# Properties

 Label 33.4.e.b Level $33$ Weight $4$ Character orbit 33.e Analytic conductor $1.947$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,Mod(4,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

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("33.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (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: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.682515625.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121$$ x^8 - 3*x^7 + 5*x^6 + 2*x^5 + 19*x^4 + 28*x^3 + 100*x^2 + 88*x + 121 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - \beta_1) q^{2}+ \cdots - 9 \beta_{7} q^{9}+O(q^{10})$$ q + (b6 + b5 + b3 - 2*b2 - b1) * q^2 + 3*b3 * q^3 + (-4*b7 - 3*b6 + 3*b4 + 4*b3 - 2*b2 + 2) * q^4 + (-2*b6 + 3*b5 + 4*b3 - 7*b2 + 4) * q^5 + (-3*b6 - 3*b3 + 3*b2 - 3) * q^6 + (6*b6 - 6*b4 - 4*b2 - 15*b1 + 4) * q^7 + (-2*b7 - 8*b6 - 2*b5 - 2*b4 - 5*b3 + 2*b2 + 8*b1 + 2) * q^8 - 9*b7 * q^9 $$q + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - \beta_1) q^{2}+ \cdots + (198 \beta_{7} - 18 \beta_{6} + \cdots - 99) q^{99}+O(q^{100})$$ q + (b6 + b5 + b3 - 2*b2 - b1) * q^2 + 3*b3 * q^3 + (-4*b7 - 3*b6 + 3*b4 + 4*b3 - 2*b2 + 2) * q^4 + (-2*b6 + 3*b5 + 4*b3 - 7*b2 + 4) * q^5 + (-3*b6 - 3*b3 + 3*b2 - 3) * q^6 + (6*b6 - 6*b4 - 4*b2 - 15*b1 + 4) * q^7 + (-2*b7 - 8*b6 - 2*b5 - 2*b4 - 5*b3 + 2*b2 + 8*b1 + 2) * q^8 - 9*b7 * q^9 + (11*b7 + 6*b4 + 11*b2 - 3) * q^10 + (-11*b7 + 14*b6 - 4*b5 - 2*b4 + 11*b3 + 4*b2 + 3*b1 - 11) * q^11 + (-6*b7 - 9*b5 - 9*b4 + 3*b2 + 9*b1 - 6) * q^12 + (46*b7 + 18*b6 + 18*b5 + 15*b4 - 25*b3 + 7*b2 - 3*b1) * q^13 + (13*b7 - 14*b6 - 2*b5 - 2*b4 - 35*b3 + 2*b2 + 14*b1 - 13) * q^14 + (-9*b6 + 9*b4 + 12*b2 + 15*b1 - 12) * q^15 + (3*b6 - 18*b5 + 17*b3 - 23*b2 + 17) * q^16 + (25*b6 - 11*b3 + 26*b2 - 11) * q^17 + (-9*b2 + 9*b1 + 9) * q^18 + (-34*b7 + 15*b6 - 27*b5 - 27*b4 - 27*b3 + 27*b2 - 15*b1 + 34) * q^19 + (-29*b7 - 31*b6 - 31*b5 + b4 + 23*b3 + 8*b2 + 32*b1) * q^20 + (12*b7 + 18*b5 - 27*b4 - 6*b2 - 18*b1 - 12) * q^21 + (11*b7 - 12*b6 + 27*b5 + 30*b4 - 11*b3 - 16*b2 - 12*b1 - 55) * q^22 + (51*b7 + 39*b5 - 18*b4 + 12*b2 - 39*b1 - 21) * q^23 + (15*b7 + 6*b6 + 6*b5 + 24*b4 + 6*b3 - 12*b2 + 18*b1) * q^24 + (-34*b7 - 3*b6 - 28*b3 + 3*b1 + 34) * q^25 + (-b7 - 11*b6 + 11*b4 + b3 + 83*b2 - 76*b1 - 83) * q^26 - 27*b2 * q^27 + (-18*b6 + 39*b5 + 98*b3 - 102*b2 + 98) * q^28 + (-169*b7 - 26*b6 + 26*b4 + 169*b3 - 181*b2 + 29*b1 + 181) * q^29 + (-33*b7 - 18*b5 - 18*b4 + 24*b3 + 18*b2 + 33) * q^30 + (49*b7 - 12*b6 - 12*b5 - 27*b4 + 20*b3 - 8*b2 - 15*b1) * q^31 + (-26*b7 - 41*b5 - 5*b4 + 15*b2 + 41*b1 + 4) * q^32 + (-33*b7 + 12*b6 + 6*b5 + 3*b4 - 33*b3 - 39*b2 - 54*b1) * q^33 + (-b7 + 39*b5 + 24*b4 - 40*b2 - 39*b1 - 63) * q^34 + (238*b7 - 29*b6 - 29*b5 - 8*b4 - 119*b3 + 148*b2 + 21*b1) * q^35 + (18*b7 + 27*b6 + 27*b5 + 27*b4 - 36*b3 - 27*b2 - 27*b1 - 18) * q^36 + (-12*b7 - 21*b6 + 21*b4 + 12*b3 + 161*b2 + 117*b1 - 161) * q^37 + (85*b6 + 88*b5 + 43*b3 - 276*b2 + 43) * q^38 + (-54*b6 - 45*b5 + 75*b3 + 108*b2 + 75) * q^39 + (-136*b7 + 39*b6 - 39*b4 + 136*b3 - 82*b2 - 21*b1 + 82) * q^40 + (-99*b7 + 97*b6 + 114*b5 + 114*b4 - 2*b3 - 114*b2 - 97*b1 + 99) * q^41 + (105*b7 + 6*b6 + 6*b5 + 42*b4 - 39*b3 + 33*b2 + 36*b1) * q^42 + (21*b7 - 135*b5 - 66*b4 + 156*b2 + 135*b1 - 175) * q^43 + (44*b7 + 8*b6 - 51*b5 - 97*b4 - 286*b3 + 282*b2 - 36*b1 - 132) * q^44 + (-36*b7 - 27*b5 + 18*b4 - 9*b2 + 27*b1 + 36) * q^45 + (81*b7 - 63*b6 - 63*b5 - 15*b4 - 120*b3 + 183*b2 + 48*b1) * q^46 + (177*b7 - 20*b6 - 87*b5 - 87*b4 - 215*b3 + 87*b2 + 20*b1 - 177) * q^47 + (72*b7 + 54*b6 - 54*b4 - 72*b3 + 123*b2 - 63*b1 - 123) * q^48 + (-51*b6 + 45*b5 - 299*b3 - 58*b2 - 299) * q^49 + (68*b6 + 34*b5 + 71*b3 - 133*b2 + 71) * q^50 + (-45*b7 + 45*b3 - 78*b2 - 75*b1 + 78) * q^51 + (-403*b7 - 138*b6 - 135*b5 - 135*b4 + 45*b3 + 135*b2 + 138*b1 + 403) * q^52 + (33*b7 - 38*b6 - 38*b5 + 15*b4 + 172*b3 - 134*b2 + 53*b1) * q^53 + (27*b7 + 27*b4 + 27*b2 - 27) * q^54 + (66*b7 + 30*b6 + 15*b5 + 123*b4 - 253*b3 + 18*b2 + 63*b1 - 176) * q^55 + (266*b7 - 50*b5 + 89*b4 + 316*b2 + 50*b1 + 48) * q^56 + (81*b7 + 81*b6 + 81*b5 - 45*b4 + 102*b3 - 183*b2 - 126*b1) * q^57 + (106*b7 + 18*b6 + 135*b5 + 135*b4 + 73*b3 - 135*b2 - 18*b1 - 106) * q^58 + (-433*b7 - 68*b6 + 68*b4 + 433*b3 - 130*b2 + 74*b1 + 130) * q^59 + (93*b6 - 3*b5 - 69*b3 - 15*b2 - 69) * q^60 + (21*b6 + 3*b5 + 238*b3 - 12*b2 + 238) * q^61 + (38*b7 + 4*b6 - 4*b4 - 38*b3 + 14*b2 + 5*b1 - 14) * q^62 + (-36*b7 - 54*b6 + 81*b5 + 81*b4 - 81*b2 + 54*b1 + 36) * q^63 + (151*b7 + 120*b6 + 120*b5 - 108*b4 + 268*b3 - 388*b2 - 228*b1) * q^64 + (30*b7 + 269*b5 + 3*b4 - 239*b2 - 269*b1 + 1) * q^65 + (-81*b6 - 90*b5 - 45*b4 - 132*b3 + 90*b2 + 117*b1 + 33) * q^66 + (186*b7 - 3*b5 + 33*b4 + 189*b2 + 3*b1 - 151) * q^67 + (-113*b7 + 11*b6 + 11*b5 - 47*b4 - 292*b3 + 281*b2 - 58*b1) * q^68 + (-153*b7 - 117*b6 + 54*b5 + 54*b4 + 90*b3 - 54*b2 + 117*b1 + 153) * q^69 + (-190*b7 + 177*b6 - 177*b4 + 190*b3 - 124*b2 - 222*b1 + 124) * q^70 + (-161*b6 - 369*b5 + 190*b3 - 10*b2 + 190) * q^71 + (-18*b6 - 72*b5 - 18*b3 + 135*b2 - 18) * q^72 + (-429*b7 + 18*b6 - 18*b4 + 429*b3 - 679*b2 + 39*b1 + 679) * q^73 + (-128*b7 + 19*b6 - 11*b5 - 11*b4 + 157*b3 + 11*b2 - 19*b1 + 128) * q^74 + (84*b7 + 9*b4 + 102*b3 - 102*b2 + 9*b1) * q^75 + (265*b7 + 333*b5 + 318*b4 - 68*b2 - 333*b1 - 171) * q^76 + (-660*b7 - 105*b6 + 184*b5 - 18*b4 + 231*b3 - 228*b2 + 49*b1 - 77) * q^77 + (-252*b7 - 33*b5 - 261*b4 - 219*b2 + 33*b1 + 249) * q^78 + (667*b7 - 99*b6 - 99*b5 + 45*b4 - 307*b3 + 406*b2 + 144*b1) * q^79 + (401*b7 - 10*b6 - 52*b5 - 52*b4 - 310*b3 + 52*b2 + 10*b1 - 401) * q^80 + (81*b7 - 81*b3 + 81*b2 - 81) * q^81 + (-321*b6 - 129*b5 - 304*b3 + 482*b2 - 304) * q^82 + (-167*b6 - 27*b5 - 491*b3 + 377*b2 - 491) * q^83 + (-105*b7 - 117*b6 + 117*b4 + 105*b3 + 189*b2 + 171*b1 - 189) * q^84 + (71*b7 + 27*b6 + 144*b5 + 144*b4 - 344*b3 - 144*b2 - 27*b1 - 71) * q^85 + (-93*b7 + 227*b6 + 227*b5 + 111*b4 + 296*b3 - 523*b2 - 116*b1) * q^86 + (36*b7 - 78*b5 + 9*b4 + 114*b2 + 78*b1 - 543) * q^87 + (231*b7 + 84*b6 - 90*b5 + 21*b4 - 209*b3 - 53*b2 + 216*b1 + 176) * q^88 + (495*b7 - 345*b5 - 282*b4 + 840*b2 + 345*b1 - 267) * q^89 + (-72*b7 + 54*b6 + 54*b5 + 99*b3 - 153*b2 - 54*b1) * q^90 + (-339*b7 + 501*b6 - 396*b5 - 396*b4 - 82*b3 + 396*b2 - 501*b1 + 339) * q^91 + (-519*b7 - 66*b6 + 66*b4 + 519*b3 - 258*b2 + 405*b1 + 258) * q^92 + (36*b6 + 81*b5 - 60*b3 + 126*b2 - 60) * q^93 + (252*b6 - 3*b5 + 185*b3 - 226*b2 + 185) * q^94 + (473*b7 + 22*b6 - 22*b4 - 473*b3 - 140*b2 + 63*b1 + 140) * q^95 + (78*b7 + 123*b6 + 15*b5 + 15*b4 - 66*b3 - 15*b2 - 123*b1 - 78) * q^96 + (-374*b7 + 6*b6 + 6*b5 + 177*b4 - 148*b3 + 142*b2 + 171*b1) * q^97 + (97*b7 - 401*b5 + b4 + 498*b2 + 401*b1 + 445) * q^98 + (198*b7 - 18*b6 - 9*b5 - 153*b4 - 99*b3 + 9*b2 - 18*b1 - 99) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{2} - 6 q^{3} - 16 q^{4} + 9 q^{5} - 18 q^{6} + 3 q^{7} + 36 q^{8} - 18 q^{9}+O(q^{10})$$ 8 * q - 6 * q^2 - 6 * q^3 - 16 * q^4 + 9 * q^5 - 18 * q^6 + 3 * q^7 + 36 * q^8 - 18 * q^9 $$8 q - 6 q^{2} - 6 q^{3} - 16 q^{4} + 9 q^{5} - 18 q^{6} + 3 q^{7} + 36 q^{8} - 18 q^{9} + 8 q^{10} - 87 q^{11} - 18 q^{12} + 171 q^{13} + 12 q^{14} - 63 q^{15} + 44 q^{16} + 36 q^{17} + 81 q^{18} + 324 q^{19} - 87 q^{20} - 66 q^{21} - 521 q^{22} - 84 q^{23} + 18 q^{24} + 263 q^{25} - 774 q^{26} - 54 q^{27} + 387 q^{28} + 393 q^{29} + 204 q^{30} + 15 q^{31} + 102 q^{32} - 216 q^{33} - 712 q^{34} + 1002 q^{35} - 144 q^{36} - 747 q^{37} - 36 q^{38} + 513 q^{39} + 41 q^{40} + 159 q^{41} + 396 q^{42} - 644 q^{43} + 219 q^{44} + 216 q^{45} + 753 q^{46} - 351 q^{47} - 423 q^{48} - 1967 q^{49} + 330 q^{50} + 63 q^{51} + 2871 q^{52} - 531 q^{53} - 162 q^{54} - 716 q^{55} + 1470 q^{56} - 453 q^{57} - 1205 q^{58} - 1002 q^{59} - 261 q^{60} + 1449 q^{61} + 99 q^{62} + 27 q^{63} - 1118 q^{64} - 954 q^{65} + 897 q^{66} - 518 q^{67} + 873 q^{68} + 693 q^{69} + 26 q^{70} + 429 q^{71} + 54 q^{72} + 2547 q^{73} + 468 q^{74} - 231 q^{75} - 2276 q^{76} - 2697 q^{77} + 1638 q^{78} + 2805 q^{79} - 1620 q^{80} - 162 q^{81} - 1631 q^{82} - 2553 q^{83} - 1509 q^{84} - 197 q^{85} - 1713 q^{86} - 3906 q^{87} + 2866 q^{88} + 1788 q^{89} - 648 q^{90} + 2885 q^{91} + 423 q^{92} + 45 q^{93} + 1159 q^{94} + 3009 q^{95} - 504 q^{96} + 9 q^{97} + 5550 q^{98} + 27 q^{99}+O(q^{100})$$ 8 * q - 6 * q^2 - 6 * q^3 - 16 * q^4 + 9 * q^5 - 18 * q^6 + 3 * q^7 + 36 * q^8 - 18 * q^9 + 8 * q^10 - 87 * q^11 - 18 * q^12 + 171 * q^13 + 12 * q^14 - 63 * q^15 + 44 * q^16 + 36 * q^17 + 81 * q^18 + 324 * q^19 - 87 * q^20 - 66 * q^21 - 521 * q^22 - 84 * q^23 + 18 * q^24 + 263 * q^25 - 774 * q^26 - 54 * q^27 + 387 * q^28 + 393 * q^29 + 204 * q^30 + 15 * q^31 + 102 * q^32 - 216 * q^33 - 712 * q^34 + 1002 * q^35 - 144 * q^36 - 747 * q^37 - 36 * q^38 + 513 * q^39 + 41 * q^40 + 159 * q^41 + 396 * q^42 - 644 * q^43 + 219 * q^44 + 216 * q^45 + 753 * q^46 - 351 * q^47 - 423 * q^48 - 1967 * q^49 + 330 * q^50 + 63 * q^51 + 2871 * q^52 - 531 * q^53 - 162 * q^54 - 716 * q^55 + 1470 * q^56 - 453 * q^57 - 1205 * q^58 - 1002 * q^59 - 261 * q^60 + 1449 * q^61 + 99 * q^62 + 27 * q^63 - 1118 * q^64 - 954 * q^65 + 897 * q^66 - 518 * q^67 + 873 * q^68 + 693 * q^69 + 26 * q^70 + 429 * q^71 + 54 * q^72 + 2547 * q^73 + 468 * q^74 - 231 * q^75 - 2276 * q^76 - 2697 * q^77 + 1638 * q^78 + 2805 * q^79 - 1620 * q^80 - 162 * q^81 - 1631 * q^82 - 2553 * q^83 - 1509 * q^84 - 197 * q^85 - 1713 * q^86 - 3906 * q^87 + 2866 * q^88 + 1788 * q^89 - 648 * q^90 + 2885 * q^91 + 423 * q^92 + 45 * q^93 + 1159 * q^94 + 3009 * q^95 - 504 * q^96 + 9 * q^97 + 5550 * q^98 + 27 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + \cdots + 440220 ) / 1168519$$ (528*v^7 + 2098*v^6 - 15725*v^5 + 33439*v^4 + 71401*v^3 - 332708*v^2 + 319181*v + 440220) / 1168519 $$\beta_{3}$$ $$=$$ $$( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + \cdots + 528473 ) / 1168519$$ (5794*v^7 - 9973*v^6 - 30517*v^5 + 195125*v^4 - 61888*v^3 + 104068*v^2 + 501961*v + 528473) / 1168519 $$\beta_{4}$$ $$=$$ $$( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + \cdots - 701074 ) / 1168519$$ (7409*v^7 - 59487*v^6 + 183537*v^5 - 171974*v^4 - 58164*v^3 - 77439*v^2 + 18601*v - 701074) / 1168519 $$\beta_{5}$$ $$=$$ $$( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + \cdots + 2809884 ) / 1168519$$ (8817*v^7 + 16927*v^6 - 106264*v^5 + 200474*v^4 + 521745*v^3 + 380907*v^2 + 2179908*v + 2809884) / 1168519 $$\beta_{6}$$ $$=$$ $$( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} + \cdots - 2305776 ) / 1168519$$ (-11971*v^7 + 3536*v^6 + 58156*v^5 - 228404*v^4 - 102852*v^3 - 979996*v^2 - 1085964*v - 2305776) / 1168519 $$\beta_{7}$$ $$=$$ $$( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} + \cdots + 665808 ) / 1168519$$ (-13790*v^7 + 57068*v^6 - 113608*v^5 + 65418*v^4 - 266949*v^3 + 6060*v^2 - 742824*v + 665808) / 1168519
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1$$ b6 + b5 + b3 - 5*b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1$$ b7 + 6*b6 + 6*b5 + 2*b4 + 4*b3 - 10*b2 - 4*b1 $$\nu^{4}$$ $$=$$ $$12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12$$ 12*b7 + 10*b6 + 13*b5 + 13*b4 + 14*b3 - 13*b2 - 10*b1 - 12 $$\nu^{5}$$ $$=$$ $$43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62$$ 43*b7 + 25*b5 + 49*b4 + 18*b2 - 25*b1 - 62 $$\nu^{6}$$ $$=$$ $$97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221$$ 97*b7 - 92*b6 + 92*b4 - 97*b3 + 221*b2 - 44*b1 - 221 $$\nu^{7}$$ $$=$$ $$-449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412$$ -449*b6 - 260*b5 - 412*b3 + 896*b2 - 412

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1 + \beta_{2} - \beta_{3} + \beta_{7}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.581882 + 1.79085i −0.390899 − 1.20306i 2.51217 − 1.82520i −1.20316 + 0.874145i 0.581882 − 1.79085i −0.390899 + 1.20306i 2.51217 + 1.82520i −1.20316 − 0.874145i
−2.02339 1.47008i 0.927051 2.85317i −0.539165 1.65938i −8.44146 + 6.13308i −6.07016 + 4.41023i −10.1220 31.1524i −7.53140 + 23.1793i −7.28115 5.29007i 26.0964
4.2 0.523388 + 0.380264i 0.927051 2.85317i −2.34280 7.21040i 9.01441 6.54935i 1.57016 1.14079i 8.07696 + 24.8583i 3.11499 9.58696i −7.28115 5.29007i 7.20851
16.1 −1.45957 4.49208i −2.42705 1.76336i −11.5763 + 8.41069i 1.86000 5.72450i −4.37870 + 13.4762i −8.05785 + 5.85437i 24.1083 + 17.5157i 2.78115 + 8.55951i −28.4297
16.2 −0.0404346 0.124445i −2.42705 1.76336i 6.45828 4.69222i 2.06705 6.36172i −0.121304 + 0.373335i 11.6029 8.43002i −1.69194 1.22926i 2.78115 + 8.55951i −0.875265
25.1 −2.02339 + 1.47008i 0.927051 + 2.85317i −0.539165 + 1.65938i −8.44146 6.13308i −6.07016 4.41023i −10.1220 + 31.1524i −7.53140 23.1793i −7.28115 + 5.29007i 26.0964
25.2 0.523388 0.380264i 0.927051 + 2.85317i −2.34280 + 7.21040i 9.01441 + 6.54935i 1.57016 + 1.14079i 8.07696 24.8583i 3.11499 + 9.58696i −7.28115 + 5.29007i 7.20851
31.1 −1.45957 + 4.49208i −2.42705 + 1.76336i −11.5763 8.41069i 1.86000 + 5.72450i −4.37870 13.4762i −8.05785 5.85437i 24.1083 17.5157i 2.78115 8.55951i −28.4297
31.2 −0.0404346 + 0.124445i −2.42705 + 1.76336i 6.45828 + 4.69222i 2.06705 + 6.36172i −0.121304 0.373335i 11.6029 + 8.43002i −1.69194 + 1.22926i 2.78115 8.55951i −0.875265
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.2 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 33.4.e.b 8
3.b odd 2 1 99.4.f.b 8
11.c even 5 1 inner 33.4.e.b 8
11.c even 5 1 363.4.a.p 4
11.d odd 10 1 363.4.a.t 4
33.f even 10 1 1089.4.a.z 4
33.h odd 10 1 99.4.f.b 8
33.h odd 10 1 1089.4.a.bg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.b 8 1.a even 1 1 trivial
33.4.e.b 8 11.c even 5 1 inner
99.4.f.b 8 3.b odd 2 1
99.4.f.b 8 33.h odd 10 1
363.4.a.p 4 11.c even 5 1
363.4.a.t 4 11.d odd 10 1
1089.4.a.z 4 33.f even 10 1
1089.4.a.bg 4 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 6T_{2}^{7} + 34T_{2}^{6} + 72T_{2}^{5} + 49T_{2}^{4} - 96T_{2}^{3} + 51T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 6 T^{7} + \cdots + 1$$
$3$ $$(T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2}$$
$5$ $$T^{8} - 9 T^{7} + \cdots + 21911761$$
$7$ $$T^{8} + \cdots + 14957045401$$
$11$ $$T^{8} + \cdots + 3138428376721$$
$13$ $$T^{8} + \cdots + 153370490192656$$
$17$ $$T^{8} + \cdots + 16499324068096$$
$19$ $$T^{8} + \cdots + 23\!\cdots\!96$$
$23$ $$(T^{4} + 42 T^{3} + \cdots - 46471644)^{2}$$
$29$ $$T^{8} + \cdots + 26\!\cdots\!16$$
$31$ $$T^{8} + \cdots + 2860289355121$$
$37$ $$T^{8} + \cdots + 19\!\cdots\!16$$
$41$ $$T^{8} + \cdots + 11\!\cdots\!36$$
$43$ $$(T^{4} + 322 T^{3} + \cdots + 5520039844)^{2}$$
$47$ $$T^{8} + \cdots + 20\!\cdots\!16$$
$53$ $$T^{8} + \cdots + 27\!\cdots\!21$$
$59$ $$T^{8} + \cdots + 36\!\cdots\!61$$
$61$ $$T^{8} + \cdots + 16\!\cdots\!16$$
$67$ $$(T^{4} + 259 T^{3} + \cdots + 1798706704)^{2}$$
$71$ $$T^{8} + \cdots + 12\!\cdots\!76$$
$73$ $$T^{8} + \cdots + 27\!\cdots\!56$$
$79$ $$T^{8} + \cdots + 11\!\cdots\!81$$
$83$ $$T^{8} + \cdots + 82\!\cdots\!21$$
$89$ $$(T^{4} - 894 T^{3} + \cdots - 245710544796)^{2}$$
$97$ $$T^{8} + \cdots + 98\!\cdots\!81$$