# Properties

 Label 441.4.e.z Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(226,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not 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: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000$$ x^16 + 74*x^14 + 4007*x^12 + 91050*x^10 + 1502189*x^8 + 12598332*x^6 + 74261084*x^4 + 22070000*x^2 + 6250000 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{8}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + (\beta_{7} + 9 \beta_{2} - 9) q^{4} - \beta_{6} q^{5} + (2 \beta_{15} - 11 \beta_{12} + 11 \beta_{4}) q^{8}+O(q^{10})$$ q - b4 * q^2 + (b7 + 9*b2 - 9) * q^4 - b6 * q^5 + (2*b15 - 11*b12 + 11*b4) * q^8 $$q - \beta_{4} q^{2} + (\beta_{7} + 9 \beta_{2} - 9) q^{4} - \beta_{6} q^{5} + (2 \beta_{15} - 11 \beta_{12} + 11 \beta_{4}) q^{8} + ( - 3 \beta_{14} - 2 \beta_{13}) q^{10} + (\beta_{15} + 7 \beta_{12} + \beta_{9}) q^{11} + ( - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{10} + 3 \beta_{8}) q^{13} + ( - 9 \beta_{7} - 105 \beta_{2} + 9 \beta_1) q^{16} + ( - 2 \beta_{11} + \beta_{6} + 2 \beta_{5} + \beta_{3}) q^{17} + (6 \beta_{10} + 2 \beta_{8}) q^{19} + (5 \beta_{5} - 14 \beta_{3}) q^{20} + ( - 4 \beta_1 + 124) q^{22} + ( - 3 \beta_{9} + 7 \beta_{4}) q^{23} + (4 \beta_{7} + 69 \beta_{2} - 69) q^{25} + (5 \beta_{11} + 18 \beta_{6}) q^{26} + ( - 9 \beta_{15} - 13 \beta_{12} + 13 \beta_{4}) q^{29} + ( - 2 \beta_{14} + 10 \beta_{13}) q^{31} + ( - 2 \beta_{15} + 107 \beta_{12} - 2 \beta_{9}) q^{32} + (\beta_{14} - 18 \beta_{13} + \beta_{10} + 18 \beta_{8}) q^{34} + (24 \beta_{7} + 4 \beta_{2} - 24 \beta_1) q^{37} + ( - 4 \beta_{11} - 32 \beta_{6} + 4 \beta_{5} - 32 \beta_{3}) q^{38} + ( - 23 \beta_{10} + 62 \beta_{8}) q^{40} + (10 \beta_{5} - \beta_{3}) q^{41} + (20 \beta_1 + 260) q^{43} + (16 \beta_{9} - 108 \beta_{4}) q^{44} + ( - 16 \beta_{7} - 104 \beta_{2} + 104) q^{46} + ( - 2 \beta_{11} - 30 \beta_{6}) q^{47} + (8 \beta_{15} - 109 \beta_{12} + 109 \beta_{4}) q^{50} + (43 \beta_{14} + 62 \beta_{13}) q^{52} + ( - 14 \beta_{15} + 34 \beta_{12} - 14 \beta_{9}) q^{53} + (26 \beta_{14} - 6 \beta_{13} + 26 \beta_{10} + 6 \beta_{8}) q^{55} + (14 \beta_{7} - 266 \beta_{2} - 14 \beta_1) q^{58} + ( - 10 \beta_{11} - 2 \beta_{6} + 10 \beta_{5} - 2 \beta_{3}) q^{59} + (22 \beta_{10} + 55 \beta_{8}) q^{61} + ( - 8 \beta_{5} + 8 \beta_{3}) q^{62} + ( - 41 \beta_1 + 969) q^{64} + ( - 17 \beta_{9} + 191 \beta_{4}) q^{65} + ( - 40 \beta_{7} - 20 \beta_{2} + 20) q^{67} + (\beta_{11} + 38 \beta_{6}) q^{68} + (9 \beta_{15} + 163 \beta_{12} - 163 \beta_{4}) q^{71} + ( - 28 \beta_{14} + 23 \beta_{13}) q^{73} + (48 \beta_{15} - 244 \beta_{12} + 48 \beta_{9}) q^{74} + ( - 52 \beta_{14} - 120 \beta_{13} - 52 \beta_{10} + 120 \beta_{8}) q^{76} + ( - 28 \beta_{7} - 688 \beta_{2} + 28 \beta_1) q^{79} + (45 \beta_{11} + 150 \beta_{6} - 45 \beta_{5} + 150 \beta_{3}) q^{80} + ( - 13 \beta_{10} + 102 \beta_{8}) q^{82} + ( - 16 \beta_{5} - 52 \beta_{3}) q^{83} + (64 \beta_1 + 326) q^{85} + ( - 40 \beta_{9} - 60 \beta_{4}) q^{86} + (124 \beta_{7} + 764 \beta_{2} - 764) q^{88} + ( - 10 \beta_{11} - 7 \beta_{6}) q^{89} + ( - 8 \beta_{15} + 208 \beta_{12} - 208 \beta_{4}) q^{92} + ( - 92 \beta_{14} - 80 \beta_{13}) q^{94} + ( - 26 \beta_{15} - 342 \beta_{12} - 26 \beta_{9}) q^{95} + (28 \beta_{14} + 57 \beta_{13} + 28 \beta_{10} - 57 \beta_{8}) q^{97}+O(q^{100})$$ q - b4 * q^2 + (b7 + 9*b2 - 9) * q^4 - b6 * q^5 + (2*b15 - 11*b12 + 11*b4) * q^8 + (-3*b14 - 2*b13) * q^10 + (b15 + 7*b12 + b9) * q^11 + (-2*b14 - 3*b13 - 2*b10 + 3*b8) * q^13 + (-9*b7 - 105*b2 + 9*b1) * q^16 + (-2*b11 + b6 + 2*b5 + b3) * q^17 + (6*b10 + 2*b8) * q^19 + (5*b5 - 14*b3) * q^20 + (-4*b1 + 124) * q^22 + (-3*b9 + 7*b4) * q^23 + (4*b7 + 69*b2 - 69) * q^25 + (5*b11 + 18*b6) * q^26 + (-9*b15 - 13*b12 + 13*b4) * q^29 + (-2*b14 + 10*b13) * q^31 + (-2*b15 + 107*b12 - 2*b9) * q^32 + (b14 - 18*b13 + b10 + 18*b8) * q^34 + (24*b7 + 4*b2 - 24*b1) * q^37 + (-4*b11 - 32*b6 + 4*b5 - 32*b3) * q^38 + (-23*b10 + 62*b8) * q^40 + (10*b5 - b3) * q^41 + (20*b1 + 260) * q^43 + (16*b9 - 108*b4) * q^44 + (-16*b7 - 104*b2 + 104) * q^46 + (-2*b11 - 30*b6) * q^47 + (8*b15 - 109*b12 + 109*b4) * q^50 + (43*b14 + 62*b13) * q^52 + (-14*b15 + 34*b12 - 14*b9) * q^53 + (26*b14 - 6*b13 + 26*b10 + 6*b8) * q^55 + (14*b7 - 266*b2 - 14*b1) * q^58 + (-10*b11 - 2*b6 + 10*b5 - 2*b3) * q^59 + (22*b10 + 55*b8) * q^61 + (-8*b5 + 8*b3) * q^62 + (-41*b1 + 969) * q^64 + (-17*b9 + 191*b4) * q^65 + (-40*b7 - 20*b2 + 20) * q^67 + (b11 + 38*b6) * q^68 + (9*b15 + 163*b12 - 163*b4) * q^71 + (-28*b14 + 23*b13) * q^73 + (48*b15 - 244*b12 + 48*b9) * q^74 + (-52*b14 - 120*b13 - 52*b10 + 120*b8) * q^76 + (-28*b7 - 688*b2 + 28*b1) * q^79 + (45*b11 + 150*b6 - 45*b5 + 150*b3) * q^80 + (-13*b10 + 102*b8) * q^82 + (-16*b5 - 52*b3) * q^83 + (64*b1 + 326) * q^85 + (-40*b9 - 60*b4) * q^86 + (124*b7 + 764*b2 - 764) * q^88 + (-10*b11 - 7*b6) * q^89 + (-8*b15 + 208*b12 - 208*b4) * q^92 + (-92*b14 - 80*b13) * q^94 + (-26*b15 - 342*b12 - 26*b9) * q^95 + (28*b14 + 57*b13 + 28*b10 - 57*b8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 68 q^{4}+O(q^{10})$$ 16 * q - 68 * q^4 $$16 q - 68 q^{4} - 804 q^{16} + 1952 q^{22} - 536 q^{25} - 64 q^{37} + 4320 q^{43} + 768 q^{46} - 2184 q^{58} + 15176 q^{64} - 5392 q^{79} + 5728 q^{85} - 5616 q^{88}+O(q^{100})$$ 16 * q - 68 * q^4 - 804 * q^16 + 1952 * q^22 - 536 * q^25 - 64 * q^37 + 4320 * q^43 + 768 * q^46 - 2184 * q^58 + 15176 * q^64 - 5392 * q^79 + 5728 * q^85 - 5616 * q^88

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000$$ :

 $$\beta_{1}$$ $$=$$ $$( 5955326 \nu^{14} + 420756261 \nu^{12} + 21224761594 \nu^{10} + 399577053316 \nu^{8} + 4077769899992 \nu^{6} + \cdots + 20\!\cdots\!85 ) / 156438029994849$$ (5955326*v^14 + 420756261*v^12 + 21224761594*v^10 + 399577053316*v^8 + 4077769899992*v^6 + 22188556375664*v^4 + 6595574220000*v^2 + 2060425458941185) / 156438029994849 $$\beta_{2}$$ $$=$$ $$( 2249798084357 \nu^{14} + 165523687535043 \nu^{12} + \cdots + 48\!\cdots\!00 ) / 46\!\cdots\!00$$ (2249798084357*v^14 + 165523687535043*v^12 + 8950261034338249*v^10 + 201417793651814725*v^8 + 3315118047702508223*v^6 + 27333084737355121649*v^4 + 163490536811521715988*v^2 + 48588317482855240000) / 46253068953138052500 $$\beta_{3}$$ $$=$$ $$( 5596355476439 \nu^{14} + 378608216872794 \nu^{12} + \cdots - 12\!\cdots\!68 ) / 10\!\cdots\!44$$ (5596355476439*v^14 + 378608216872794*v^12 + 19945391869847941*v^10 + 375491657481782674*v^8 + 5545670366939508311*v^6 + 20851091776877154296*v^4 + 6198011310607830000*v^2 - 1244227061134685320568) / 104790953020229571744 $$\beta_{4}$$ $$=$$ $$( 11310865459517 \nu^{15} + 773166243923574 \nu^{13} + \cdots - 47\!\cdots\!76 \nu ) / 13\!\cdots\!00$$ (11310865459517*v^15 + 773166243923574*v^13 + 40311885999197623*v^11 + 758910979981201222*v^9 + 10456380827878555493*v^7 + 42142407637473355688*v^5 + 12526879742717490000*v^3 - 4701473085998115583976*v) / 1309886912752869646800 $$\beta_{5}$$ $$=$$ $$( 81945585249437 \nu^{14} + \cdots - 11\!\cdots\!80 ) / 26\!\cdots\!60$$ (81945585249437*v^14 + 5641408644608886*v^12 + 292053786912774103*v^10 + 5498200348095991942*v^8 + 71663640280722515861*v^6 + 305315651577077926568*v^4 + 90755432954299890000*v^2 - 11287233145670329435880) / 261977382550573929360 $$\beta_{6}$$ $$=$$ $$( 36\!\cdots\!39 \nu^{14} + \cdots + 81\!\cdots\!00 ) / 65\!\cdots\!00$$ (36549057905883239*v^14 + 2716062717018068436*v^12 + 147193283270968061173*v^10 + 3368536871004748877200*v^8 + 55670660323382905812671*v^6 + 474549060694699086636098*v^4 + 2756768003598295094681076*v^2 + 819299223684428072230000) / 65494345637643482340000 $$\beta_{7}$$ $$=$$ $$( - 10\!\cdots\!57 \nu^{14} + \cdots - 10\!\cdots\!00 ) / 16\!\cdots\!00$$ (-1035048282591857*v^14 - 75366198338390343*v^12 - 4075230308964056149*v^10 - 90616156314394050925*v^8 - 1509438606757895941523*v^6 - 12445292369887636979549*v^4 - 76073073971134978813188*v^2 - 1063284696474873437500) / 1637358640941087058500 $$\beta_{8}$$ $$=$$ $$( 79176058216619 \nu^{15} + \cdots - 23\!\cdots\!32 \nu ) / 13\!\cdots\!00$$ (79176058216619*v^15 + 5412163707465018*v^13 + 282183201994383361*v^11 + 5312376859868408554*v^9 + 73194665795149888451*v^7 + 294996853462313489816*v^5 + 87688158199022430000*v^3 - 23741103212716721560232*v) / 1309886912752869646800 $$\beta_{9}$$ $$=$$ $$( 38872184755339 \nu^{15} + \cdots - 21\!\cdots\!40 \nu ) / 52\!\cdots\!20$$ (38872184755339*v^15 + 2625734874661842*v^13 + 138540334159707041*v^11 + 2608158317528460074*v^9 + 39176637776066606827*v^7 + 144831309468040223896*v^5 + 43051275387340830000*v^3 - 21780492554335386764440*v) / 523954765101147858720 $$\beta_{10}$$ $$=$$ $$( 4417491293917 \nu^{15} + 300026796677634 \nu^{13} + \cdots - 15\!\cdots\!96 \nu ) / 59\!\cdots\!00$$ (4417491293917*v^15 + 300026796677634*v^13 + 15743923935811223*v^11 + 296394883214191622*v^9 + 4277010047478401233*v^7 + 16458839468079877288*v^5 + 4892409197285490000*v^3 - 1566391769961532269496*v) / 59540314216039529400 $$\beta_{11}$$ $$=$$ $$( - 63\!\cdots\!73 \nu^{14} + \cdots - 14\!\cdots\!00 ) / 32\!\cdots\!00$$ (-63275195548744273*v^14 - 4765069144825024452*v^12 - 259282498216755770411*v^10 - 6055965732126021373400*v^8 - 100600434716506560273097*v^6 - 865142366320131816159886*v^4 - 5007028502862878323309932*v^2 - 1488079696206944407610000) / 32747172818821741170000 $$\beta_{12}$$ $$=$$ $$( - 13\!\cdots\!73 \nu^{15} + \cdots - 30\!\cdots\!00 \nu ) / 81\!\cdots\!00$$ (-139708174970895173*v^15 - 10345479671980724052*v^13 - 560339201436145870711*v^11 - 12745643623835228145400*v^9 - 210342767559956131671197*v^7 - 1761815992670299711407686*v^5 - 10401239765058335770397532*v^3 - 3091194738786765780610000*v) / 818679320470543529250000 $$\beta_{13}$$ $$=$$ $$( - 69\!\cdots\!11 \nu^{15} + \cdots - 15\!\cdots\!00 \nu ) / 81\!\cdots\!00$$ (-699207242144433911*v^15 - 51909972818273240664*v^13 - 2813437067898512043877*v^11 - 64263840733386752590300*v^9 - 1061656246809352152868679*v^7 - 8946142749733798407542702*v^5 - 52552200844460809781869524*v^3 - 15618270635381596228270000*v) / 818679320470543529250000 $$\beta_{14}$$ $$=$$ $$( 46\!\cdots\!33 \nu^{15} + \cdots + 10\!\cdots\!00 \nu ) / 37\!\cdots\!00$$ (46524892852974133*v^15 + 3445959372421567092*v^13 + 186630777641743075931*v^11 + 4247201545491490453400*v^9 + 70098339932879564514637*v^7 + 589712610118527035764906*v^5 + 3466603523571558801590172*v^3 + 1030256822982238427810000*v) / 37212696385024705875000 $$\beta_{15}$$ $$=$$ $$( - 560405796289847 \nu^{15} + \cdots - 58\!\cdots\!00 \nu ) / 27\!\cdots\!00$$ (-560405796289847*v^15 - 41256773136579303*v^13 - 2230852239373189429*v^11 - 50245648889503936225*v^9 - 826293053591415900083*v^7 - 6812770383049970560829*v^5 - 39835642226376958682898*v^3 - 582060611643123437500*v) / 270637791891088770000
 $$\nu$$ $$=$$ $$( \beta_{8} - 7\beta_{4} ) / 7$$ (b8 - 7*b4) / 7 $$\nu^{2}$$ $$=$$ $$( 2\beta_{11} + 7\beta_{7} + 4\beta_{6} - 2\beta_{5} + 4\beta_{3} + 133\beta_{2} - 133 ) / 7$$ (2*b11 + 7*b7 + 4*b6 - 2*b5 + 4*b3 + 133*b2 - 133) / 7 $$\nu^{3}$$ $$=$$ $$( 14\beta_{15} + 21\beta_{14} + 44\beta_{13} - 231\beta_{12} + 21\beta_{10} - 44\beta_{8} + 231\beta_{4} ) / 7$$ (14*b15 + 21*b14 + 44*b13 - 231*b12 + 21*b10 - 44*b8 + 231*b4) / 7 $$\nu^{4}$$ $$=$$ $$( -92\beta_{11} - 315\beta_{7} - 296\beta_{6} - 4599\beta_{2} + 315\beta_1 ) / 7$$ (-92*b11 - 315*b7 - 296*b6 - 4599*b2 + 315*b1) / 7 $$\nu^{5}$$ $$=$$ $$( -742\beta_{15} - 1295\beta_{14} - 2034\beta_{13} + 9373\beta_{12} - 742\beta_{9} ) / 7$$ (-742*b15 - 1295*b14 - 2034*b13 + 9373*b12 - 742*b9) / 7 $$\nu^{6}$$ $$=$$ $$( 4350\beta_{5} - 15364\beta_{3} - 14189\beta _1 + 191877 ) / 7$$ (4350*b5 - 15364*b3 - 14189*b1 + 191877) / 7 $$\nu^{7}$$ $$=$$ $$( -64631\beta_{10} + 35042\beta_{9} + 95558\beta_{8} - 414659\beta_{4} ) / 7$$ (-64631*b10 + 35042*b9 + 95558*b8 - 414659*b4) / 7 $$\nu^{8}$$ $$=$$ $$( 204408 \beta_{11} + 649047 \beta_{7} + 737424 \beta_{6} - 204408 \beta_{5} + 737424 \beta_{3} + 8599591 \beta_{2} - 8599591 ) / 7$$ (204408*b11 + 649047*b7 + 737424*b6 - 204408*b5 + 737424*b3 + 8599591*b2 - 8599591) / 7 $$\nu^{9}$$ $$=$$ $$( 1626702 \beta_{15} + 3065727 \beta_{14} + 4469278 \beta_{13} - 18937597 \beta_{12} + 3065727 \beta_{10} - 4469278 \beta_{8} + 18937597 \beta_{4} ) / 7$$ (1626702*b15 + 3065727*b14 + 4469278*b13 - 18937597*b12 + 3065727*b10 - 4469278*b8 + 18937597*b4) / 7 $$\nu^{10}$$ $$=$$ $$( -9543218\beta_{11} - 29949157\beta_{7} - 34602748\beta_{6} - 394769893\beta_{2} + 29949157\beta_1 ) / 7$$ (-9543218*b11 - 29949157*b7 - 34602748*b6 - 394769893*b2 + 29949157*b1) / 7 $$\nu^{11}$$ $$=$$ $$( -75414626\beta_{15} - 143300619\beta_{14} - 208198022\beta_{13} + 874415451\beta_{12} - 75414626\beta_{9} ) / 7$$ (-75414626*b15 - 143300619*b14 - 208198022*b13 + 874415451*b12 - 75414626*b9) / 7 $$\nu^{12}$$ $$=$$ $$( 444094580\beta_{5} - 1612220888\beta_{3} - 1387260567\beta _1 + 18262565559 ) / 7$$ (444094580*b5 - 1612220888*b3 - 1387260567*b1 + 18262565559) / 7 $$\nu^{13}$$ $$=$$ $$( -6668017811\beta_{10} + 3498552862\beta_{9} + 9679785270\beta_{8} - 40524590533\beta_{4} ) / 7$$ (-6668017811*b10 + 3498552862*b9 + 9679785270*b8 - 40524590533*b4) / 7 $$\nu^{14}$$ $$=$$ $$2948235786 \beta_{11} + 9193754963 \beta_{7} + 10706353996 \beta_{6} - 2948235786 \beta_{5} + 10706353996 \beta_{3} + 120992910771 \beta_{2} + \cdots - 120992910771$$ 2948235786*b11 + 9193754963*b7 + 10706353996*b6 - 2948235786*b5 + 10706353996*b3 + 120992910771*b2 - 120992910771 $$\nu^{15}$$ $$=$$ $$( 162381746450 \beta_{15} + 309827369159 \beta_{14} + 449677106846 \beta_{13} - 1880447860739 \beta_{12} + 309827369159 \beta_{10} - 449677106846 \beta_{8} + \cdots + 1880447860739 \beta_{4} ) / 7$$ (162381746450*b15 + 309827369159*b14 + 449677106846*b13 - 1880447860739*b12 + 309827369159*b10 - 449677106846*b8 + 1880447860739*b4) / 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −3.40706 + 5.90120i −1.99285 + 3.45171i −0.272818 + 0.472535i −1.68703 + 2.92202i 0.272818 − 0.472535i 1.68703 − 2.92202i 3.40706 − 5.90120i 1.99285 − 3.45171i −3.40706 − 5.90120i −1.99285 − 3.45171i −0.272818 − 0.472535i −1.68703 − 2.92202i 0.272818 + 0.472535i 1.68703 + 2.92202i 3.40706 + 5.90120i 1.99285 + 3.45171i
−2.69995 + 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 71.0573 0 −42.0566 72.8441i
226.2 −2.69995 + 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 71.0573 0 42.0566 + 72.8441i
226.3 −0.979925 + 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 −23.8298 0 −11.6510 20.1801i
226.4 −0.979925 + 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 −23.8298 0 11.6510 + 20.1801i
226.5 0.979925 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 23.8298 0 11.6510 + 20.1801i
226.6 0.979925 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 23.8298 0 −11.6510 20.1801i
226.7 2.69995 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 −71.0573 0 42.0566 + 72.8441i
226.8 2.69995 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 −71.0573 0 −42.0566 72.8441i
361.1 −2.69995 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 71.0573 0 −42.0566 + 72.8441i
361.2 −2.69995 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 71.0573 0 42.0566 72.8441i
361.3 −0.979925 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 −23.8298 0 −11.6510 + 20.1801i
361.4 −0.979925 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 −23.8298 0 11.6510 20.1801i
361.5 0.979925 + 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 23.8298 0 11.6510 20.1801i
361.6 0.979925 + 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 23.8298 0 −11.6510 + 20.1801i
361.7 2.69995 + 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 −71.0573 0 42.0566 72.8441i
361.8 2.69995 + 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 −71.0573 0 −42.0566 + 72.8441i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.z 16
3.b odd 2 1 inner 441.4.e.z 16
7.b odd 2 1 inner 441.4.e.z 16
7.c even 3 1 441.4.a.x 8
7.c even 3 1 inner 441.4.e.z 16
7.d odd 6 1 441.4.a.x 8
7.d odd 6 1 inner 441.4.e.z 16
21.c even 2 1 inner 441.4.e.z 16
21.g even 6 1 441.4.a.x 8
21.g even 6 1 inner 441.4.e.z 16
21.h odd 6 1 441.4.a.x 8
21.h odd 6 1 inner 441.4.e.z 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.x 8 7.c even 3 1
441.4.a.x 8 7.d odd 6 1
441.4.a.x 8 21.g even 6 1
441.4.a.x 8 21.h odd 6 1
441.4.e.z 16 1.a even 1 1 trivial
441.4.e.z 16 3.b odd 2 1 inner
441.4.e.z 16 7.b odd 2 1 inner
441.4.e.z 16 7.c even 3 1 inner
441.4.e.z 16 7.d odd 6 1 inner
441.4.e.z 16 21.c even 2 1 inner
441.4.e.z 16 21.g even 6 1 inner
441.4.e.z 16 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{8} + 33T_{2}^{6} + 977T_{2}^{4} + 3696T_{2}^{2} + 12544$$ T2^8 + 33*T2^6 + 977*T2^4 + 3696*T2^2 + 12544 $$T_{5}^{8} + 384T_{5}^{6} + 113156T_{5}^{4} + 13171200T_{5}^{2} + 1176490000$$ T5^8 + 384*T5^6 + 113156*T5^4 + 13171200*T5^2 + 1176490000 $$T_{13}^{4} - 5268T_{13}^{2} + 4900$$ T13^4 - 5268*T13^2 + 4900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 33 T^{6} + 977 T^{4} + \cdots + 12544)^{2}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 384 T^{6} + 113156 T^{4} + \cdots + 1176490000)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} + 2348 T^{6} + \cdots + 1836567040000)^{2}$$
$13$ $$(T^{4} - 5268 T^{2} + 4900)^{4}$$
$17$ $$(T^{8} + 10496 T^{6} + \cdots + 66171521929744)^{2}$$
$19$ $$(T^{8} + 23080 T^{6} + \cdots + 17\!\cdots\!16)^{2}$$
$23$ $$(T^{8} + 6012 T^{6} + \cdots + 81603616374784)^{2}$$
$29$ $$(T^{4} - 53088 T^{2} + 16601200)^{4}$$
$31$ $$(T^{8} + 19464 T^{6} + \cdots + 26\!\cdots\!76)^{2}$$
$37$ $$(T^{4} + 16 T^{3} + 92496 T^{2} + \cdots + 8508217600)^{4}$$
$41$ $$(T^{4} - 233264 T^{2} + \cdots + 10038958300)^{4}$$
$43$ $$(T^{2} - 540 T + 8800)^{8}$$
$47$ $$(T^{8} + 335128 T^{6} + \cdots + 36\!\cdots\!44)^{2}$$
$53$ $$(T^{8} + 133376 T^{6} + \cdots + 19\!\cdots\!24)^{2}$$
$59$ $$(T^{8} + 231096 T^{6} + \cdots + 11\!\cdots\!00)^{2}$$
$61$ $$(T^{8} + 745844 T^{6} + \cdots + 15\!\cdots\!56)^{2}$$
$67$ $$(T^{4} + 256400 T^{2} + \cdots + 65740960000)^{4}$$
$71$ $$(T^{4} - 959388 T^{2} + \cdots + 187904819200)^{4}$$
$73$ $$(T^{8} + 535668 T^{6} + \cdots + 30\!\cdots\!00)^{2}$$
$79$ $$(T^{4} + 1348 T^{3} + \cdots + 108004249600)^{4}$$
$83$ $$(T^{4} - 1919232 T^{2} + \cdots + 351934815232)^{4}$$
$89$ $$(T^{8} + 231776 T^{6} + \cdots + 17\!\cdots\!00)^{2}$$
$97$ $$(T^{4} - 1382388 T^{2} + \cdots + 35588822500)^{4}$$