# Properties

 Label 33.2.e.b Level $33$ Weight $2$ Character orbit 33.e Analytic conductor $0.264$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,2,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, 2, names="a")

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

chi := DirichletCharacter("33.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.263506326670$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
0.309017 + 0.224514i −0.309017 + 0.951057i −0.572949 1.76336i −1.30902 + 0.951057i −0.309017 + 0.224514i −0.309017 0.951057i 0.454915 1.40008i −0.809017 0.587785i −0.618034
16.1 −0.809017 2.48990i 0.809017 + 0.587785i −3.92705 + 2.85317i −0.190983 + 0.587785i 0.809017 2.48990i 0.809017 0.587785i 6.04508 + 4.39201i 0.309017 + 0.951057i 1.61803
25.1 0.309017 0.224514i −0.309017 0.951057i −0.572949 + 1.76336i −1.30902 0.951057i −0.309017 0.224514i −0.309017 + 0.951057i 0.454915 + 1.40008i −0.809017 + 0.587785i −0.618034
31.1 −0.809017 + 2.48990i 0.809017 0.587785i −3.92705 2.85317i −0.190983 0.587785i 0.809017 + 2.48990i 0.809017 + 0.587785i 6.04508 4.39201i 0.309017 0.951057i 1.61803
 $$n$$: e.g. 2-40 or 990-1000 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 33.2.e.b 4
3.b odd 2 1 99.2.f.a 4
4.b odd 2 1 528.2.y.b 4
5.b even 2 1 825.2.n.c 4
5.c odd 4 2 825.2.bx.d 8
9.c even 3 2 891.2.n.c 8
9.d odd 6 2 891.2.n.b 8
11.b odd 2 1 363.2.e.f 4
11.c even 5 1 inner 33.2.e.b 4
11.c even 5 1 363.2.a.d 2
11.c even 5 2 363.2.e.k 4
11.d odd 10 1 363.2.a.i 2
11.d odd 10 2 363.2.e.b 4
11.d odd 10 1 363.2.e.f 4
33.f even 10 1 1089.2.a.l 2
33.h odd 10 1 99.2.f.a 4
33.h odd 10 1 1089.2.a.t 2
44.g even 10 1 5808.2.a.ci 2
44.h odd 10 1 528.2.y.b 4
44.h odd 10 1 5808.2.a.cj 2
55.h odd 10 1 9075.2.a.u 2
55.j even 10 1 825.2.n.c 4
55.j even 10 1 9075.2.a.cb 2
55.k odd 20 2 825.2.bx.d 8
99.m even 15 2 891.2.n.c 8
99.n odd 30 2 891.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 1.a even 1 1 trivial
33.2.e.b 4 11.c even 5 1 inner
99.2.f.a 4 3.b odd 2 1
99.2.f.a 4 33.h odd 10 1
363.2.a.d 2 11.c even 5 1
363.2.a.i 2 11.d odd 10 1
363.2.e.b 4 11.d odd 10 2
363.2.e.f 4 11.b odd 2 1
363.2.e.f 4 11.d odd 10 1
363.2.e.k 4 11.c even 5 2
528.2.y.b 4 4.b odd 2 1
528.2.y.b 4 44.h odd 10 1
825.2.n.c 4 5.b even 2 1
825.2.n.c 4 55.j even 10 1
825.2.bx.d 8 5.c odd 4 2
825.2.bx.d 8 55.k odd 20 2
891.2.n.b 8 9.d odd 6 2
891.2.n.b 8 99.n odd 30 2
891.2.n.c 8 9.c even 3 2
891.2.n.c 8 99.m even 15 2
1089.2.a.l 2 33.f even 10 1
1089.2.a.t 2 33.h odd 10 1
5808.2.a.ci 2 44.g even 10 1
5808.2.a.cj 2 44.h odd 10 1
9075.2.a.u 2 55.h odd 10 1
9075.2.a.cb 2 55.j even 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + 6 T^{2} - 4 T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$7$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$11$ $$T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 121$$
$13$ $$T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1$$
$17$ $$T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81$$
$19$ $$T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25$$
$23$ $$(T^{2} + 4 T - 1)^{2}$$
$29$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 961$$
$37$ $$T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121$$
$41$ $$T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1$$
$43$ $$(T^{2} - 45)^{2}$$
$47$ $$T^{4} - 17 T^{3} + 114 T^{2} + \cdots + 121$$
$53$ $$T^{4} - 4 T^{3} + 6 T^{2} + T + 1$$
$59$ $$T^{4} + 6 T^{3} + 76 T^{2} + \cdots + 5041$$
$61$ $$T^{4} + 21 T^{3} + 306 T^{2} + \cdots + 9801$$
$67$ $$(T^{2} + 3 T - 9)^{2}$$
$71$ $$T^{4} - 15 T^{3} + 190 T^{2} + \cdots + 3025$$
$73$ $$T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$79$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$83$ $$T^{4} - 13 T^{3} + 69 T^{2} + \cdots + 121$$
$89$ $$(T^{2} + 12 T + 31)^{2}$$
$97$ $$T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81$$