# Properties

 Label 48.3.i.a Level $48$ Weight $3$ Character orbit 48.i Analytic conductor $1.308$ 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] = [48,3,Mod(5,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("48.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 48.i (of order $$4$$, 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: $$1.30790526893$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16$$ x^8 - 2*x^6 + 2*x^4 - 8*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{4} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{8} + ( - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{9}+O(q^{10})$$ q + b4 * q^2 + (-b4 + b3 + b2 + b1) * q^3 + (2*b7 + b6 + b5 + b4 - b3) * q^4 + (2*b6 - b5 + 2*b4 + b3) * q^5 + (-b7 + 2*b5 + b4 + 2*b2 - b1 + 2) * q^6 + (-b7 - b6 - b5 - b4 + b3 + b2 + b1 - 1) * q^7 + (-4*b6 - 2*b4 + 2*b3) * q^8 + (-2*b7 - 3*b6 - 5*b5 - 5*b4 + b3 - b2 + 2*b1 - 2) * q^9 $$q + \beta_{4} q^{2} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{4} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{8} + ( - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{9} + (4 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} + \cdots - 4) q^{10}+ \cdots + ( - 6 \beta_{5} + 8 \beta_{4} - 14 \beta_{3} + 16 \beta_{2} - 8 \beta_1 + 24) q^{99}+O(q^{100})$$ q + b4 * q^2 + (-b4 + b3 + b2 + b1) * q^3 + (2*b7 + b6 + b5 + b4 - b3) * q^4 + (2*b6 - b5 + 2*b4 + b3) * q^5 + (-b7 + 2*b5 + b4 + 2*b2 - b1 + 2) * q^6 + (-b7 - b6 - b5 - b4 + b3 + b2 + b1 - 1) * q^7 + (-4*b6 - 2*b4 + 2*b3) * q^8 + (-2*b7 - 3*b6 - 5*b5 - 5*b4 + b3 - b2 + 2*b1 - 2) * q^9 + (4*b7 + 3*b6 + 3*b5 + 3*b4 - 3*b3 + 8*b2 - 2*b1 - 4) * q^10 + (-2*b6 - 2*b4) * q^11 + (2*b7 + 3*b6 - b5 + 3*b4 - 5*b3 + 8) * q^12 + (b6 + b5 + b4 - b3 - 13*b2 - 2*b1 - 11) * q^13 + (2*b6 + 2*b5 + b4 - b3) * q^14 + (-4*b7 - 3*b6 + 2*b5 - 2*b4 + 3*b3 - 4*b2 - 4*b1 + 14) * q^15 + (-4*b7 - 16*b2 - 4*b1) * q^16 + (9*b6 - b5 + b4 - 9*b3) * q^17 + (-6*b7 - 2*b4 + 5*b3 - 4*b2 + 2*b1 - 12) * q^18 + (-4*b6 - 4*b5 - 4*b4 + 4*b3 + 6*b2 + 8*b1 - 2) * q^19 + (-8*b6 - 4*b5 - 10*b4 - 6*b3) * q^20 + (-2*b7 - 3*b6 - 2*b5 - 3*b4 + 2*b3 + 3*b2 - 5) * q^21 + (-4*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 - 8*b2) * q^22 + (2*b6 + 10*b5 + 10*b4 + 2*b3) * q^23 + (4*b7 - 6*b6 - 2*b5 + 4*b4 + 4*b3 + 8*b2 + 8*b1 - 8) * q^24 + (14*b7 + 14*b6 + 14*b5 + 14*b4 - 14*b3 + 17*b2 - 14*b1 + 14) * q^25 + (-4*b5 - 13*b4 + 11*b3) * q^26 + (7*b7 - 3*b6 - 2*b5 - 3*b4 + 2*b3 + 12*b2 - 5) * q^27 + (2*b7 + 8*b2 + 2*b1 + 8) * q^28 + (-7*b6 + 7*b4) * q^29 + (-4*b7 + 9*b6 - 7*b5 + 15*b4 - 5*b3 - 12*b2 - 6*b1 + 8) * q^30 + (b7 + b2 + b1 + 9) * q^31 + (8*b6 - 8*b5 + 8*b3) * q^32 + (4*b7 - 2*b5 + 2*b4 + 4*b2 + 4*b1 - 8) * q^33 + (2*b7 - 8*b6 - 8*b5 - 8*b4 + 8*b3 + 36*b2 + 18*b1 - 4) * q^34 + (-b6 + 5*b5 + b4 + 5*b3) * q^35 + (4*b7 + 15*b6 + 7*b5 - b4 - 3*b3 + 16*b2 - 2*b1 + 16) * q^36 + (-10*b7 - 5*b6 - 5*b5 - 5*b4 + 5*b3 - 19*b2 + 9) * q^37 + (16*b5 + 6*b4 + 2*b3) * q^38 + (13*b7 + 3*b6 + 16*b5 + 16*b4 - 23*b3 - 19*b2 - 13*b1 + 13) * q^39 + (-20*b7 - 16*b6 - 16*b5 - 16*b4 + 16*b3 - 32*b2 + 12*b1 - 16) * q^40 + (-b6 + 7*b5 + 7*b4 - b3) * q^41 + (-4*b7 + 3*b6 - b5 - 4*b4 - 4*b3 - 8*b2 - 2*b1 - 4) * q^42 + (-8*b7 - 4*b6 - 4*b5 - 4*b4 + 4*b3 + 26*b2 - 34) * q^43 + (8*b6 + 4*b4 + 4*b3) * q^44 + (-9*b6 + 6*b5 - 11*b4 + 26*b3 - 4*b2 + 20*b1 - 24) * q^45 + (20*b7 + 12*b6 + 12*b5 + 12*b4 - 12*b3 + 8*b2 - 4*b1 + 40) * q^46 + (24*b6 - 14*b5 + 14*b4 - 24*b3) * q^47 + (12*b7 + 24*b5 + 4*b4 - 4*b3 - 16*b2 - 4*b1) * q^48 + (-2*b7 - 2*b2 - 2*b1 + 41) * q^49 + (-28*b6 - 28*b5 - 14*b4 - 17*b3) * q^50 + (27*b6 + 3*b5 - 7*b4 - 17*b3 - 14*b2 - 20*b1 + 6) * q^51 + (-26*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 - 22*b1 - 16) * q^52 + (-8*b6 - 15*b5 - 8*b4 + 15*b3) * q^53 + (-13*b7 - 15*b6 - b5 - 13*b4 - 4*b3 - 26*b2 - 11*b1 - 22) * q^54 + (-12*b7 - 12*b6 - 12*b5 - 12*b4 + 12*b3 - 28*b2 + 12*b1 - 12) * q^55 + (-4*b6 + 4*b5 + 8*b4 - 4*b3) * q^56 + (-6*b7 - 12*b6 - 18*b5 - 18*b4 + 30*b2 + 6*b1 - 6) * q^57 + (14*b7 + 7*b6 + 7*b5 + 7*b4 - 7*b3 - 28*b2) * q^58 + (-15*b6 - 5*b5 - 15*b4 + 5*b3) * q^59 + (28*b7 + 18*b6 - 2*b5 + 16*b4 - 8*b3 + 32*b2 + 8*b1 - 32) * q^60 + (5*b6 + 5*b5 + 5*b4 - 5*b3 + 21*b2 - 10*b1 + 31) * q^61 + (-2*b6 + 2*b5 + 9*b4 + b3) * q^62 + (-b7 - 6*b6 - 4*b5 + 4*b4 + 6*b3 - b2 - b1 - 13) * q^63 + (8*b6 + 8*b5 + 8*b4 - 8*b3 + 32*b2 - 16*b1 - 32) * q^64 + (-5*b6 + 29*b5 - 29*b4 + 5*b3) * q^65 + (4*b7 - 6*b6 + 10*b5 - 6*b4 + 2*b3 - 8) * q^66 + (-15*b6 - 15*b5 - 15*b4 + 15*b3 - 14*b2 + 30*b1 - 44) * q^67 + (-4*b6 + 36*b5 + 12*b4 - 16*b3) * q^68 + (-24*b7 - 6*b6 + 6*b5 - 6*b4 - 6*b3 - 12*b2 - 12) * q^69 + (2*b7 + 6*b6 + 6*b5 + 6*b4 - 6*b3 - 4*b2 - 10*b1 + 20) * q^70 + (16*b6 - 26*b5 - 26*b4 + 16*b3) * q^71 + (-8*b7 - 12*b6 - 8*b5 + 6*b4 - 10*b3 + 48*b2 + 16) * q^72 + (-14*b7 - 14*b6 - 14*b5 - 14*b4 + 14*b3 - 22*b2 + 14*b1 - 14) * q^73 + (20*b6 + 19*b4 + 9*b3) * q^74 + (-3*b7 + 42*b6 - 3*b5 + 42*b4 + 3*b3 - 42*b2 + 39) * q^75 + (12*b7 + 8*b6 + 8*b5 + 8*b4 - 8*b3 - 4*b1 + 64) * q^76 + (-2*b6 - 2*b5 + 2*b4 - 2*b3) * q^77 + (6*b7 - 33*b6 - 33*b5 - 20*b4 + 26*b3 - 40*b2 + 20*b1 + 12) * q^78 + (-13*b7 - 13*b2 - 13*b1 - 17) * q^79 + (40*b6 + 24*b5 + 16*b4 + 24*b3) * q^80 + (8*b7 - 6*b6 - 22*b5 + 22*b4 + 6*b3 + 8*b2 + 8*b1 + 23) * q^81 + (14*b7 + 6*b6 + 6*b5 + 6*b4 - 6*b3 - 4*b2 + 2*b1 + 28) * q^82 + (-18*b6 - 20*b5 + 18*b4 - 20*b3) * q^83 + (-6*b7 - 12*b5 - 10*b4 + 10*b3 + 16*b2 + 10*b1) * q^84 + (44*b7 + 22*b6 + 22*b5 + 22*b4 - 22*b3 + 36*b2 + 8) * q^85 + (16*b6 - 26*b4 - 34*b3) * q^86 + (21*b5 + 21*b4 + 42*b2) * q^87 + (8*b7 + 8*b6 + 8*b5 + 8*b4 - 8*b3 + 32*b2 - 8*b1) * q^88 + (18*b6 + 10*b5 + 10*b4 + 18*b3) * q^89 + (-2*b7 + 15*b6 + 55*b5 + 11*b4 + 9*b3 + 4*b2 - 32*b1 + 64) * q^90 + (26*b7 + 13*b6 + 13*b5 + 13*b4 - 13*b3 - 6*b2 + 32) * q^91 + (-40*b6 - 8*b5 + 16*b4 + 8*b3) * q^92 + (-3*b5 - 10*b4 + 7*b3 + 13*b2 + 10*b1 + 3) * q^93 + (28*b7 - 10*b6 - 10*b5 - 10*b4 + 10*b3 + 96*b2 + 48*b1 - 56) * q^94 + (-26*b6 + 22*b5 - 22*b4 + 26*b3) * q^95 + (-8*b7 - 24*b6 - 8*b5 - 16*b4 + 24*b3 - 32*b2 - 8*b1 + 64) * q^96 + (-8*b7 - 8*b2 - 8*b1 - 60) * q^97 + (4*b6 - 4*b5 + 41*b4 - 2*b3) * q^98 + (-6*b5 + 8*b4 - 14*b3 + 16*b2 - 8*b1 + 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} - 8 q^{4} + 16 q^{6}+O(q^{10})$$ 8 * q + 4 * q^3 - 8 * q^4 + 16 * q^6 $$8 q + 4 q^{3} - 8 q^{4} + 16 q^{6} - 56 q^{10} + 56 q^{12} - 96 q^{13} + 112 q^{15} - 64 q^{18} + 16 q^{19} - 32 q^{21} + 16 q^{22} - 48 q^{24} - 68 q^{27} + 64 q^{28} + 56 q^{30} + 72 q^{31} - 64 q^{33} + 32 q^{34} + 104 q^{36} + 112 q^{37} - 24 q^{42} - 240 q^{43} - 112 q^{45} + 224 q^{46} - 64 q^{48} + 328 q^{49} - 32 q^{51} - 112 q^{52} - 168 q^{54} - 56 q^{58} - 336 q^{60} + 208 q^{61} - 104 q^{63} - 320 q^{64} - 80 q^{66} - 232 q^{67} + 112 q^{70} + 160 q^{72} + 324 q^{75} + 448 q^{76} + 152 q^{78} - 136 q^{79} + 184 q^{81} + 176 q^{82} + 64 q^{84} - 112 q^{85} - 64 q^{88} + 392 q^{90} + 152 q^{91} + 64 q^{93} - 368 q^{94} + 512 q^{96} - 480 q^{97} + 160 q^{99}+O(q^{100})$$ 8 * q + 4 * q^3 - 8 * q^4 + 16 * q^6 - 56 * q^10 + 56 * q^12 - 96 * q^13 + 112 * q^15 - 64 * q^18 + 16 * q^19 - 32 * q^21 + 16 * q^22 - 48 * q^24 - 68 * q^27 + 64 * q^28 + 56 * q^30 + 72 * q^31 - 64 * q^33 + 32 * q^34 + 104 * q^36 + 112 * q^37 - 24 * q^42 - 240 * q^43 - 112 * q^45 + 224 * q^46 - 64 * q^48 + 328 * q^49 - 32 * q^51 - 112 * q^52 - 168 * q^54 - 56 * q^58 - 336 * q^60 + 208 * q^61 - 104 * q^63 - 320 * q^64 - 80 * q^66 - 232 * q^67 + 112 * q^70 + 160 * q^72 + 324 * q^75 + 448 * q^76 + 152 * q^78 - 136 * q^79 + 184 * q^81 + 176 * q^82 + 64 * q^84 - 112 * q^85 - 64 * q^88 + 392 * q^90 + 152 * q^91 + 64 * q^93 - 368 * q^94 + 512 * q^96 - 480 * q^97 + 160 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 2\nu^{5} + 10\nu^{3} + 24\nu^{2} + 8\nu ) / 24$$ (v^7 + 2*v^5 + 10*v^3 + 24*v^2 + 8*v) / 24 $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 12$$ (v^6 + 2*v^4 - 2*v^2 - 4) / 12 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 2\nu^{5} + 2\nu^{3} - 8\nu ) / 12$$ (-v^7 - 2*v^5 + 2*v^3 - 8*v) / 12 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 2\nu^{5} + 2\nu^{3} + 16\nu ) / 12$$ (-v^7 - 2*v^5 + 2*v^3 + 16*v) / 12 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 4\nu^{5} + 2\nu^{3} + 4\nu ) / 12$$ (-v^7 + 4*v^5 + 2*v^3 + 4*v) / 12 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - \nu^{5} + 4\nu^{3} - 10\nu ) / 6$$ (v^7 - v^5 + 4*v^3 - 10*v) / 6 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} - 8\nu^{6} - 2\nu^{5} + 8\nu^{4} - 10\nu^{3} - 8\nu^{2} - 8\nu + 32 ) / 24$$ (-v^7 - 8*v^6 - 2*v^5 + 8*v^4 - 10*v^3 - 8*v^2 - 8*v + 32) / 24
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} ) / 2$$ (b4 - b3) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_1 ) / 2$$ (-b6 - b5 - b4 + b3 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4}$$ b6 + b5 + b4 $$\nu^{4}$$ $$=$$ $$\beta_{7} + 4\beta_{2} + \beta_1$$ b7 + 4*b2 + b1 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - \beta_{4} - \beta_{3}$$ 2*b5 - b4 - b3 $$\nu^{6}$$ $$=$$ $$-2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4\beta_{2} + 4$$ -2*b7 - b6 - b5 - b4 + b3 + 4*b2 + 4 $$\nu^{7}$$ $$=$$ $$2\beta_{6} - 2\beta_{5} - 6\beta_{3}$$ 2*b6 - 2*b5 - 6*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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
 −1.38255 + 0.297594i 0.767178 + 1.18804i −0.767178 − 1.18804i 1.38255 − 0.297594i −1.38255 − 0.297594i 0.767178 − 1.18804i −0.767178 + 1.18804i 1.38255 + 0.297594i
−1.68014 1.08495i 2.90783 0.737922i 1.64575 + 3.64575i 1.57472 1.57472i −5.68618 1.91505i 3.64575i 1.19038 7.91094i 7.91094 4.29150i −4.35425 + 0.937254i
5.2 −0.420861 + 1.95522i −2.77809 1.13234i −3.64575 1.64575i −6.28651 + 6.28651i 3.38317 4.95522i 1.64575i 4.75216 6.43560i 6.43560 + 6.29150i −9.64575 14.9373i
5.3 0.420861 1.95522i 1.13234 + 2.77809i −3.64575 1.64575i 6.28651 6.28651i 5.90834 1.04478i 1.64575i −4.75216 + 6.43560i −6.43560 + 6.29150i −9.64575 14.9373i
5.4 1.68014 + 1.08495i 0.737922 2.90783i 1.64575 + 3.64575i −1.57472 + 1.57472i 4.39467 4.08495i 3.64575i −1.19038 + 7.91094i −7.91094 4.29150i −4.35425 + 0.937254i
29.1 −1.68014 + 1.08495i 2.90783 + 0.737922i 1.64575 3.64575i 1.57472 + 1.57472i −5.68618 + 1.91505i 3.64575i 1.19038 + 7.91094i 7.91094 + 4.29150i −4.35425 0.937254i
29.2 −0.420861 1.95522i −2.77809 + 1.13234i −3.64575 + 1.64575i −6.28651 6.28651i 3.38317 + 4.95522i 1.64575i 4.75216 + 6.43560i 6.43560 6.29150i −9.64575 + 14.9373i
29.3 0.420861 + 1.95522i 1.13234 2.77809i −3.64575 + 1.64575i 6.28651 + 6.28651i 5.90834 + 1.04478i 1.64575i −4.75216 6.43560i −6.43560 6.29150i −9.64575 + 14.9373i
29.4 1.68014 1.08495i 0.737922 + 2.90783i 1.64575 3.64575i −1.57472 1.57472i 4.39467 + 4.08495i 3.64575i −1.19038 7.91094i −7.91094 + 4.29150i −4.35425 0.937254i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.4 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.i.a 8
3.b odd 2 1 inner 48.3.i.a 8
4.b odd 2 1 192.3.i.a 8
8.b even 2 1 384.3.i.a 8
8.d odd 2 1 384.3.i.b 8
12.b even 2 1 192.3.i.a 8
16.e even 4 1 inner 48.3.i.a 8
16.e even 4 1 384.3.i.a 8
16.f odd 4 1 192.3.i.a 8
16.f odd 4 1 384.3.i.b 8
24.f even 2 1 384.3.i.b 8
24.h odd 2 1 384.3.i.a 8
48.i odd 4 1 inner 48.3.i.a 8
48.i odd 4 1 384.3.i.a 8
48.k even 4 1 192.3.i.a 8
48.k even 4 1 384.3.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.a 8 1.a even 1 1 trivial
48.3.i.a 8 3.b odd 2 1 inner
48.3.i.a 8 16.e even 4 1 inner
48.3.i.a 8 48.i odd 4 1 inner
192.3.i.a 8 4.b odd 2 1
192.3.i.a 8 12.b even 2 1
192.3.i.a 8 16.f odd 4 1
192.3.i.a 8 48.k even 4 1
384.3.i.a 8 8.b even 2 1
384.3.i.a 8 16.e even 4 1
384.3.i.a 8 24.h odd 2 1
384.3.i.a 8 48.i odd 4 1
384.3.i.b 8 8.d odd 2 1
384.3.i.b 8 16.f odd 4 1
384.3.i.b 8 24.f even 2 1
384.3.i.b 8 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6272T_{5}^{4} + 153664$$ acting on $$S_{3}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{6} + 8 T^{4} + 64 T^{2} + \cdots + 256$$
$3$ $$T^{8} - 4 T^{7} + 8 T^{6} + 12 T^{5} + \cdots + 6561$$
$5$ $$T^{8} + 6272 T^{4} + 153664$$
$7$ $$(T^{4} + 16 T^{2} + 36)^{2}$$
$11$ $$T^{8} + 2048 T^{4} + 16384$$
$13$ $$(T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 75076)^{2}$$
$17$ $$(T^{4} + 920 T^{2} + 103968)^{2}$$
$19$ $$(T^{4} - 8 T^{3} + 32 T^{2} + 1728 T + 46656)^{2}$$
$23$ $$(T^{4} - 1120 T^{2} + 225792)^{2}$$
$29$ $$T^{8} + 614656 T^{4} + \cdots + 29884728384$$
$31$ $$(T^{2} - 18 T + 74)^{4}$$
$37$ $$(T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 1764)^{2}$$
$41$ $$(T^{4} - 664 T^{2} + 2592)^{2}$$
$43$ $$(T^{4} + 120 T^{3} + 7200 T^{2} + \cdots + 2483776)^{2}$$
$47$ $$(T^{4} + 8144 T^{2} + 14193792)^{2}$$
$53$ $$T^{8} + 18284288 T^{4} + \cdots + 18847994944$$
$59$ $$T^{8} + 3520000 T^{4} + \cdots + 2624400000000$$
$61$ $$(T^{4} - 104 T^{3} + 5408 T^{2} + \cdots + 1004004)^{2}$$
$67$ $$(T^{4} + 116 T^{3} + 6728 T^{2} + \cdots + 2155024)^{2}$$
$71$ $$(T^{4} - 16560 T^{2} + 24668288)^{2}$$
$73$ $$(T^{4} + 3712 T^{2} + 788544)^{2}$$
$79$ $$(T^{2} + 34 T - 894)^{4}$$
$83$ $$T^{8} + 135735296 T^{4} + \cdots + 10\!\cdots\!04$$
$89$ $$(T^{4} - 6240 T^{2} + 184832)^{2}$$
$97$ $$(T^{2} + 120 T + 3152)^{4}$$