# Properties

 Label 126.2.f.d Level $126$ Weight $2$ Character orbit 126.f Analytic conductor $1.006$ 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] = [126,2,Mod(43,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("126.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 126.f (of order $$3$$, degree $$2$$, 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: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ 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: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2}$$ b3 + 3*b2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 2\beta _1 + 3$$ -2*b3 + 2*b1 + 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 1.68614 − 0.396143i −1.18614 + 1.26217i 1.68614 + 0.396143i −1.18614 − 1.26217i
0.500000 0.866025i −1.68614 + 0.396143i −0.500000 0.866025i −2.18614 3.78651i −0.500000 + 1.65831i −0.500000 + 0.866025i −1.00000 2.68614 1.33591i −4.37228
43.2 0.500000 0.866025i 1.18614 1.26217i −0.500000 0.866025i 0.686141 + 1.18843i −0.500000 1.65831i −0.500000 + 0.866025i −1.00000 −0.186141 2.99422i 1.37228
85.1 0.500000 + 0.866025i −1.68614 0.396143i −0.500000 + 0.866025i −2.18614 + 3.78651i −0.500000 1.65831i −0.500000 0.866025i −1.00000 2.68614 + 1.33591i −4.37228
85.2 0.500000 + 0.866025i 1.18614 + 1.26217i −0.500000 + 0.866025i 0.686141 1.18843i −0.500000 + 1.65831i −0.500000 0.866025i −1.00000 −0.186141 + 2.99422i 1.37228
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.f.d 4
3.b odd 2 1 378.2.f.c 4
4.b odd 2 1 1008.2.r.f 4
7.b odd 2 1 882.2.f.k 4
7.c even 3 1 882.2.e.l 4
7.c even 3 1 882.2.h.m 4
7.d odd 6 1 882.2.e.k 4
7.d odd 6 1 882.2.h.n 4
9.c even 3 1 inner 126.2.f.d 4
9.c even 3 1 1134.2.a.k 2
9.d odd 6 1 378.2.f.c 4
9.d odd 6 1 1134.2.a.n 2
12.b even 2 1 3024.2.r.f 4
21.c even 2 1 2646.2.f.j 4
21.g even 6 1 2646.2.e.m 4
21.g even 6 1 2646.2.h.l 4
21.h odd 6 1 2646.2.e.n 4
21.h odd 6 1 2646.2.h.k 4
36.f odd 6 1 1008.2.r.f 4
36.f odd 6 1 9072.2.a.bm 2
36.h even 6 1 3024.2.r.f 4
36.h even 6 1 9072.2.a.bb 2
63.g even 3 1 882.2.e.l 4
63.h even 3 1 882.2.h.m 4
63.i even 6 1 2646.2.h.l 4
63.j odd 6 1 2646.2.h.k 4
63.k odd 6 1 882.2.e.k 4
63.l odd 6 1 882.2.f.k 4
63.l odd 6 1 7938.2.a.bh 2
63.n odd 6 1 2646.2.e.n 4
63.o even 6 1 2646.2.f.j 4
63.o even 6 1 7938.2.a.bs 2
63.s even 6 1 2646.2.e.m 4
63.t odd 6 1 882.2.h.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 1.a even 1 1 trivial
126.2.f.d 4 9.c even 3 1 inner
378.2.f.c 4 3.b odd 2 1
378.2.f.c 4 9.d odd 6 1
882.2.e.k 4 7.d odd 6 1
882.2.e.k 4 63.k odd 6 1
882.2.e.l 4 7.c even 3 1
882.2.e.l 4 63.g even 3 1
882.2.f.k 4 7.b odd 2 1
882.2.f.k 4 63.l odd 6 1
882.2.h.m 4 7.c even 3 1
882.2.h.m 4 63.h even 3 1
882.2.h.n 4 7.d odd 6 1
882.2.h.n 4 63.t odd 6 1
1008.2.r.f 4 4.b odd 2 1
1008.2.r.f 4 36.f odd 6 1
1134.2.a.k 2 9.c even 3 1
1134.2.a.n 2 9.d odd 6 1
2646.2.e.m 4 21.g even 6 1
2646.2.e.m 4 63.s even 6 1
2646.2.e.n 4 21.h odd 6 1
2646.2.e.n 4 63.n odd 6 1
2646.2.f.j 4 21.c even 2 1
2646.2.f.j 4 63.o even 6 1
2646.2.h.k 4 21.h odd 6 1
2646.2.h.k 4 63.j odd 6 1
2646.2.h.l 4 21.g even 6 1
2646.2.h.l 4 63.i even 6 1
3024.2.r.f 4 12.b even 2 1
3024.2.r.f 4 36.h even 6 1
7938.2.a.bh 2 63.l odd 6 1
7938.2.a.bs 2 63.o even 6 1
9072.2.a.bb 2 36.h even 6 1
9072.2.a.bm 2 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3T_{5}^{3} + 15T_{5}^{2} - 18T_{5} + 36$$ acting on $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + T^{3} - 2 T^{2} + 3 T + 9$$
$5$ $$T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36$$
$13$ $$(T^{2} + 2 T + 4)^{2}$$
$17$ $$(T^{2} + 3 T - 6)^{2}$$
$19$ $$(T - 5)^{4}$$
$23$ $$T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144$$
$29$ $$T^{4} - 6 T^{3} + 60 T^{2} + 144 T + 576$$
$31$ $$(T^{2} + 2 T + 4)^{2}$$
$37$ $$(T - 2)^{4}$$
$41$ $$T^{4} + 15 T^{3} + 177 T^{2} + \cdots + 2304$$
$43$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 6 T - 24)^{2}$$
$59$ $$T^{4} - 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$61$ $$T^{4} - 11 T^{3} + 165 T^{2} + \cdots + 1936$$
$67$ $$T^{4} + 13 T^{3} + 201 T^{2} + \cdots + 1024$$
$71$ $$(T^{2} - 3 T - 72)^{2}$$
$73$ $$(T^{2} - 7 T - 62)^{2}$$
$79$ $$T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$83$ $$T^{4} + 12 T^{3} + 240 T^{2} + \cdots + 9216$$
$89$ $$(T^{2} - 18 T + 48)^{2}$$
$97$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$