2*b8 - 3*b7 + 5*b6 + b5 - 2*b4 + 2*b3 + b2 + 7) * q^59 + (-2*b11 - b10 + 2*b9 - b8 + 2*b5 + b1 + 1) * q^61 + (-b11 + b10 - 10*b8 + 2*b7 - b6 + b4 - 4*b3 - b2 + b1 - 1) * q^63 + (-2*b8 - 3*b7 + b5 - b4 + b2 + 2) * q^65 + (2*b10 + 2*b9 + 3*b8 + 3*b5 - 3*b4 + b2 + 3*b1 - 3) * q^67 + (-b11 + b10 - 4*b8 + 2*b6 + 2*b4 - 5*b3 - 2*b2 + b1 - 2) * q^69 + (-b11 - b10 + b9 + 2*b8 - 4*b6 - b5 + b3 + b2 - 1) * q^71 + (b11 + 4*b9 - 2*b8 - b7 + 4*b6 - b5 + 2*b4 - 2*b3 - b2 + 2) * q^73 + (-b7 - b4 + 1) * q^75 + (b11 + b10 + 4*b8 + 5*b7 - b6 - b5 + 5*b4 + b1 - 3) * q^77 + (b11 + b10 + b9 + 7*b8 + 2*b7 - 5*b6 + 2*b5 + b4 + 2*b3 + b1 - 2) * q^79 + (3*b9 + 4*b8 - 5*b7 + 2*b6 - 4*b4 + 4*b3 + 6) * q^81 + (-2*b11 + b10 + 2*b9 - 2*b8 - 3*b6 - 4*b5 - 3*b3 - 3*b2 + 3*b1 - 1) * q^83 + (-b11 - b10 - b6 - b4 + b3 + b2 + b1 + 1) * q^85 + (b10 + b9 + 6*b8 + 7*b7 - b5 + b4 - 4*b2 - 5) * q^87 + (3*b10 + 3*b9 + 2*b8 + 4*b7 - 2*b5 + 2*b4 - 4*b2 + b1 - 8) * q^89 + (b11 - 3*b10 - 5*b8 - 9*b7 + 5*b6 - 4*b4 - 2*b3 + 4*b2 - b1 + 4) * q^91 + (b11 - b9 + 2*b8 + 2*b6 - 2*b5 - 3*b3 - 3*b2 - b1 - 5) * q^93 + (b9 - b8 + b7 + b5 + b4 - b3 + b2 - 1) * q^95 + (-b9 - 6*b8 - 2*b7 + 8*b6 + 2*b5 - 2*b4 + 2*b3 - b1 + 2) * q^97 + (b11 + 2*b10 - 6*b8 - b7 + 2*b6 + b5 - b3 - 3*b2 + b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - q^{3} + 3 q^{5} - q^{7} - 10 q^{9}+O(q^{10})$$ 12 * q - q^3 + 3 * q^5 - q^7 - 10 * q^9 $$12 q - q^{3} + 3 q^{5} - q^{7} - 10 q^{9} + 4 q^{11} + 18 q^{13} + q^{15} + 3 q^{17} + 4 q^{19} - 28 q^{21} - 18 q^{23} - 3 q^{25} + 23 q^{27} + 15 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{35} + 6 q^{37} - 33 q^{39} + 2 q^{41} - 36 q^{43} - 10 q^{45} - 16 q^{47} - 16 q^{49} - 10 q^{51} + 19 q^{53} + 6 q^{55} + 62 q^{57} + 46 q^{59} + 18 q^{61} - 7 q^{63} + 2 q^{65} - 44 q^{67} - q^{69} - 6 q^{71} + 25 q^{73} + 4 q^{75} + 10 q^{77} + 19 q^{79} + 30 q^{81} - 3 q^{85} + 6 q^{87} - 50 q^{89} - 46 q^{91} - 37 q^{93} - 4 q^{95} - 31 q^{97} + 79 q^{99}+O(q^{100})$$ 12 * q - q^3 + 3 * q^5 - q^7 - 10 * q^9 + 4 * q^11 + 18 * q^13 + q^15 + 3 * q^17 + 4 * q^19 - 28 * q^21 - 18 * q^23 - 3 * q^25 + 23 * q^27 + 15 * q^29 - 8 * q^31 + 4 * q^33 + 6 * q^35 + 6 * q^37 - 33 * q^39 + 2 * q^41 - 36 * q^43 - 10 * q^45 - 16 * q^47 - 16 * q^49 - 10 * q^51 + 19 * q^53 + 6 * q^55 + 62 * q^57 + 46 * q^59 + 18 * q^61 - 7 * q^63 + 2 * q^65 - 44 * q^67 - q^69 - 6 * q^71 + 25 * q^73 + 4 * q^75 + 10 * q^77 + 19 * q^79 + 30 * q^81 - 3 * q^85 + 6 * q^87 - 50 * q^89 - 46 * q^91 - 37 * q^93 - 4 * q^95 - 31 * q^97 + 79 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 3497434625 \nu^{11} - 2215518483724 \nu^{10} + 702407416002 \nu^{9} + \cdots - 708170152940520 ) / 249616873443718$$ (3497434625*v^11 - 2215518483724*v^10 + 702407416002*v^9 - 8423802073100*v^8 - 14573743515344*v^7 - 66609322567911*v^6 + 148817803652752*v^5 - 581250414645334*v^4 - 8021760458332*v^3 - 1258466920099962*v^2 + 77330441076607*v - 708170152940520) / 249616873443718 $$\beta_{2}$$ $$=$$ $$( 469750424879 \nu^{11} - 1250785586077 \nu^{10} + 1635686014444 \nu^{9} + \cdots - 518561109011779 ) / 499233746887436$$ (469750424879*v^11 - 1250785586077*v^10 + 1635686014444*v^9 + 670802485458*v^8 + 4322069337726*v^7 - 62881183501887*v^6 + 206852040696993*v^5 - 204123089805384*v^4 + 21492679830862*v^3 - 264896462451694*v^2 + 16227833704539*v - 518561109011779) / 499233746887436 $$\beta_{3}$$ $$=$$ $$( - 2167683167019 \nu^{11} + 2869118757525 \nu^{10} - 9747395928812 \nu^{9} + \cdots + 19849150430431 ) / 499233746887436$$ (-2167683167019*v^11 + 2869118757525*v^10 - 9747395928812*v^9 - 7371577071234*v^8 - 48593848103822*v^7 + 201845578827663*v^6 - 743446314206713*v^5 + 577766807427048*v^4 - 1279290819013694*v^3 + 703552963734814*v^2 - 848031061166511*v + 19849150430431) / 499233746887436 $$\beta_{4}$$ $$=$$ $$( - 11524681849762 \nu^{11} + 12737270254139 \nu^{10} - 59052962251434 \nu^{9} + \cdots + 472887038077187 ) / 499233746887436$$ (-11524681849762*v^11 + 12737270254139*v^10 - 59052962251434*v^9 - 42393450210186*v^8 - 314010264277656*v^7 + 956570196886864*v^6 - 3966226985040665*v^5 + 3083095896356116*v^4 - 9051734864032572*v^3 + 3541336067437778*v^2 - 7827084768984622*v + 472887038077187) / 499233746887436 $$\beta_{5}$$ $$=$$ $$( 19849150430431 \nu^{11} - 17681467263412 \nu^{10} + 96376633394630 \nu^{9} + \cdots - 5482407342022 ) / 499233746887436$$ (19849150430431*v^11 - 17681467263412*v^10 + 96376633394630*v^9 + 89143997650536*v^8 + 563147789123302*v^7 - 1559187336761089*v^6 + 6447619815366722*v^5 - 3921104036944572*v^4 + 14944268829169994*v^3 - 4715152610976468*v^2 + 13908426739189063*v - 5482407342022) / 499233746887436 $$\beta_{6}$$ $$=$$ $$( - 19849150430431 \nu^{11} + 17681467263412 \nu^{10} - 96376633394630 \nu^{9} + \cdots + 5482407342022 ) / 499233746887436$$ (-19849150430431*v^11 + 17681467263412*v^10 - 96376633394630*v^9 - 89143997650536*v^8 - 563147789123302*v^7 + 1559187336761089*v^6 - 6447619815366722*v^5 + 3921104036944572*v^4 - 14944268829169994*v^3 + 4715152610976468*v^2 - 13409192992301627*v + 5482407342022) / 499233746887436 $$\beta_{7}$$ $$=$$ $$( 30450870201976 \nu^{11} - 30852043974105 \nu^{10} + 152997188989378 \nu^{9} + \cdots - 818952899033937 ) / 499233746887436$$ (30450870201976*v^11 - 30852043974105*v^10 + 152997188989378*v^9 + 120361682510210*v^8 + 851647463443536*v^7 - 2470720229161238*v^6 + 10248085748121643*v^5 - 7285880862536400*v^4 + 23944873590288560*v^3 - 9212898166171606*v^2 + 21794377296591832*v - 818952899033937) / 499233746887436 $$\beta_{8}$$ $$=$$ $$( - 30972658508064 \nu^{11} + 29675899538053 \nu^{10} - 153987797348370 \nu^{9} + \cdots + 508820315621185 ) / 499233746887436$$ (-30972658508064*v^11 + 29675899538053*v^10 - 153987797348370*v^9 - 130560545648930*v^8 - 872958310599776*v^7 + 2468713303188270*v^6 - 10283920435335347*v^5 + 6871906840957360*v^4 - 23979268328027712*v^3 + 8112680095427350*v^2 - 21726712280937280*v + 508820315621185) / 499233746887436 $$\beta_{9}$$ $$=$$ $$( 88405871699823 \nu^{11} - 83922796851166 \nu^{10} + 438655411068042 \nu^{9} + \cdots - 23\!\cdots\!76 ) / 499233746887436$$ (88405871699823*v^11 - 83922796851166*v^10 + 438655411068042*v^9 + 371258807599044*v^8 + 2497388207471794*v^7 - 7040382256171265*v^6 + 29252344157672984*v^5 - 19487904560619272*v^4 + 68610501460025506*v^3 - 24186893326668496*v^2 + 61555067941453715*v - 2314644923794976) / 499233746887436 $$\beta_{10}$$ $$=$$ $$( - 89746069507409 \nu^{11} + 87864404405688 \nu^{10} - 443314313150206 \nu^{9} + \cdots + 14\!\cdots\!34 ) / 499233746887436$$ (-89746069507409*v^11 + 87864404405688*v^10 - 443314313150206*v^9 - 372313991895452*v^8 - 2508422791852306*v^7 + 7230008310244831*v^6 - 29830683748782198*v^5 + 20143571933506260*v^4 - 68672999460482794*v^3 + 25092836486941108*v^2 - 61610590299253393*v + 1442843847065734) / 499233746887436 $$\beta_{11}$$ $$=$$ $$( 12823391623530 \nu^{11} - 12647557096509 \nu^{10} + 63908603030210 \nu^{9} + \cdots - 340671582608193 ) / 45384886080676$$ (12823391623530*v^11 - 12647557096509*v^10 + 63908603030210*v^9 + 52451648717282*v^8 + 359629895384696*v^7 - 1033401867435936*v^6 + 4281968742053927*v^5 - 2946130858565152*v^4 + 9957419369419212*v^3 - 3688973244252970*v^2 + 9062757904790182*v - 340671582608193) / 45384886080676
 $$\nu$$ $$=$$ $$\beta_{6} + \beta_{5}$$ b6 + b5 $$\nu^{2}$$ $$=$$ $$-\beta_{11} - 3\beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_1$$ -b11 - 3*b8 + b7 - b6 + b3 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{9} - 6\beta_{8} - 2\beta_{7} + \beta_{6} - 5\beta_{5} + 6\beta_{4} - 6\beta_{3} - 5\beta_{2} - 5$$ b11 - b9 - 6*b8 - 2*b7 + b6 - 5*b5 + 6*b4 - 6*b3 - 5*b2 - 5 $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} - 7\beta_{9} + 24\beta_{8} + 24\beta_{7} - 24\beta_{6} - 9\beta_{4} - 7\beta _1 - 24$$ b11 + b10 - 7*b9 + 24*b8 + 24*b7 - 24*b6 - 9*b4 - 7*b1 - 24 $$\nu^{5}$$ $$=$$ $$10\beta_{10} + 10\beta_{9} - 11\beta_{8} - 3\beta_{7} - 8\beta_{5} + 8\beta_{4} + 32\beta_{2} + 2\beta _1 + 55$$ 10*b10 + 10*b9 - 11*b8 - 3*b7 - 8*b5 + 8*b4 + 32*b2 + 2*b1 + 55 $$\nu^{6}$$ $$=$$ $$- 10 \beta_{11} - 48 \beta_{10} + 10 \beta_{9} + 62 \beta_{8} + 164 \beta_{6} + 70 \beta_{5} + \cdots - 65$$ -10*b11 - 48*b10 + 10*b9 + 62*b8 + 164*b6 + 70*b5 - 3*b3 - 3*b2 - 38*b1 - 65 $$\nu^{7}$$ $$=$$ $$- 83 \beta_{11} - 55 \beta_{10} - 110 \beta_{8} + 44 \beta_{7} - 99 \beta_{6} - 55 \beta_{4} + \cdots + 55$$ -83*b11 - 55*b10 - 110*b8 + 44*b7 - 99*b6 - 55*b4 + 277*b3 + 55*b2 + 83*b1 + 55 $$\nu^{8}$$ $$=$$ $$332 \beta_{11} - 83 \beta_{9} - 569 \beta_{8} - 1052 \beta_{7} + 458 \beta_{6} - 525 \beta_{5} + \cdots - 111$$ 332*b11 - 83*b9 - 569*b8 - 1052*b7 + 458*b6 - 525*b5 + 569*b4 - 569*b3 - 525*b2 - 111 $$\nu^{9}$$ $$=$$ $$371 \beta_{11} + 371 \beta_{10} - 652 \beta_{9} + 3240 \beta_{8} + 1739 \beta_{7} - 2865 \beta_{6} + \cdots - 1739$$ 371*b11 + 371*b10 - 652*b9 + 3240*b8 + 1739*b7 - 2865*b6 + 375*b5 - 1953*b4 + 375*b3 - 652*b1 - 1739 $$\nu^{10}$$ $$=$$ $$2324 \beta_{10} + 2324 \beta_{9} - 3253 \beta_{8} - 2797 \beta_{7} - 456 \beta_{5} + 456 \beta_{4} + \cdots + 11142$$ 2324*b10 + 2324*b9 - 3253*b8 - 2797*b7 - 456*b5 + 456*b4 + 3888*b2 + 1676*b1 + 11142 $$\nu^{11}$$ $$=$$ $$- 2516 \beta_{11} - 4992 \beta_{10} + 2516 \beta_{9} + 4036 \beta_{8} + 21113 \beta_{6} + 11317 \beta_{5} + \cdots - 6641$$ -2516*b11 - 4992*b10 + 2516*b9 + 4036*b8 + 21113*b6 + 11317*b5 - 2605*b3 - 2605*b2 - 2476*b1 - 6641

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$221$$ $$321$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 1.85498 − 1.34772i 0.0307040 − 0.0223078i −2.19470 + 1.59454i 1.85498 + 1.34772i 0.0307040 + 0.0223078i −2.19470 − 1.59454i 0.830630 + 2.55642i 0.377272 + 1.16112i −0.398885 − 1.22764i 0.830630 − 2.55642i 0.377272 − 1.16112i −0.398885 + 1.22764i
0 −1.01756 + 3.13172i 0 0.809017 0.587785i 0 1.08622 + 3.34304i 0 −6.34518 4.61004i 0
81.2 0 −0.320745 + 0.987151i 0 0.809017 0.587785i 0 −0.804092 2.47474i 0 1.55546 + 1.13011i 0
81.3 0 0.529284 1.62897i 0 0.809017 0.587785i 0 1.14492 + 3.52372i 0 0.0536500 + 0.0389790i 0
201.1 0 −1.01756 3.13172i 0 0.809017 + 0.587785i 0 1.08622 3.34304i 0 −6.34518 + 4.61004i 0
201.2 0 −0.320745 0.987151i 0 0.809017 + 0.587785i 0 −0.804092 + 2.47474i 0 1.55546 1.13011i 0
201.3 0 0.529284 + 1.62897i 0 0.809017 + 0.587785i 0 1.14492 3.52372i 0 0.0536500 0.0389790i 0
361.1 0 −1.36560 + 0.992167i 0 −0.309017 0.951057i 0 0.156923 + 0.114011i 0 −0.0465816 + 0.143363i 0
361.2 0 −0.178694 + 0.129829i 0 −0.309017 0.951057i 0 1.68900 + 1.22713i 0 −0.911975 + 2.80677i 0
361.3 0 1.85331 1.34651i 0 −0.309017 0.951057i 0 −3.77298 2.74123i 0 0.694624 2.13783i 0
401.1 0 −1.36560 0.992167i 0 −0.309017 + 0.951057i 0 0.156923 0.114011i 0 −0.0465816 0.143363i 0
401.2 0 −0.178694 0.129829i 0 −0.309017 + 0.951057i 0 1.68900 1.22713i 0 −0.911975 2.80677i 0
401.3 0 1.85331 + 1.34651i 0 −0.309017 + 0.951057i 0 −3.77298 + 2.74123i 0 0.694624 + 2.13783i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.y.c 12
4.b odd 2 1 880.2.bo.i 12
11.c even 5 1 inner 440.2.y.c 12
11.c even 5 1 4840.2.a.bb 6
11.d odd 10 1 4840.2.a.ba 6
44.g even 10 1 9680.2.a.dd 6
44.h odd 10 1 880.2.bo.i 12
44.h odd 10 1 9680.2.a.dc 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.c 12 1.a even 1 1 trivial
440.2.y.c 12 11.c even 5 1 inner
880.2.bo.i 12 4.b odd 2 1
880.2.bo.i 12 44.h odd 10 1
4840.2.a.ba 6 11.d odd 10 1
4840.2.a.bb 6 11.c even 5 1
9680.2.a.dc 6 44.h odd 10 1
9680.2.a.dd 6 44.g even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + T_{3}^{11} + 10 T_{3}^{10} - 6 T_{3}^{9} + 29 T_{3}^{8} + 54 T_{3}^{7} + 165 T_{3}^{6} + \cdots + 25$$ acting on $$S_{2}^{\mathrm{new}}(440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + T^{11} + \cdots + 25$$
$5$ $$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{3}$$
$7$ $$T^{12} + T^{11} + \cdots + 4096$$
$11$ $$T^{12} - 4 T^{11} + \cdots + 1771561$$
$13$ $$T^{12} - 18 T^{11} + \cdots + 6400$$
$17$ $$T^{12} - 3 T^{11} + \cdots + 366025$$
$19$ $$T^{12} - 4 T^{11} + \cdots + 95863681$$
$23$ $$(T^{6} + 9 T^{5} + \cdots + 5956)^{2}$$
$29$ $$T^{12} - 15 T^{11} + \cdots + 1210000$$
$31$ $$T^{12} + 8 T^{11} + \cdots + 12931216$$
$37$ $$T^{12} - 6 T^{11} + \cdots + 3610000$$
$41$ $$T^{12} - 2 T^{11} + \cdots + 14250625$$
$43$ $$(T^{6} + 18 T^{5} + \cdots + 3751)^{2}$$
$47$ $$T^{12} + 16 T^{11} + \cdots + 86192656$$
$53$ $$T^{12} + \cdots + 39921638416$$
$59$ $$T^{12} + \cdots + 138274081$$
$61$ $$T^{12} - 18 T^{11} + \cdots + 43824400$$
$67$ $$(T^{6} + 22 T^{5} + \cdots + 126475)^{2}$$
$71$ $$T^{12} + 6 T^{11} + \cdots + 384400$$
$73$ $$T^{12} + \cdots + 692275584961$$
$79$ $$T^{12} - 19 T^{11} + \cdots + 45212176$$
$83$ $$T^{12} + 180 T^{10} + \cdots + 1890625$$
$89$ $$(T^{6} + 25 T^{5} + \cdots - 22429)^{2}$$
$97$ $$T^{12} + \cdots + 27124443025$$