# Properties

 Label 1078.2.i.a Level $1078$ Weight $2$ Character orbit 1078.i Analytic conductor $8.608$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(901,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("1078.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.i (of order $$6$$, degree $$2$$, not 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: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{48})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2}) q^{3} + \beta_{3} q^{4} + ( - \beta_{13} - \beta_{2}) q^{5} + ( - \beta_{15} - \beta_{11}) q^{6} + \beta_{5} q^{8} + (2 \beta_{14} - 2 \beta_{6} - \beta_{3} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b10 + b7 - b2) * q^3 + b3 * q^4 + (-b13 - b2) * q^5 + (-b15 - b11) * q^6 + b5 * q^8 + (2*b14 - 2*b6 - b3 + 1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2}) q^{3} + \beta_{3} q^{4} + ( - \beta_{13} - \beta_{2}) q^{5} + ( - \beta_{15} - \beta_{11}) q^{6} + \beta_{5} q^{8} + (2 \beta_{14} - 2 \beta_{6} - \beta_{3} + 1) q^{9} + ( - \beta_{11} + \beta_{8} - \beta_{4}) q^{10} + ( - \beta_{6} - 3 \beta_{5} + 3 \beta_1) q^{11} + ( - \beta_{13} - \beta_{2}) q^{12} + 3 \beta_{15} q^{13} + (2 \beta_{14} + 4) q^{15} + (\beta_{3} - 1) q^{16} + (2 \beta_{9} - \beta_{5} + \beta_1) q^{18} - 3 \beta_{8} q^{19} + ( - \beta_{13} + \beta_{10} - \beta_{7}) q^{20} + ( - \beta_{12} + \beta_{9} + 3) q^{22} + (3 \beta_{14} - 3 \beta_{6} + 2 \beta_{3} - 2) q^{23} + ( - \beta_{11} + \beta_{8} - \beta_{4}) q^{24} + (2 \beta_{6} - \beta_{3}) q^{25} + 3 \beta_{2} q^{26} + ( - 2 \beta_{13} + 2 \beta_{10} + 2 \beta_{7}) q^{27} + (3 \beta_{12} - 3 \beta_{9} + 6 \beta_{5}) q^{29} + (2 \beta_{12} + 4 \beta_1) q^{30} + ( - 2 \beta_{10} + 3 \beta_{7} - 3 \beta_{2}) q^{31} + (\beta_{5} - \beta_1) q^{32} + ( - 3 \beta_{15} - 3 \beta_{8} + 3 \beta_{4} + 2 \beta_{2}) q^{33} + (2 \beta_{14} + 1) q^{36} + (6 \beta_{3} - 6) q^{37} - 3 \beta_{10} q^{38} + ( - 6 \beta_{9} + 6 \beta_{5} - 6 \beta_1) q^{39} + (\beta_{15} + \beta_{8} - \beta_{4}) q^{40} + 6 \beta_{15} q^{41} + ( - 3 \beta_{12} + 3 \beta_{9}) q^{43} + (\beta_{14} - \beta_{6} + 3 \beta_1) q^{44} + ( - \beta_{10} + 5 \beta_{7} - 5 \beta_{2}) q^{45} + (3 \beta_{9} + 2 \beta_{5} - 2 \beta_1) q^{46} + ( - \beta_{13} + 4 \beta_{2}) q^{47} + ( - \beta_{13} + \beta_{10} - \beta_{7}) q^{48} + (2 \beta_{12} - 2 \beta_{9} - \beta_{5}) q^{50} + 3 \beta_{4} q^{52} + (6 \beta_{6} + 4 \beta_{3}) q^{53} + ( - 2 \beta_{15} + 2 \beta_{8} + 2 \beta_{4}) q^{54} + ( - 3 \beta_{15} - 3 \beta_{11} + 2 \beta_{7}) q^{55} - 6 \beta_{5} q^{57} + ( - 3 \beta_{14} + 3 \beta_{6} + 6 \beta_{3} - 6) q^{58} + ( - \beta_{10} - 7 \beta_{7} + 7 \beta_{2}) q^{59} + (2 \beta_{6} + 4 \beta_{3}) q^{60} - 3 \beta_{8} q^{61} + ( - 3 \beta_{15} - 2 \beta_{11}) q^{62} - q^{64} + ( - 6 \beta_{12} - 6 \beta_1) q^{65} + ( - 3 \beta_{10} + 3 \beta_{7} + 2 \beta_{4} - 3 \beta_{2}) q^{66} + (4 \beta_{6} - 6 \beta_{3}) q^{67} + ( - 2 \beta_{13} + 2 \beta_{10} + 4 \beta_{7}) q^{69} + 2 q^{71} + (2 \beta_{12} + \beta_1) q^{72} + (6 \beta_{11} - 6 \beta_{8} - 6 \beta_{4}) q^{73} + (6 \beta_{5} - 6 \beta_1) q^{74} + (\beta_{13} - 3 \beta_{2}) q^{75} - 3 \beta_{11} q^{76} + ( - 6 \beta_{14} - 6) q^{78} - 6 \beta_{12} q^{79} + (\beta_{10} - \beta_{7} + \beta_{2}) q^{80} + (2 \beta_{6} + 3 \beta_{3}) q^{81} + 6 \beta_{2} q^{82} + ( - 3 \beta_{15} + 6 \beta_{11}) q^{83} + (3 \beta_{14} - 3 \beta_{6}) q^{86} + ( - 6 \beta_{11} + 6 \beta_{8} - 12 \beta_{4}) q^{87} + (\beta_{9} + 3 \beta_{3}) q^{88} + (4 \beta_{13} + 5 \beta_{2}) q^{89} + ( - 5 \beta_{15} - \beta_{11}) q^{90} + (3 \beta_{14} - 2) q^{92} + (6 \beta_{14} - 6 \beta_{6} - 10 \beta_{3} + 10) q^{93} + ( - \beta_{11} + \beta_{8} + 4 \beta_{4}) q^{94} + ( - 6 \beta_{5} + 6 \beta_1) q^{95} + (\beta_{15} + \beta_{8} - \beta_{4}) q^{96} + ( - 3 \beta_{13} + 3 \beta_{10} + 4 \beta_{7}) q^{97} + ( - \beta_{14} - 6 \beta_{12} + 6 \beta_{9} - 3 \beta_{5} - 4) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b10 + b7 - b2) * q^3 + b3 * q^4 + (-b13 - b2) * q^5 + (-b15 - b11) * q^6 + b5 * q^8 + (2*b14 - 2*b6 - b3 + 1) * q^9 + (-b11 + b8 - b4) * q^10 + (-b6 - 3*b5 + 3*b1) * q^11 + (-b13 - b2) * q^12 + 3*b15 * q^13 + (2*b14 + 4) * q^15 + (b3 - 1) * q^16 + (2*b9 - b5 + b1) * q^18 - 3*b8 * q^19 + (-b13 + b10 - b7) * q^20 + (-b12 + b9 + 3) * q^22 + (3*b14 - 3*b6 + 2*b3 - 2) * q^23 + (-b11 + b8 - b4) * q^24 + (2*b6 - b3) * q^25 + 3*b2 * q^26 + (-2*b13 + 2*b10 + 2*b7) * q^27 + (3*b12 - 3*b9 + 6*b5) * q^29 + (2*b12 + 4*b1) * q^30 + (-2*b10 + 3*b7 - 3*b2) * q^31 + (b5 - b1) * q^32 + (-3*b15 - 3*b8 + 3*b4 + 2*b2) * q^33 + (2*b14 + 1) * q^36 + (6*b3 - 6) * q^37 - 3*b10 * q^38 + (-6*b9 + 6*b5 - 6*b1) * q^39 + (b15 + b8 - b4) * q^40 + 6*b15 * q^41 + (-3*b12 + 3*b9) * q^43 + (b14 - b6 + 3*b1) * q^44 + (-b10 + 5*b7 - 5*b2) * q^45 + (3*b9 + 2*b5 - 2*b1) * q^46 + (-b13 + 4*b2) * q^47 + (-b13 + b10 - b7) * q^48 + (2*b12 - 2*b9 - b5) * q^50 + 3*b4 * q^52 + (6*b6 + 4*b3) * q^53 + (-2*b15 + 2*b8 + 2*b4) * q^54 + (-3*b15 - 3*b11 + 2*b7) * q^55 - 6*b5 * q^57 + (-3*b14 + 3*b6 + 6*b3 - 6) * q^58 + (-b10 - 7*b7 + 7*b2) * q^59 + (2*b6 + 4*b3) * q^60 - 3*b8 * q^61 + (-3*b15 - 2*b11) * q^62 - q^64 + (-6*b12 - 6*b1) * q^65 + (-3*b10 + 3*b7 + 2*b4 - 3*b2) * q^66 + (4*b6 - 6*b3) * q^67 + (-2*b13 + 2*b10 + 4*b7) * q^69 + 2 * q^71 + (2*b12 + b1) * q^72 + (6*b11 - 6*b8 - 6*b4) * q^73 + (6*b5 - 6*b1) * q^74 + (b13 - 3*b2) * q^75 - 3*b11 * q^76 + (-6*b14 - 6) * q^78 - 6*b12 * q^79 + (b10 - b7 + b2) * q^80 + (2*b6 + 3*b3) * q^81 + 6*b2 * q^82 + (-3*b15 + 6*b11) * q^83 + (3*b14 - 3*b6) * q^86 + (-6*b11 + 6*b8 - 12*b4) * q^87 + (b9 + 3*b3) * q^88 + (4*b13 + 5*b2) * q^89 + (-5*b15 - b11) * q^90 + (3*b14 - 2) * q^92 + (6*b14 - 6*b6 - 10*b3 + 10) * q^93 + (-b11 + b8 + 4*b4) * q^94 + (-6*b5 + 6*b1) * q^95 + (b15 + b8 - b4) * q^96 + (-3*b13 + 3*b10 + 4*b7) * q^97 + (-b14 - 6*b12 + 6*b9 - 3*b5 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + 8 q^{9}+O(q^{10})$$ 16 * q + 8 * q^4 + 8 * q^9 $$16 q + 8 q^{4} + 8 q^{9} + 64 q^{15} - 8 q^{16} + 48 q^{22} - 16 q^{23} - 8 q^{25} + 16 q^{36} - 48 q^{37} + 32 q^{53} - 48 q^{58} + 32 q^{60} - 16 q^{64} - 48 q^{67} + 32 q^{71} - 96 q^{78} + 24 q^{81} + 24 q^{88} - 32 q^{92} + 80 q^{93} - 64 q^{99}+O(q^{100})$$ 16 * q + 8 * q^4 + 8 * q^9 + 64 * q^15 - 8 * q^16 + 48 * q^22 - 16 * q^23 - 8 * q^25 + 16 * q^36 - 48 * q^37 + 32 * q^53 - 48 * q^58 + 32 * q^60 - 16 * q^64 - 48 * q^67 + 32 * q^71 - 96 * q^78 + 24 * q^81 + 24 * q^88 - 32 * q^92 + 80 * q^93 - 64 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{48}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{48}^{7} + \zeta_{48}$$ v^7 + v $$\beta_{3}$$ $$=$$ $$\zeta_{48}^{8}$$ v^8 $$\beta_{4}$$ $$=$$ $$\zeta_{48}^{11} + \zeta_{48}^{5}$$ v^11 + v^5 $$\beta_{5}$$ $$=$$ $$\zeta_{48}^{12}$$ v^12 $$\beta_{6}$$ $$=$$ $$\zeta_{48}^{14} + \zeta_{48}^{2}$$ v^14 + v^2 $$\beta_{7}$$ $$=$$ $$\zeta_{48}^{15} + \zeta_{48}^{9}$$ v^15 + v^9 $$\beta_{8}$$ $$=$$ $$-\zeta_{48}^{7} + \zeta_{48}$$ -v^7 + v $$\beta_{9}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{2}$$ -v^14 + v^2 $$\beta_{10}$$ $$=$$ $$-\zeta_{48}^{11} + \zeta_{48}^{5}$$ -v^11 + v^5 $$\beta_{11}$$ $$=$$ $$-\zeta_{48}^{15} + \zeta_{48}^{9}$$ -v^15 + v^9 $$\beta_{12}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6}$$ -v^14 + v^10 + v^6 $$\beta_{13}$$ $$=$$ $$\zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3}$$ v^13 - v^11 + v^3 $$\beta_{14}$$ $$=$$ $$-\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2}$$ -v^10 + v^6 + v^2 $$\beta_{15}$$ $$=$$ $$-\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3}$$ -v^13 + v^5 + v^3
 $$\zeta_{48}$$ $$=$$ $$( \beta_{8} + \beta_{2} ) / 2$$ (b8 + b2) / 2 $$\zeta_{48}^{2}$$ $$=$$ $$( \beta_{9} + \beta_{6} ) / 2$$ (b9 + b6) / 2 $$\zeta_{48}^{3}$$ $$=$$ $$( \beta_{15} + \beta_{13} - \beta_{10} ) / 2$$ (b15 + b13 - b10) / 2 $$\zeta_{48}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{48}^{5}$$ $$=$$ $$( \beta_{10} + \beta_{4} ) / 2$$ (b10 + b4) / 2 $$\zeta_{48}^{6}$$ $$=$$ $$( \beta_{14} + \beta_{12} - \beta_{9} ) / 2$$ (b14 + b12 - b9) / 2 $$\zeta_{48}^{7}$$ $$=$$ $$( -\beta_{8} + \beta_{2} ) / 2$$ (-b8 + b2) / 2 $$\zeta_{48}^{8}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{48}^{9}$$ $$=$$ $$( \beta_{11} + \beta_{7} ) / 2$$ (b11 + b7) / 2 $$\zeta_{48}^{10}$$ $$=$$ $$( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2$$ (-b14 + b12 + b6) / 2 $$\zeta_{48}^{11}$$ $$=$$ $$( -\beta_{10} + \beta_{4} ) / 2$$ (-b10 + b4) / 2 $$\zeta_{48}^{12}$$ $$=$$ $$\beta_{5}$$ b5 $$\zeta_{48}^{13}$$ $$=$$ $$( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2$$ (-b15 + b13 + b4) / 2 $$\zeta_{48}^{14}$$ $$=$$ $$( -\beta_{9} + \beta_{6} ) / 2$$ (-b9 + b6) / 2 $$\zeta_{48}^{15}$$ $$=$$ $$( -\beta_{11} + \beta_{7} ) / 2$$ (-b11 + b7) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$1 - \beta_{3}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 0.608761 + 0.793353i −0.793353 + 0.608761i 0.793353 − 0.608761i −0.608761 − 0.793353i 0.991445 + 0.130526i 0.130526 − 0.991445i −0.130526 + 0.991445i −0.991445 − 0.130526i 0.608761 − 0.793353i −0.793353 − 0.608761i 0.793353 + 0.608761i −0.608761 + 0.793353i 0.991445 − 0.130526i 0.130526 + 0.991445i −0.130526 − 0.991445i −0.991445 + 0.130526i
−0.866025 0.500000i −2.26303 + 1.30656i 0.500000 + 0.866025i −2.26303 1.30656i 2.61313 0 1.00000i 1.91421 3.31552i 1.30656 + 2.26303i
901.2 −0.866025 0.500000i −0.937379 + 0.541196i 0.500000 + 0.866025i −0.937379 0.541196i 1.08239 0 1.00000i −0.914214 + 1.58346i 0.541196 + 0.937379i
901.3 −0.866025 0.500000i 0.937379 0.541196i 0.500000 + 0.866025i 0.937379 + 0.541196i −1.08239 0 1.00000i −0.914214 + 1.58346i −0.541196 0.937379i
901.4 −0.866025 0.500000i 2.26303 1.30656i 0.500000 + 0.866025i 2.26303 + 1.30656i −2.61313 0 1.00000i 1.91421 3.31552i −1.30656 2.26303i
901.5 0.866025 + 0.500000i −2.26303 + 1.30656i 0.500000 + 0.866025i −2.26303 1.30656i −2.61313 0 1.00000i 1.91421 3.31552i −1.30656 2.26303i
901.6 0.866025 + 0.500000i −0.937379 + 0.541196i 0.500000 + 0.866025i −0.937379 0.541196i −1.08239 0 1.00000i −0.914214 + 1.58346i −0.541196 0.937379i
901.7 0.866025 + 0.500000i 0.937379 0.541196i 0.500000 + 0.866025i 0.937379 + 0.541196i 1.08239 0 1.00000i −0.914214 + 1.58346i 0.541196 + 0.937379i
901.8 0.866025 + 0.500000i 2.26303 1.30656i 0.500000 + 0.866025i 2.26303 + 1.30656i 2.61313 0 1.00000i 1.91421 3.31552i 1.30656 + 2.26303i
1011.1 −0.866025 + 0.500000i −2.26303 1.30656i 0.500000 0.866025i −2.26303 + 1.30656i 2.61313 0 1.00000i 1.91421 + 3.31552i 1.30656 2.26303i
1011.2 −0.866025 + 0.500000i −0.937379 0.541196i 0.500000 0.866025i −0.937379 + 0.541196i 1.08239 0 1.00000i −0.914214 1.58346i 0.541196 0.937379i
1011.3 −0.866025 + 0.500000i 0.937379 + 0.541196i 0.500000 0.866025i 0.937379 0.541196i −1.08239 0 1.00000i −0.914214 1.58346i −0.541196 + 0.937379i
1011.4 −0.866025 + 0.500000i 2.26303 + 1.30656i 0.500000 0.866025i 2.26303 1.30656i −2.61313 0 1.00000i 1.91421 + 3.31552i −1.30656 + 2.26303i
1011.5 0.866025 0.500000i −2.26303 1.30656i 0.500000 0.866025i −2.26303 + 1.30656i −2.61313 0 1.00000i 1.91421 + 3.31552i −1.30656 + 2.26303i
1011.6 0.866025 0.500000i −0.937379 0.541196i 0.500000 0.866025i −0.937379 + 0.541196i −1.08239 0 1.00000i −0.914214 1.58346i −0.541196 + 0.937379i
1011.7 0.866025 0.500000i 0.937379 + 0.541196i 0.500000 0.866025i 0.937379 0.541196i 1.08239 0 1.00000i −0.914214 1.58346i 0.541196 0.937379i
1011.8 0.866025 0.500000i 2.26303 + 1.30656i 0.500000 0.866025i 2.26303 1.30656i 2.61313 0 1.00000i 1.91421 + 3.31552i 1.30656 2.26303i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.i.a 16
7.b odd 2 1 inner 1078.2.i.a 16
7.c even 3 1 1078.2.c.a 8
7.c even 3 1 inner 1078.2.i.a 16
7.d odd 6 1 1078.2.c.a 8
7.d odd 6 1 inner 1078.2.i.a 16
11.b odd 2 1 inner 1078.2.i.a 16
77.b even 2 1 inner 1078.2.i.a 16
77.h odd 6 1 1078.2.c.a 8
77.h odd 6 1 inner 1078.2.i.a 16
77.i even 6 1 1078.2.c.a 8
77.i even 6 1 inner 1078.2.i.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.a 8 7.c even 3 1
1078.2.c.a 8 7.d odd 6 1
1078.2.c.a 8 77.h odd 6 1
1078.2.c.a 8 77.i even 6 1
1078.2.i.a 16 1.a even 1 1 trivial
1078.2.i.a 16 7.b odd 2 1 inner
1078.2.i.a 16 7.c even 3 1 inner
1078.2.i.a 16 7.d odd 6 1 inner
1078.2.i.a 16 11.b odd 2 1 inner
1078.2.i.a 16 77.b even 2 1 inner
1078.2.i.a 16 77.h odd 6 1 inner
1078.2.i.a 16 77.i even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 8T_{3}^{6} + 56T_{3}^{4} - 64T_{3}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$(T^{8} - 8 T^{6} + 56 T^{4} - 64 T^{2} + \cdots + 64)^{2}$$
$5$ $$(T^{8} - 8 T^{6} + 56 T^{4} - 64 T^{2} + \cdots + 64)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} - 14 T^{6} + 75 T^{4} - 1694 T^{2} + \cdots + 14641)^{2}$$
$13$ $$(T^{4} - 36 T^{2} + 162)^{4}$$
$17$ $$T^{16}$$
$19$ $$(T^{8} + 36 T^{6} + 1134 T^{4} + \cdots + 26244)^{2}$$
$23$ $$(T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196)^{4}$$
$29$ $$(T^{4} + 108 T^{2} + 324)^{4}$$
$31$ $$(T^{8} - 52 T^{6} + 2606 T^{4} + \cdots + 9604)^{2}$$
$37$ $$(T^{2} + 6 T + 36)^{8}$$
$41$ $$(T^{4} - 144 T^{2} + 2592)^{4}$$
$43$ $$(T^{2} + 18)^{8}$$
$47$ $$(T^{8} - 68 T^{6} + 3566 T^{4} + \cdots + 1119364)^{2}$$
$53$ $$(T^{4} - 8 T^{3} + 120 T^{2} + 448 T + 3136)^{4}$$
$59$ $$(T^{8} - 200 T^{6} + 32312 T^{4} + \cdots + 59105344)^{2}$$
$61$ $$(T^{8} + 36 T^{6} + 1134 T^{4} + \cdots + 26244)^{2}$$
$67$ $$(T^{4} + 12 T^{3} + 140 T^{2} + 48 T + 16)^{4}$$
$71$ $$(T - 2)^{16}$$
$73$ $$(T^{8} + 288 T^{6} + 72576 T^{4} + \cdots + 107495424)^{2}$$
$79$ $$(T^{4} - 72 T^{2} + 5184)^{4}$$
$83$ $$(T^{4} - 180 T^{2} + 162)^{4}$$
$89$ $$(T^{8} - 164 T^{6} + 24974 T^{4} + \cdots + 3694084)^{2}$$
$97$ $$(T^{4} + 100 T^{2} + 1922)^{4}$$