# Properties

 Label 48.2.j.a Level $48$ Weight $2$ Character orbit 48.j Analytic conductor $0.383$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,2,Mod(13,48)]

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, 3, 0]))

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

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

chi := DirichletCharacter("48.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 48.j (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: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5$$ 2*v^7 - 7*v^6 + 24*v^5 - 42*v^4 + 59*v^3 - 48*v^2 + 24*v - 5 $$\beta_{2}$$ $$=$$ $$-2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8$$ -2*v^7 + 7*v^6 - 24*v^5 + 43*v^4 - 61*v^3 + 54*v^2 - 29*v + 8 $$\beta_{3}$$ $$=$$ $$-3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11$$ -3*v^7 + 10*v^6 - 35*v^5 + 60*v^4 - 87*v^3 + 73*v^2 - 42*v + 11 $$\beta_{4}$$ $$=$$ $$-3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13$$ -3*v^7 + 11*v^6 - 38*v^5 + 70*v^4 - 102*v^3 + 91*v^2 - 53*v + 13 $$\beta_{5}$$ $$=$$ $$5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19$$ 5*v^7 - 17*v^6 + 60*v^5 - 105*v^4 + 155*v^3 - 133*v^2 + 77*v - 19 $$\beta_{6}$$ $$=$$ $$-5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23$$ -5*v^7 + 18*v^6 - 63*v^5 + 115*v^4 - 170*v^3 + 152*v^2 - 89*v + 23 $$\beta_{7}$$ $$=$$ $$-8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31$$ -8*v^7 + 28*v^6 - 98*v^5 + 175*v^4 - 256*v^3 + 223*v^2 - 126*v + 31
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (-b7 + b6 - b5 - b4 - b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2$$ (-b7 + 3*b6 + b5 - 3*b4 + b3 + b2 - b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2$$ (5*b7 - b6 + 7*b5 - b4 + 5*b3 - 3*b2 + 3*b1 - 2) / 2 $$\nu^{4}$$ $$=$$ $$( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2$$ (11*b7 - 15*b6 + 3*b5 + 11*b4 - b3 - 5*b2 + 9*b1 + 14) / 2 $$\nu^{5}$$ $$=$$ $$( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2$$ (-13*b7 - 11*b6 - 29*b5 + 17*b4 - 23*b3 + 13*b2 - 3*b1 + 18) / 2 $$\nu^{6}$$ $$=$$ $$( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2$$ (-67*b7 + 59*b6 - 41*b5 - 29*b4 - 15*b3 + 37*b2 - 47*b1 - 48) / 2 $$\nu^{7}$$ $$=$$ $$( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2$$ (-7*b7 + 113*b6 + 97*b5 - 105*b4 + 91*b3 - 31*b2 - 39*b1 - 122) / 2

## Character values

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

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.5 + 2.10607i 0.5 − 1.44392i 0.5 − 0.691860i 0.5 + 0.0297061i 0.5 − 2.10607i 0.5 + 1.44392i 0.5 + 0.691860i 0.5 − 0.0297061i
−1.34277 0.443806i 0.707107 + 0.707107i 1.60607 + 1.19186i 1.27133 1.27133i −0.635665 1.26330i 0.158942i −1.62764 2.31318i 1.00000i −2.27133 + 1.14288i
13.2 −0.167452 1.40426i −0.707107 0.707107i −1.94392 + 0.470294i 1.74912 1.74912i −0.874559 + 1.11137i 2.55765i 0.985930 + 2.65103i 1.00000i −2.74912 2.16333i
13.3 0.635665 1.26330i 0.707107 + 0.707107i −1.19186 1.60607i −2.68554 + 2.68554i 1.34277 0.443806i 2.15894i −2.78658 + 0.484753i 1.00000i 1.68554 + 5.09976i
13.4 0.874559 + 1.11137i −0.707107 0.707107i −0.470294 + 1.94392i −0.334904 + 0.334904i 0.167452 1.40426i 4.55765i −2.57172 + 1.17740i 1.00000i −0.665096 0.0793096i
37.1 −1.34277 + 0.443806i 0.707107 0.707107i 1.60607 1.19186i 1.27133 + 1.27133i −0.635665 + 1.26330i 0.158942i −1.62764 + 2.31318i 1.00000i −2.27133 1.14288i
37.2 −0.167452 + 1.40426i −0.707107 + 0.707107i −1.94392 0.470294i 1.74912 + 1.74912i −0.874559 1.11137i 2.55765i 0.985930 2.65103i 1.00000i −2.74912 + 2.16333i
37.3 0.635665 + 1.26330i 0.707107 0.707107i −1.19186 + 1.60607i −2.68554 2.68554i 1.34277 + 0.443806i 2.15894i −2.78658 0.484753i 1.00000i 1.68554 5.09976i
37.4 0.874559 1.11137i −0.707107 + 0.707107i −0.470294 1.94392i −0.334904 0.334904i 0.167452 + 1.40426i 4.55765i −2.57172 1.17740i 1.00000i −0.665096 + 0.0793096i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.j.a 8
3.b odd 2 1 144.2.k.b 8
4.b odd 2 1 192.2.j.a 8
8.b even 2 1 384.2.j.b 8
8.d odd 2 1 384.2.j.a 8
12.b even 2 1 576.2.k.b 8
16.e even 4 1 inner 48.2.j.a 8
16.e even 4 1 384.2.j.b 8
16.f odd 4 1 192.2.j.a 8
16.f odd 4 1 384.2.j.a 8
24.f even 2 1 1152.2.k.f 8
24.h odd 2 1 1152.2.k.c 8
32.g even 8 1 3072.2.a.i 4
32.g even 8 1 3072.2.a.t 4
32.g even 8 2 3072.2.d.f 8
32.h odd 8 1 3072.2.a.n 4
32.h odd 8 1 3072.2.a.o 4
32.h odd 8 2 3072.2.d.i 8
48.i odd 4 1 144.2.k.b 8
48.i odd 4 1 1152.2.k.c 8
48.k even 4 1 576.2.k.b 8
48.k even 4 1 1152.2.k.f 8
96.o even 8 1 9216.2.a.x 4
96.o even 8 1 9216.2.a.bn 4
96.p odd 8 1 9216.2.a.y 4
96.p odd 8 1 9216.2.a.bo 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 1.a even 1 1 trivial
48.2.j.a 8 16.e even 4 1 inner
144.2.k.b 8 3.b odd 2 1
144.2.k.b 8 48.i odd 4 1
192.2.j.a 8 4.b odd 2 1
192.2.j.a 8 16.f odd 4 1
384.2.j.a 8 8.d odd 2 1
384.2.j.a 8 16.f odd 4 1
384.2.j.b 8 8.b even 2 1
384.2.j.b 8 16.e even 4 1
576.2.k.b 8 12.b even 2 1
576.2.k.b 8 48.k even 4 1
1152.2.k.c 8 24.h odd 2 1
1152.2.k.c 8 48.i odd 4 1
1152.2.k.f 8 24.f even 2 1
1152.2.k.f 8 48.k even 4 1
3072.2.a.i 4 32.g even 8 1
3072.2.a.n 4 32.h odd 8 1
3072.2.a.o 4 32.h odd 8 1
3072.2.a.t 4 32.g even 8 1
3072.2.d.f 8 32.g even 8 2
3072.2.d.i 8 32.h odd 8 2
9216.2.a.x 4 96.o even 8 1
9216.2.a.y 4 96.p odd 8 1
9216.2.a.bn 4 96.o even 8 1
9216.2.a.bo 4 96.p odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{6} + \cdots + 16$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8} - 16 T^{5} + \cdots + 64$$
$7$ $$T^{8} + 32 T^{6} + \cdots + 16$$
$11$ $$T^{8} + 8 T^{7} + \cdots + 1024$$
$13$ $$T^{8} - 64 T^{5} + \cdots + 16$$
$17$ $$(T^{4} - 32 T^{2} + \cdots + 16)^{2}$$
$19$ $$T^{8} + 8 T^{7} + \cdots + 256$$
$23$ $$(T^{2} + 8)^{4}$$
$29$ $$T^{8} + 16 T^{7} + \cdots + 61504$$
$31$ $$(T^{4} - 12 T^{3} + \cdots - 28)^{2}$$
$37$ $$T^{8} + 16 T^{7} + \cdots + 1106704$$
$41$ $$T^{8} + 128 T^{6} + \cdots + 12544$$
$43$ $$T^{8} + 8 T^{7} + \cdots + 12544$$
$47$ $$(T^{2} - 8)^{4}$$
$53$ $$T^{8} - 16 T^{7} + \cdots + 18496$$
$59$ $$(T^{2} - 8 T + 32)^{4}$$
$61$ $$T^{8} - 16 T^{7} + \cdots + 1106704$$
$67$ $$T^{8} + 16 T^{7} + \cdots + 65536$$
$71$ $$T^{8} + 128 T^{6} + \cdots + 4096$$
$73$ $$T^{8} + 256 T^{6} + \cdots + 4096$$
$79$ $$(T^{4} + 12 T^{3} + \cdots - 10108)^{2}$$
$83$ $$T^{8} + 40 T^{7} + \cdots + 1024$$
$89$ $$T^{8} + 464 T^{6} + \cdots + 3625216$$
$97$ $$(T^{4} - 224 T^{2} + \cdots + 512)^{2}$$