# Properties

 Label 33.3.b.b Level $33$ Weight $3$ Character orbit 33.b Analytic conductor $0.899$ 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] = [33,3,Mod(23,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("33.23");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.b (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{8} + ( - 5 \beta_{3} + 3 \beta_{2} - 3) q^{9}+O(q^{10})$$ q + (b3 - b2 - b1) * q^2 + (-b2 + b1 - 1) * q^3 + (b3 + b1 + 1) * q^4 + (b3 - b1) * q^5 + (-b3 + 3*b2 + 3*b1) * q^6 + (-2*b3 - 2*b1) * q^7 + (3*b3 - b2 - 3*b1) * q^8 + (-5*b3 + 3*b2 - 3) * q^9 $$q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{8} + ( - 5 \beta_{3} + 3 \beta_{2} - 3) q^{9} - 2 q^{10} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + ( - 5 \beta_{3} + 3 \beta_1 + 3) q^{12} + ( - 4 \beta_{3} - 4 \beta_1 - 4) q^{13} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1) q^{14} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{15} + (5 \beta_{3} + 5 \beta_1 - 3) q^{16} + ( - 2 \beta_{3} + 16 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{3} - 7 \beta_{2} - 5 \beta_1 + 8) q^{18} + (6 \beta_{3} + 6 \beta_1 - 6) q^{19} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{20} + (10 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 8) q^{21} + ( - \beta_{3} - \beta_1 + 5) q^{22} + ( - \beta_{3} - 16 \beta_{2} + \beta_1) q^{23} + (\beta_{3} + 5 \beta_{2} + 7 \beta_1 + 8) q^{24} + ( - \beta_{3} - \beta_1 + 21) q^{25} + (4 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{26} + (16 \beta_{3} - 8 \beta_{2} - 16 \beta_1 - 5) q^{27} - 16 q^{28} + ( - 24 \beta_{3} + 4 \beta_{2} + 24 \beta_1) q^{29} + (2 \beta_{2} - 2 \beta_1 + 2) q^{30} + (5 \beta_{3} + 5 \beta_1 + 14) q^{31} + ( - \beta_{3} + 19 \beta_{2} + \beta_1) q^{32} + ( - 4 \beta_{2} - 5 \beta_1 - 4) q^{33} + ( - 16 \beta_{3} - 16 \beta_1 + 20) q^{34} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{35} + (\beta_{3} - 8 \beta_{2} - 7 \beta_1 - 23) q^{36} + ( - 7 \beta_{3} - 7 \beta_1 - 26) q^{37} + ( - 18 \beta_{3} + 30 \beta_{2} + 18 \beta_1) q^{38} + (20 \beta_{3} - 12 \beta_1 - 12) q^{39} + ( - 2 \beta_{3} - 2 \beta_1 - 10) q^{40} + (4 \beta_{3} + 12 \beta_{2} - 4 \beta_1) q^{41} + ( - 12 \beta_{3} + 20 \beta_{2} + 16 \beta_1 - 16) q^{42} + (4 \beta_{3} + 4 \beta_1 + 26) q^{43} + ( - \beta_{3} - 5 \beta_{2} + \beta_1) q^{44} + ( - 6 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 4) q^{45} + (16 \beta_{3} + 16 \beta_1 - 14) q^{46} + ( - 22 \beta_{3} + 14 \beta_{2} + 22 \beta_1) q^{47} + ( - 25 \beta_{3} + 8 \beta_{2} + 7 \beta_1 + 23) q^{48} + ( - 4 \beta_{3} - 4 \beta_1 - 17) q^{49} + (23 \beta_{3} - 25 \beta_{2} - 23 \beta_1) q^{50} + (30 \beta_{3} - 34 \beta_{2} - 20 \beta_1 + 56) q^{51} + ( - 4 \beta_{3} - 4 \beta_1 - 36) q^{52} + (30 \beta_{3} - 38 \beta_{2} - 30 \beta_1) q^{53} + (3 \beta_{3} + 5 \beta_{2} + 13 \beta_1 - 40) q^{54} + (\beta_{3} + \beta_1 + 6) q^{55} - 16 \beta_{2} q^{56} + ( - 30 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 30) q^{57} + ( - 4 \beta_{3} - 4 \beta_1 + 52) q^{58} + (23 \beta_{3} + 2 \beta_{2} - 23 \beta_1) q^{59} + (6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 16) q^{60} + (20 \beta_{3} + 20 \beta_1 + 4) q^{61} + (4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{62} + ( - 12 \beta_{3} + 22 \beta_{2} + 14 \beta_1 + 40) q^{63} + (\beta_{3} + \beta_1 + 9) q^{64} + ( - 8 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{65} + (5 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 9) q^{66} + (17 \beta_{3} + 17 \beta_1 - 6) q^{67} + (44 \beta_{3} - 20 \beta_{2} - 44 \beta_1) q^{68} + ( - 33 \beta_{3} + 31 \beta_{2} + 14 \beta_1 - 68) q^{69} + (4 \beta_{3} + 4 \beta_1) q^{70} + (3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{71} + ( - 13 \beta_{3} - 17 \beta_{2} + 5 \beta_1 + 16) q^{72} + ( - 6 \beta_{3} - 6 \beta_1 - 74) q^{73} + ( - 12 \beta_{3} - 2 \beta_{2} + 12 \beta_1) q^{74} + (5 \beta_{3} - 22 \beta_{2} + 19 \beta_1 - 25) q^{75} + ( - 6 \beta_{3} - 6 \beta_1 + 42) q^{76} + ( - 2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{77} + ( - 20 \beta_{3} + 28 \beta_{2} + 20 \beta_1 - 32) q^{78} + ( - 26 \beta_{3} - 26 \beta_1 - 32) q^{79} + (2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{80} + (37 \beta_{2} + 35 \beta_1 + 37) q^{81} + ( - 12 \beta_{3} - 12 \beta_1 + 4) q^{82} + ( - 30 \beta_{3} + 24 \beta_{2} + 30 \beta_1) q^{83} + (16 \beta_{2} - 16 \beta_1 + 16) q^{84} + ( - 14 \beta_{3} - 14 \beta_1 - 24) q^{85} + (18 \beta_{3} - 10 \beta_{2} - 18 \beta_1) q^{86} + ( - 16 \beta_{3} - 32 \beta_{2} - 52 \beta_1 - 80) q^{87} + (\beta_{3} + \beta_1 + 17) q^{88} + (39 \beta_{3} + 26 \beta_{2} - 39 \beta_1) q^{89} + (10 \beta_{3} - 6 \beta_{2} + 6) q^{90} + 64 q^{91} + ( - 50 \beta_{3} + 14 \beta_{2} + 50 \beta_1) q^{92} + ( - 25 \beta_{3} - 9 \beta_{2} + 24 \beta_1 + 6) q^{93} + ( - 14 \beta_{3} - 14 \beta_1 + 58) q^{94} + 12 \beta_{2} q^{95} + (37 \beta_{3} - 39 \beta_{2} - 21 \beta_1 + 72) q^{96} + ( - 33 \beta_{3} - 33 \beta_1 + 38) q^{97} + ( - 9 \beta_{3} + \beta_{2} + 9 \beta_1) q^{98} + (7 \beta_{3} + 12 \beta_{2} - 12) q^{99}+O(q^{100})$$ q + (b3 - b2 - b1) * q^2 + (-b2 + b1 - 1) * q^3 + (b3 + b1 + 1) * q^4 + (b3 - b1) * q^5 + (-b3 + 3*b2 + 3*b1) * q^6 + (-2*b3 - 2*b1) * q^7 + (3*b3 - b2 - 3*b1) * q^8 + (-5*b3 + 3*b2 - 3) * q^9 - 2 * q^10 + (-2*b3 + b2 + 2*b1) * q^11 + (-5*b3 + 3*b1 + 3) * q^12 + (-4*b3 - 4*b1 - 4) * q^13 + (4*b3 - 8*b2 - 4*b1) * q^14 + (b3 + b2 + 2*b1 + 4) * q^15 + (5*b3 + 5*b1 - 3) * q^16 + (-2*b3 + 16*b2 + 2*b1) * q^17 + (-b3 - 7*b2 - 5*b1 + 8) * q^18 + (6*b3 + 6*b1 - 6) * q^19 + (2*b3 + 2*b2 - 2*b1) * q^20 + (10*b3 - 2*b2 - 4*b1 - 8) * q^21 + (-b3 - b1 + 5) * q^22 + (-b3 - 16*b2 + b1) * q^23 + (b3 + 5*b2 + 7*b1 + 8) * q^24 + (-b3 - b1 + 21) * q^25 + (4*b3 - 12*b2 - 4*b1) * q^26 + (16*b3 - 8*b2 - 16*b1 - 5) * q^27 - 16 * q^28 + (-24*b3 + 4*b2 + 24*b1) * q^29 + (2*b2 - 2*b1 + 2) * q^30 + (5*b3 + 5*b1 + 14) * q^31 + (-b3 + 19*b2 + b1) * q^32 + (-4*b2 - 5*b1 - 4) * q^33 + (-16*b3 - 16*b1 + 20) * q^34 + (-2*b3 - 4*b2 + 2*b1) * q^35 + (b3 - 8*b2 - 7*b1 - 23) * q^36 + (-7*b3 - 7*b1 - 26) * q^37 + (-18*b3 + 30*b2 + 18*b1) * q^38 + (20*b3 - 12*b1 - 12) * q^39 + (-2*b3 - 2*b1 - 10) * q^40 + (4*b3 + 12*b2 - 4*b1) * q^41 + (-12*b3 + 20*b2 + 16*b1 - 16) * q^42 + (4*b3 + 4*b1 + 26) * q^43 + (-b3 - 5*b2 + b1) * q^44 + (-6*b3 - 5*b2 + 5*b1 + 4) * q^45 + (16*b3 + 16*b1 - 14) * q^46 + (-22*b3 + 14*b2 + 22*b1) * q^47 + (-25*b3 + 8*b2 + 7*b1 + 23) * q^48 + (-4*b3 - 4*b1 - 17) * q^49 + (23*b3 - 25*b2 - 23*b1) * q^50 + (30*b3 - 34*b2 - 20*b1 + 56) * q^51 + (-4*b3 - 4*b1 - 36) * q^52 + (30*b3 - 38*b2 - 30*b1) * q^53 + (3*b3 + 5*b2 + 13*b1 - 40) * q^54 + (b3 + b1 + 6) * q^55 - 16*b2 * q^56 + (-30*b3 + 12*b2 + 6*b1 + 30) * q^57 + (-4*b3 - 4*b1 + 52) * q^58 + (23*b3 + 2*b2 - 23*b1) * q^59 + (6*b3 - 2*b2 + 2*b1 + 16) * q^60 + (20*b3 + 20*b1 + 4) * q^61 + (4*b3 + 6*b2 - 4*b1) * q^62 + (-12*b3 + 22*b2 + 14*b1 + 40) * q^63 + (b3 + b1 + 9) * q^64 + (-8*b3 - 8*b2 + 8*b1) * q^65 + (5*b3 - 6*b2 + 3*b1 - 9) * q^66 + (17*b3 + 17*b1 - 6) * q^67 + (44*b3 - 20*b2 - 44*b1) * q^68 + (-33*b3 + 31*b2 + 14*b1 - 68) * q^69 + (4*b3 + 4*b1) * q^70 + (3*b3 + 4*b2 - 3*b1) * q^71 + (-13*b3 - 17*b2 + 5*b1 + 16) * q^72 + (-6*b3 - 6*b1 - 74) * q^73 + (-12*b3 - 2*b2 + 12*b1) * q^74 + (5*b3 - 22*b2 + 19*b1 - 25) * q^75 + (-6*b3 - 6*b1 + 42) * q^76 + (-2*b3 + 12*b2 + 2*b1) * q^77 + (-20*b3 + 28*b2 + 20*b1 - 32) * q^78 + (-26*b3 - 26*b1 - 32) * q^79 + (2*b3 + 10*b2 - 2*b1) * q^80 + (37*b2 + 35*b1 + 37) * q^81 + (-12*b3 - 12*b1 + 4) * q^82 + (-30*b3 + 24*b2 + 30*b1) * q^83 + (16*b2 - 16*b1 + 16) * q^84 + (-14*b3 - 14*b1 - 24) * q^85 + (18*b3 - 10*b2 - 18*b1) * q^86 + (-16*b3 - 32*b2 - 52*b1 - 80) * q^87 + (b3 + b1 + 17) * q^88 + (39*b3 + 26*b2 - 39*b1) * q^89 + (10*b3 - 6*b2 + 6) * q^90 + 64 * q^91 + (-50*b3 + 14*b2 + 50*b1) * q^92 + (-25*b3 - 9*b2 + 24*b1 + 6) * q^93 + (-14*b3 - 14*b1 + 58) * q^94 + 12*b2 * q^95 + (37*b3 - 39*b2 - 21*b1 + 72) * q^96 + (-33*b3 - 33*b1 + 38) * q^97 + (-9*b3 + b2 + 9*b1) * q^98 + (7*b3 + 12*b2 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 7 q^{9}+O(q^{10})$$ 4 * q - 5 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^7 - 7 * q^9 $$4 q - 5 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 7 q^{9} - 8 q^{10} + 14 q^{12} - 8 q^{13} + 13 q^{15} - 22 q^{16} + 38 q^{18} - 36 q^{19} - 38 q^{21} + 22 q^{22} + 24 q^{24} + 86 q^{25} - 20 q^{27} - 64 q^{28} + 10 q^{30} + 46 q^{31} - 11 q^{33} + 112 q^{34} - 86 q^{36} - 90 q^{37} - 56 q^{39} - 36 q^{40} - 68 q^{42} + 96 q^{43} + 17 q^{45} - 88 q^{46} + 110 q^{48} - 60 q^{49} + 214 q^{51} - 136 q^{52} - 176 q^{54} + 22 q^{55} + 144 q^{57} + 216 q^{58} + 56 q^{60} - 24 q^{61} + 158 q^{63} + 34 q^{64} - 44 q^{66} - 58 q^{67} - 253 q^{69} - 8 q^{70} + 72 q^{72} - 284 q^{73} - 124 q^{75} + 180 q^{76} - 128 q^{78} - 76 q^{79} + 113 q^{81} + 40 q^{82} + 80 q^{84} - 68 q^{85} - 252 q^{87} + 66 q^{88} + 14 q^{90} + 256 q^{91} + 25 q^{93} + 260 q^{94} + 272 q^{96} + 218 q^{97} - 55 q^{99}+O(q^{100})$$ 4 * q - 5 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^7 - 7 * q^9 - 8 * q^10 + 14 * q^12 - 8 * q^13 + 13 * q^15 - 22 * q^16 + 38 * q^18 - 36 * q^19 - 38 * q^21 + 22 * q^22 + 24 * q^24 + 86 * q^25 - 20 * q^27 - 64 * q^28 + 10 * q^30 + 46 * q^31 - 11 * q^33 + 112 * q^34 - 86 * q^36 - 90 * q^37 - 56 * q^39 - 36 * q^40 - 68 * q^42 + 96 * q^43 + 17 * q^45 - 88 * q^46 + 110 * q^48 - 60 * q^49 + 214 * q^51 - 136 * q^52 - 176 * q^54 + 22 * q^55 + 144 * q^57 + 216 * q^58 + 56 * q^60 - 24 * q^61 + 158 * q^63 + 34 * q^64 - 44 * q^66 - 58 * q^67 - 253 * q^69 - 8 * q^70 + 72 * q^72 - 284 * q^73 - 124 * q^75 + 180 * q^76 - 128 * q^78 - 76 * q^79 + 113 * q^81 + 40 * q^82 + 80 * q^84 - 68 * q^85 - 252 * q^87 + 66 * q^88 + 14 * q^90 + 256 * q^91 + 25 * q^93 + 260 * q^94 + 272 * q^96 + 218 * q^97 - 55 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 4*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3$$ (-v^3 - 2*v^2 + 2*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 2$$ (b2 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} - 3\beta_{2} + 3 ) / 2$$ (2*b3 - 3*b2 + 3) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ -2*b3 + b2 + 2*b1 + 4

## Character values

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

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i −2.68614 1.33591i −2.37228 0.792287i −3.37228 + 6.78073i 6.74456 4.10891i 5.43070 + 7.17687i −2.00000
23.2 0.792287i 0.186141 + 2.99422i 3.37228 2.52434i 2.37228 0.147477i −4.74456 5.84096i −8.93070 + 1.11469i −2.00000
23.3 0.792287i 0.186141 2.99422i 3.37228 2.52434i 2.37228 + 0.147477i −4.74456 5.84096i −8.93070 1.11469i −2.00000
23.4 2.52434i −2.68614 + 1.33591i −2.37228 0.792287i −3.37228 6.78073i 6.74456 4.10891i 5.43070 7.17687i −2.00000
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.b.b 4
3.b odd 2 1 inner 33.3.b.b 4
4.b odd 2 1 528.3.i.d 4
11.b odd 2 1 363.3.b.h 4
11.c even 5 4 363.3.h.m 16
11.d odd 10 4 363.3.h.l 16
12.b even 2 1 528.3.i.d 4
33.d even 2 1 363.3.b.h 4
33.f even 10 4 363.3.h.l 16
33.h odd 10 4 363.3.h.m 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 1.a even 1 1 trivial
33.3.b.b 4 3.b odd 2 1 inner
363.3.b.h 4 11.b odd 2 1
363.3.b.h 4 33.d even 2 1
363.3.h.l 16 11.d odd 10 4
363.3.h.l 16 33.f even 10 4
363.3.h.m 16 11.c even 5 4
363.3.h.m 16 33.h odd 10 4
528.3.i.d 4 4.b odd 2 1
528.3.i.d 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7T_{2}^{2} + 4$$ acting on $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 4$$
$3$ $$T^{4} + 5 T^{3} + 16 T^{2} + 45 T + 81$$
$5$ $$T^{4} + 7T^{2} + 4$$
$7$ $$(T^{2} - 2 T - 32)^{2}$$
$11$ $$(T^{2} + 11)^{2}$$
$13$ $$(T^{2} + 4 T - 128)^{2}$$
$17$ $$T^{4} + 1372 T^{2} + 440896$$
$19$ $$(T^{2} + 18 T - 216)^{2}$$
$23$ $$T^{4} + 1639 T^{2} + 662596$$
$29$ $$T^{4} + 3552 T^{2} + \cdots + 1937664$$
$31$ $$(T^{2} - 23 T - 74)^{2}$$
$37$ $$(T^{2} + 45 T + 102)^{2}$$
$41$ $$T^{4} + 1264 T^{2} + 295936$$
$43$ $$(T^{2} - 48 T + 444)^{2}$$
$47$ $$T^{4} + 2716 T^{2} + \cdots + 1700416$$
$53$ $$T^{4} + 8124 T^{2} + 788544$$
$59$ $$T^{4} + 4003 T^{2} + 824464$$
$61$ $$(T^{2} + 12 T - 3264)^{2}$$
$67$ $$(T^{2} + 29 T - 2174)^{2}$$
$71$ $$T^{4} + 231T^{2} + 4356$$
$73$ $$(T^{2} + 142 T + 4744)^{2}$$
$79$ $$(T^{2} + 38 T - 5216)^{2}$$
$83$ $$T^{4} + 5436 T^{2} + \cdots + 4981824$$
$89$ $$T^{4} + 20787 T^{2} + \cdots + 4112784$$
$97$ $$(T^{2} - 109 T - 6014)^{2}$$