# Properties

 Label 33.3.h.a Level $33$ Weight $3$ Character orbit 33.h Analytic conductor $0.899$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,3,Mod(5,33)]

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([5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("33.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.h (of order $$10$$, 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: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{3} - 3 \zeta_{20}^{2} q^{4} + (5 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 5 \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{6} + ( - 3 \zeta_{20}^{4} + 7 \zeta_{20}^{2} - 3) q^{7} - 7 \zeta_{20}^{3} q^{8} + ( - 8 \zeta_{20}^{7} + \zeta_{20}^{6} + 8 \zeta_{20}^{5} + 4 \zeta_{20}) q^{9} +O(q^{10})$$ q + z * q^2 + (-2*z^6 + 2*z^5 + 2*z^4 - z^3 - 2*z^2 + 2*z + 2) * q^3 - 3*z^2 * q^4 + (5*z^7 - 6*z^5 + 6*z^3 - 5*z) * q^5 + (-2*z^7 + 2*z^6 + 2*z^5 - z^4 - 2*z^3 + 2*z^2 + 2*z) * q^6 + (-3*z^4 + 7*z^2 - 3) * q^7 - 7*z^3 * q^8 + (-8*z^7 + z^6 + 8*z^5 + 4*z) * q^9 $$q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{3} - 3 \zeta_{20}^{2} q^{4} + (5 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 5 \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{6} + ( - 3 \zeta_{20}^{4} + 7 \zeta_{20}^{2} - 3) q^{7} - 7 \zeta_{20}^{3} q^{8} + ( - 8 \zeta_{20}^{7} + \zeta_{20}^{6} + 8 \zeta_{20}^{5} + 4 \zeta_{20}) q^{9} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} - 5) q^{10} + (2 \zeta_{20}^{7} - 9 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 12 \zeta_{20}) q^{11} + ( - 6 \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 6 \zeta_{20}^{3} - 6) q^{12} + (10 \zeta_{20}^{6} - 2 \zeta_{20}^{2} + 2) q^{13} + ( - 3 \zeta_{20}^{5} + 7 \zeta_{20}^{3} - 3 \zeta_{20}) q^{14} + (10 \zeta_{20}^{7} - 11 \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 8 \zeta_{20}^{2} + 2 \zeta_{20} - 11) q^{15} + 5 \zeta_{20}^{4} q^{16} + ( - 6 \zeta_{20}^{7} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{3} + 6 \zeta_{20}) q^{17} + (\zeta_{20}^{7} + 8 \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 8) q^{18} + ( - 20 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 20) q^{19} + (3 \zeta_{20}^{7} - 3 \zeta_{20}^{5} + 15 \zeta_{20}) q^{20} + (11 \zeta_{20}^{7} + 6 \zeta_{20}^{6} - 13 \zeta_{20}^{5} - 6 \zeta_{20}^{4} + 11 \zeta_{20}^{3} + 8) q^{21} + ( - 7 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 10 \zeta_{20}^{2} - 2) q^{22} + (4 \zeta_{20}^{7} + 20 \zeta_{20}^{5} + 4 \zeta_{20}^{3}) q^{23} + ( - 7 \zeta_{20}^{6} - 14 \zeta_{20}^{2} - 14 \zeta_{20} + 14) q^{24} + ( - \zeta_{20}^{6} + 12 \zeta_{20}^{4} - 12 \zeta_{20}^{2} + 1) q^{25} + (10 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{26} + ( - 7 \zeta_{20}^{7} - 7 \zeta_{20}^{5} + 22 \zeta_{20}^{4} + 7 \zeta_{20}^{3} + 7 \zeta_{20}) q^{27} + (9 \zeta_{20}^{6} - 21 \zeta_{20}^{4} + 9 \zeta_{20}^{2}) q^{28} + ( - 27 \zeta_{20}^{7} - 33 \zeta_{20}^{3} + 33 \zeta_{20}) q^{29} + (10 \zeta_{20}^{6} - 11 \zeta_{20}^{5} - 12 \zeta_{20}^{4} + 8 \zeta_{20}^{3} + 12 \zeta_{20}^{2} + \cdots - 10) q^{30} + \cdots + ( - 6 \zeta_{20}^{7} + 20 \zeta_{20}^{6} - 6 \zeta_{20}^{5} - 112 \zeta_{20}^{4} + 4 \zeta_{20}^{3} + \cdots - 32) q^{99} +O(q^{100})$$ q + z * q^2 + (-2*z^6 + 2*z^5 + 2*z^4 - z^3 - 2*z^2 + 2*z + 2) * q^3 - 3*z^2 * q^4 + (5*z^7 - 6*z^5 + 6*z^3 - 5*z) * q^5 + (-2*z^7 + 2*z^6 + 2*z^5 - z^4 - 2*z^3 + 2*z^2 + 2*z) * q^6 + (-3*z^4 + 7*z^2 - 3) * q^7 - 7*z^3 * q^8 + (-8*z^7 + z^6 + 8*z^5 + 4*z) * q^9 + (-z^6 + z^4 - 5) * q^10 + (2*z^7 - 9*z^5 + 6*z^3 - 12*z) * q^11 + (-6*z^7 + 3*z^5 - 6*z^3 - 6) * q^12 + (10*z^6 - 2*z^2 + 2) * q^13 + (-3*z^5 + 7*z^3 - 3*z) * q^14 + (10*z^7 - 11*z^4 - 2*z^3 + 8*z^2 + 2*z - 11) * q^15 + 5*z^4 * q^16 + (-6*z^7 + 6*z^5 - 6*z^3 + 6*z) * q^17 + (z^7 + 8*z^4 - 4*z^2 + 8) * q^18 + (-20*z^6 + 2*z^4 - 2*z^2 + 20) * q^19 + (3*z^7 - 3*z^5 + 15*z) * q^20 + (11*z^7 + 6*z^6 - 13*z^5 - 6*z^4 + 11*z^3 + 8) * q^21 + (-7*z^6 + 4*z^4 - 10*z^2 - 2) * q^22 + (4*z^7 + 20*z^5 + 4*z^3) * q^23 + (-7*z^6 - 14*z^2 - 14*z + 14) * q^24 + (-z^6 + 12*z^4 - 12*z^2 + 1) * q^25 + (10*z^7 - 2*z^3 + 2*z) * q^26 + (-7*z^7 - 7*z^5 + 22*z^4 + 7*z^3 + 7*z) * q^27 + (9*z^6 - 21*z^4 + 9*z^2) * q^28 + (-27*z^7 - 33*z^3 + 33*z) * q^29 + (10*z^6 - 11*z^5 - 12*z^4 + 8*z^3 + 12*z^2 - 11*z - 10) * q^30 + (24*z^6 + 11*z^2 - 11) * q^31 + 33*z^5 * q^32 + (24*z^7 - 23*z^6 - 20*z^5 - z^4 + 6*z^3 - 3*z^2 - 12*z - 5) * q^33 + 6 * q^34 + (-22*z^7 + 22*z^5 - 23*z) * q^35 + (-3*z^6 - 24*z^5 + 3*z^4 + 12*z^3 - 3*z^2 - 24*z + 3) * q^36 + (-22*z^4 + 6*z^2 - 22) * q^37 + (-20*z^7 + 2*z^5 - 2*z^3 + 20*z) * q^38 + (6*z^7 - 4*z^6 + 16*z^5 + 24*z^4 - 16*z^3 - 4*z^2 - 6*z) * q^39 + (-7*z^4 + 42*z^2 - 7) * q^40 + (-22*z^5 - 8*z^3 - 22*z) * q^41 + (6*z^7 - 2*z^6 - 6*z^5 + 11*z^2 + 8*z - 11) * q^42 + (6*z^6 - 6*z^4 - 36) * q^43 + (21*z^7 - 12*z^5 + 30*z^3 + 6*z) * q^44 + (z^7 + 44*z^6 - 6*z^5 - 44*z^4 + z^3 - 12) * q^45 + (24*z^6 + 4*z^2 - 4) * q^46 + (6*z^5 - 14*z^3 + 6*z) * q^47 + (5*z^7 + 10*z^3 + 10*z^2 - 10*z) * q^48 + (-33*z^6 + 9*z^4 - 33*z^2) * q^49 + (-z^7 + 12*z^5 - 12*z^3 + z) * q^50 + (-12*z^7 + 12*z^4 - 6*z^2 + 12) * q^51 + (-30*z^6 + 36*z^4 - 36*z^2 + 30) * q^52 + (-15*z^7 + 15*z^5 + 2*z) * q^53 + (-14*z^6 + 22*z^5 + 14*z^4 + 7) * q^54 + (-7*z^6 + 48*z^4 - 43*z^2 + 64) * q^55 + (21*z^7 - 49*z^5 + 21*z^3) * q^56 + (-22*z^7 - 40*z^6 + 22*z^5 - 36*z^2 + 56*z + 36) * q^57 + (-27*z^6 - 6*z^4 + 6*z^2 + 27) * q^58 + (30*z^7 + 5*z^3 - 5*z) * q^59 + (-30*z^7 + 33*z^6 + 36*z^5 - 24*z^4 - 36*z^3 + 33*z^2 + 30*z) * q^60 + (54*z^6 - 60*z^4 + 54*z^2) * q^61 + (24*z^7 + 11*z^3 - 11*z) * q^62 + (4*z^6 + 44*z^5 - 7*z^4 - 52*z^3 + 7*z^2 + 44*z - 4) * q^63 + 13*z^6 * q^64 + (22*z^7 - 74*z^5 + 22*z^3) * q^65 + (-23*z^7 + 4*z^6 - z^5 - 18*z^4 - 3*z^3 + 12*z^2 - 5*z - 24) * q^66 + (-66*z^6 + 66*z^4 + 30) * q^67 - 18*z * q^68 + (32*z^6 + 8*z^5 + 12*z^4 + 40*z^3 - 12*z^2 + 8*z - 32) * q^69 + (22*z^4 - 45*z^2 + 22) * q^70 + (82*z^7 - 72*z^5 + 72*z^3 - 82*z) * q^71 + (-7*z^7 - 56*z^6 + 7*z^5 + 28*z^4 - 7*z^3 - 56*z^2 + 7*z) * q^72 + (-21*z^4 - 28*z^2 - 21) * q^73 + (-22*z^5 + 6*z^3 - 22*z) * q^74 + (-13*z^7 - 2*z^6 + 13*z^5 + 22*z^2 - 21*z - 22) * q^75 + (54*z^6 - 54*z^4 - 60) * q^76 + (-46*z^7 + 64*z^5 - 61*z^3 + z) * q^77 + (-4*z^7 + 22*z^6 + 24*z^5 - 22*z^4 - 4*z^3 - 6) * q^78 + (13*z^6 + 81*z^2 - 81) * q^79 + (5*z^5 - 30*z^3 + 5*z) * q^80 + (8*z^7 + 16*z^3 + 79*z^2 - 16*z) * q^81 + (-22*z^6 - 8*z^4 - 22*z^2) * q^82 + (-49*z^7 - 50*z^5 + 50*z^3 + 49*z) * q^83 + (6*z^7 + 18*z^4 - 33*z^3 - 42*z^2 + 33*z + 18) * q^84 + (30*z^6 - 36*z^4 + 36*z^2 - 30) * q^85 + (6*z^7 - 6*z^5 - 36*z) * q^86 + (-66*z^7 - 21*z^6 + 12*z^5 + 21*z^4 - 66*z^3 + 93) * q^87 + (21*z^6 + 21*z^4 + 63*z^2 - 49) * q^88 + (-48*z^7 + 68*z^5 - 48*z^3) * q^89 + (44*z^7 - 5*z^6 - 44*z^5 + z^2 - 12*z - 1) * q^90 + (46*z^6 - 90*z^4 + 90*z^2 - 46) * q^91 + (-72*z^7 - 12*z^3 + 12*z) * q^92 + (46*z^7 + 22*z^6 - 9*z^5 + 26*z^4 + 9*z^3 + 22*z^2 - 46*z) * q^93 + (6*z^6 - 14*z^4 + 6*z^2) * q^94 + (82*z^7 + 88*z^3 - 88*z) * q^95 + (33*z^6 + 33*z^4 + 66*z^3 - 33*z^2 - 33) * q^96 + (-39*z^6 - 99*z^2 + 99) * q^97 + (-33*z^7 + 9*z^5 - 33*z^3) * q^98 + (-6*z^7 + 20*z^6 - 6*z^5 - 112*z^4 + 4*z^3 + 16*z^2 + 3*z - 32) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 8 * q + 4 * q^3 - 6 * q^4 + 10 * q^6 - 4 * q^7 + 2 * q^9 $$8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9} - 44 q^{10} - 48 q^{12} + 32 q^{13} - 50 q^{15} - 10 q^{16} + 40 q^{18} + 112 q^{19} + 88 q^{21} - 58 q^{22} + 70 q^{24} - 42 q^{25} - 44 q^{27} + 78 q^{28} - 12 q^{30} - 18 q^{31} - 90 q^{33} + 48 q^{34} + 6 q^{36} - 120 q^{37} - 64 q^{39} + 42 q^{40} - 70 q^{42} - 264 q^{43} + 80 q^{45} + 24 q^{46} + 20 q^{48} - 150 q^{49} + 60 q^{51} + 36 q^{52} + 316 q^{55} + 136 q^{57} + 186 q^{58} + 180 q^{60} + 336 q^{61} + 4 q^{63} + 26 q^{64} - 124 q^{66} - 24 q^{67} - 240 q^{69} + 42 q^{70} - 280 q^{72} - 182 q^{73} - 136 q^{75} - 264 q^{76} + 40 q^{78} - 460 q^{79} + 158 q^{81} - 72 q^{82} + 24 q^{84} - 36 q^{85} + 660 q^{87} - 266 q^{88} - 16 q^{90} + 84 q^{91} + 36 q^{93} + 52 q^{94} - 330 q^{96} + 516 q^{97} + 40 q^{99}+O(q^{100})$$ 8 * q + 4 * q^3 - 6 * q^4 + 10 * q^6 - 4 * q^7 + 2 * q^9 - 44 * q^10 - 48 * q^12 + 32 * q^13 - 50 * q^15 - 10 * q^16 + 40 * q^18 + 112 * q^19 + 88 * q^21 - 58 * q^22 + 70 * q^24 - 42 * q^25 - 44 * q^27 + 78 * q^28 - 12 * q^30 - 18 * q^31 - 90 * q^33 + 48 * q^34 + 6 * q^36 - 120 * q^37 - 64 * q^39 + 42 * q^40 - 70 * q^42 - 264 * q^43 + 80 * q^45 + 24 * q^46 + 20 * q^48 - 150 * q^49 + 60 * q^51 + 36 * q^52 + 316 * q^55 + 136 * q^57 + 186 * q^58 + 180 * q^60 + 336 * q^61 + 4 * q^63 + 26 * q^64 - 124 * q^66 - 24 * q^67 - 240 * q^69 + 42 * q^70 - 280 * q^72 - 182 * q^73 - 136 * q^75 - 264 * q^76 + 40 * q^78 - 460 * q^79 + 158 * q^81 - 72 * q^82 + 24 * q^84 - 36 * q^85 + 660 * q^87 - 266 * q^88 - 16 * q^90 + 84 * q^91 + 36 * q^93 + 52 * q^94 - 330 * q^96 + 516 * q^97 + 40 * q^99

## Character values

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

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-\zeta_{20}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.951057 + 0.309017i 0.951057 − 0.309017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i
−0.951057 + 0.309017i 0.303706 + 2.98459i −2.42705 + 1.76336i 4.16750 + 1.35410i −1.21113 2.74466i 1.73607 1.26133i 4.11450 5.66312i −8.81553 + 1.81288i −4.38197
5.2 0.951057 0.309017i 2.93236 0.633446i −2.42705 + 1.76336i −4.16750 1.35410i 2.59310 1.50859i 1.73607 1.26133i −4.11450 + 5.66312i 8.19749 3.71499i −4.38197
14.1 −0.587785 0.809017i −2.74466 1.21113i 0.927051 2.85317i 3.88998 5.35410i 0.633446 + 2.93236i −2.73607 + 8.42075i −6.65740 + 2.16312i 6.06633 + 6.64828i −6.61803
14.2 0.587785 + 0.809017i 1.50859 2.59310i 0.927051 2.85317i −3.88998 + 5.35410i 2.98459 0.303706i −2.73607 + 8.42075i 6.65740 2.16312i −4.44829 7.82385i −6.61803
20.1 −0.951057 0.309017i 0.303706 2.98459i −2.42705 1.76336i 4.16750 1.35410i −1.21113 + 2.74466i 1.73607 + 1.26133i 4.11450 + 5.66312i −8.81553 1.81288i −4.38197
20.2 0.951057 + 0.309017i 2.93236 + 0.633446i −2.42705 1.76336i −4.16750 + 1.35410i 2.59310 + 1.50859i 1.73607 + 1.26133i −4.11450 5.66312i 8.19749 + 3.71499i −4.38197
26.1 −0.587785 + 0.809017i −2.74466 + 1.21113i 0.927051 + 2.85317i 3.88998 + 5.35410i 0.633446 2.93236i −2.73607 8.42075i −6.65740 2.16312i 6.06633 6.64828i −6.61803
26.2 0.587785 0.809017i 1.50859 + 2.59310i 0.927051 + 2.85317i −3.88998 5.35410i 2.98459 + 0.303706i −2.73607 8.42075i 6.65740 + 2.16312i −4.44829 + 7.82385i −6.61803
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.h.a 8
3.b odd 2 1 inner 33.3.h.a 8
11.b odd 2 1 363.3.h.g 8
11.c even 5 1 inner 33.3.h.a 8
11.c even 5 1 363.3.b.f 4
11.c even 5 2 363.3.h.h 8
11.d odd 10 1 363.3.b.g 4
11.d odd 10 1 363.3.h.g 8
11.d odd 10 2 363.3.h.i 8
33.d even 2 1 363.3.h.g 8
33.f even 10 1 363.3.b.g 4
33.f even 10 1 363.3.h.g 8
33.f even 10 2 363.3.h.i 8
33.h odd 10 1 inner 33.3.h.a 8
33.h odd 10 1 363.3.b.f 4
33.h odd 10 2 363.3.h.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.a 8 1.a even 1 1 trivial
33.3.h.a 8 3.b odd 2 1 inner
33.3.h.a 8 11.c even 5 1 inner
33.3.h.a 8 33.h odd 10 1 inner
363.3.b.f 4 11.c even 5 1
363.3.b.f 4 33.h odd 10 1
363.3.b.g 4 11.d odd 10 1
363.3.b.g 4 33.f even 10 1
363.3.h.g 8 11.b odd 2 1
363.3.h.g 8 11.d odd 10 1
363.3.h.g 8 33.d even 2 1
363.3.h.g 8 33.f even 10 1
363.3.h.h 8 11.c even 5 2
363.3.h.h 8 33.h odd 10 2
363.3.h.i 8 11.d odd 10 2
363.3.h.i 8 33.f even 10 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$3$ $$T^{8} - 4 T^{7} + 7 T^{6} + 8 T^{5} + \cdots + 6561$$
$5$ $$T^{8} - 4 T^{6} + 1446 T^{4} + \cdots + 707281$$
$7$ $$(T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361)^{2}$$
$11$ $$T^{8} - 316 T^{6} + \cdots + 214358881$$
$13$ $$(T^{4} - 16 T^{3} + 136 T^{2} + \cdots + 13456)^{2}$$
$17$ $$T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616$$
$19$ $$(T^{4} - 56 T^{3} + 1336 T^{2} + \cdots + 80656)^{2}$$
$23$ $$(T^{4} + 1008 T^{2} + 215296)^{2}$$
$29$ $$T^{8} + 261 T^{6} + \cdots + 2449228130001$$
$31$ $$(T^{4} + 9 T^{3} + 796 T^{2} - 8786 T + 36481)^{2}$$
$37$ $$(T^{4} + 60 T^{3} + 1840 T^{2} + \cdots + 336400)^{2}$$
$41$ $$T^{8} + 1124 T^{6} + \cdots + 3544535296$$
$43$ $$(T^{2} + 66 T + 1044)^{4}$$
$47$ $$T^{8} - 496 T^{6} + \cdots + 33362176$$
$53$ $$T^{8} + 101 T^{6} + \cdots + 3969126001$$
$59$ $$T^{8} - 100 T^{6} + \cdots + 276281640625$$
$61$ $$(T^{4} - 168 T^{3} + 12024 T^{2} + \cdots + 6533136)^{2}$$
$67$ $$(T^{2} + 6 T - 5436)^{4}$$
$71$ $$T^{8} - 9904 T^{6} + \cdots + 11\!\cdots\!56$$
$73$ $$(T^{4} + 91 T^{3} + 3136 T^{2} + \cdots + 866761)^{2}$$
$79$ $$(T^{4} + 230 T^{3} + 25360 T^{2} + \cdots + 55428025)^{2}$$
$83$ $$T^{8} - 12004 T^{6} + \cdots + 22\!\cdots\!01$$
$89$ $$(T^{4} + 9632 T^{2} + 891136)^{2}$$
$97$ $$(T^{4} - 258 T^{3} + 36864 T^{2} + \cdots + 147403881)^{2}$$