# Properties

 Label 1280.2.o.v Level $1280$ Weight $2$ Character orbit 1280.o Analytic conductor $10.221$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(127,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1280.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1280.o (of order $$4$$, 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: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.74906393903104.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 12x^{10} + 50x^{8} + 90x^{6} + 69x^{4} + 18x^{2} + 1$$ x^12 + 12*x^10 + 50*x^8 + 90*x^6 + 69*x^4 + 18*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 640) 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{9}+O(q^{10})$$ q - b2 * q^3 + b7 * q^5 + (-b10 - b8 - b6 - b4 + b2 + 1) * q^7 + (-b11 - b10 - b5 - b4 - b3 + b2) * q^9 $$q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + 8 \beta_{10} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100})$$ q - b2 * q^3 + b7 * q^5 + (-b10 - b8 - b6 - b4 + b2 + 1) * q^7 + (-b11 - b10 - b5 - b4 - b3 + b2) * q^9 + (-b7 - b6 + b1) * q^11 + (b11 + b9 + b7 - b1) * q^13 + (b10 - b9 - b6 - b5 - b2 + b1 - 1) * q^15 + (b10 - b9 + b8 - b7 - b6 + 2*b3 + 1) * q^17 + (-b11 + b9 - b8 - b3 + b2) * q^19 + (2*b10 - b9 + b8 + 2*b3 - 2*b2) * q^21 + (b11 - b10 + b8 - b6 + b5 + b3 - b1 - 1) * q^23 + (b9 + b8 + b7 + b6 - b5 + b4 + b3 - b2 - b1 - 1) * q^25 + (b11 + 2*b10 + b9 + b8 + b7 - b6 + 2*b5 + 2*b3 - b1 + 2) * q^27 + (-b9 - b8 - b7 - b6 + 2*b5 - 2*b4 + 2) * q^29 + (2*b10 - b9 + b8 + b7 - b6 + b5 + b4 + b3 - b2) * q^31 + (b9 + b8 - b7 + b6 + 2*b2) * q^33 + (-2*b10 + 2*b9 + b7 - b6 - 2*b4 + b2 - b1 - 2) * q^35 + (2*b10 + b8 + b6 - 2) * q^37 + (-b9 - b8 - b7 - b6 - b5 + b4 - b3 - b2 + 2) * q^39 + (b5 - b4 - b3 - b2 - b1) * q^41 + (-2*b10 + 2*b9 - 2*b7 + 2*b4 + b2 + 2) * q^43 + (-b11 - 2*b10 + b9 - 2*b5 - 2*b4 + 4*b2 - b1) * q^45 + (-b11 - b10 - b9 + b7 + b4 + b2 - b1 + 1) * q^47 + (b11 - b10 + 2*b7 - 2*b6 + b5 + b4 - b3 + b2) * q^49 + (2*b9 + 2*b8 - 2*b5 + 2*b4 - b3 - b2 - 4) * q^51 + (-b11 - b8 + b6 - 4*b3 + b1) * q^53 + (-b11 + 2*b10 - b7 - b6 + b5 + b4 - 2*b3 + 2*b2 - 2) * q^55 + (-b11 - b9 - b8 - b7 + b6 + 2*b5 + b1) * q^57 + (b11 + 4*b10 + b9 - b8 + 2*b7 - 2*b6 - 2*b5 - 2*b4 + b3 - b2) * q^59 + (-b7 + b6 + 2*b5 + 2*b4 + 2*b3 - 2*b2) * q^61 + (-b11 - 3*b10 - 2*b9 + b8 - 2*b7 - b6 - b5 - b3 + b1 - 3) * q^63 + (b10 + 2*b6 + b5 - b4 + b3 - 3*b2 + b1 - 1) * q^65 + (2*b10 + 2*b9 + 2*b7 - 2*b5 - b3 + 2) * q^67 + (-b7 - b6 - 2*b5 + 2*b4 + 2) * q^69 + (2*b10 - b9 + b8 + b7 - b6 - b5 - b4 - b3 + b2) * q^71 + (b11 + b10 - b9 - b8 + b7 - b6 + 2*b4 + b1 - 1) * q^73 + (-b11 - 4*b10 + b9 - b8 - b7 - b6 - 2*b5 + 2*b4 - 2*b3 + b2 + b1 - 4) * q^75 + (b11 + 4*b4 + b1) * q^77 + (-2*b7 - 2*b6 + 2*b5 - 2*b4 + 4) * q^79 + (-2*b9 - 2*b8 - 2*b7 - 2*b6 - b5 + b4 - 3*b3 - 3*b2 + b1 - 1) * q^81 + (b11 - 6*b10 + b9 - 3*b8 - b7 - 3*b6 - 2*b4 + b2 + b1 + 6) * q^83 + (b11 - 2*b10 - b8 - b7 - 2*b6 + 2*b5 + 2*b4 + 2*b3 + 2*b2 + b1 - 2) * q^85 + (-2*b10 - 2*b9 + 2*b7 - 2*b4 - 4*b2 + 2) * q^87 + (2*b11 + 2*b9 - 2*b8 - 2*b5 - 2*b4 - 2*b3 + 2*b2) * q^89 + (2*b9 + 2*b8 + 2*b7 + 2*b6 + 2*b5 - 2*b4 - b3 - b2 - 2*b1 - 4) * q^91 + (-2*b11 - 2*b10 - b9 + b8 - b7 - b6 - 4*b5 - 4*b3 + 2*b1 - 2) * q^93 + (b11 + 4*b10 + b9 - b8 + 2*b7 - 2*b5 - 2*b4 - b3 + b2 - 2*b1 - 2) * q^95 + (b11 - b10 + 3*b9 - 3*b8 + 3*b7 + 3*b6 - 2*b5 - b1 - 1) * q^97 + (b11 + 8*b10 - b9 + b8 + 4*b5 + 4*b4 + 2*b3 - 2*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{5} + 8 q^{7}+O(q^{10})$$ 12 * q + 4 * q^5 + 8 * q^7 $$12 q + 4 q^{5} + 8 q^{7} - 8 q^{11} + 4 q^{13} - 16 q^{15} + 4 q^{17} - 16 q^{23} - 4 q^{25} + 24 q^{27} + 16 q^{29} - 24 q^{35} - 20 q^{37} + 16 q^{39} + 16 q^{43} + 16 q^{47} - 48 q^{51} + 4 q^{53} - 32 q^{55} - 48 q^{63} - 4 q^{65} + 32 q^{67} + 16 q^{69} - 12 q^{73} - 56 q^{75} + 32 q^{79} - 28 q^{81} + 56 q^{83} - 36 q^{85} + 32 q^{87} - 32 q^{91} - 32 q^{93} - 16 q^{95} + 12 q^{97}+O(q^{100})$$ 12 * q + 4 * q^5 + 8 * q^7 - 8 * q^11 + 4 * q^13 - 16 * q^15 + 4 * q^17 - 16 * q^23 - 4 * q^25 + 24 * q^27 + 16 * q^29 - 24 * q^35 - 20 * q^37 + 16 * q^39 + 16 * q^43 + 16 * q^47 - 48 * q^51 + 4 * q^53 - 32 * q^55 - 48 * q^63 - 4 * q^65 + 32 * q^67 + 16 * q^69 - 12 * q^73 - 56 * q^75 + 32 * q^79 - 28 * q^81 + 56 * q^83 - 36 * q^85 + 32 * q^87 - 32 * q^91 - 32 * q^93 - 16 * q^95 + 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 12x^{10} + 50x^{8} + 90x^{6} + 69x^{4} + 18x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{8} - 10\nu^{6} - 29\nu^{4} - 23\nu^{2}$$ -v^8 - 10*v^6 - 29*v^4 - 23*v^2 $$\beta_{2}$$ $$=$$ $$( \nu^{10} + 11\nu^{8} + 39\nu^{6} + 52\nu^{4} + 23\nu^{2} + 2\nu + 2 ) / 2$$ (v^10 + 11*v^8 + 39*v^6 + 52*v^4 + 23*v^2 + 2*v + 2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{10} + 11\nu^{8} + 39\nu^{6} + 52\nu^{4} + 23\nu^{2} - 2\nu + 2 ) / 2$$ (v^10 + 11*v^8 + 39*v^6 + 52*v^4 + 23*v^2 - 2*v + 2) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{11} + \nu^{10} + 11 \nu^{9} + 12 \nu^{8} + 40 \nu^{7} + 49 \nu^{6} + 60 \nu^{5} + 81 \nu^{4} + \cdots + 6 ) / 2$$ (v^11 + v^10 + 11*v^9 + 12*v^8 + 40*v^7 + 49*v^6 + 60*v^5 + 81*v^4 + 40*v^3 + 48*v^2 + 13*v + 6) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{11} - \nu^{10} + 11 \nu^{9} - 12 \nu^{8} + 40 \nu^{7} - 49 \nu^{6} + 60 \nu^{5} - 81 \nu^{4} + \cdots - 6 ) / 2$$ (v^11 - v^10 + 11*v^9 - 12*v^8 + 40*v^7 - 49*v^6 + 60*v^5 - 81*v^4 + 40*v^3 - 48*v^2 + 13*v - 6) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{11} + \nu^{10} + 12 \nu^{9} + 11 \nu^{8} + 50 \nu^{7} + 40 \nu^{6} + 89 \nu^{5} + 60 \nu^{4} + \cdots + 7 ) / 2$$ (v^11 + v^10 + 12*v^9 + 11*v^8 + 50*v^7 + 40*v^6 + 89*v^5 + 60*v^4 + 61*v^3 + 38*v^2 + 5*v + 7) / 2 $$\beta_{7}$$ $$=$$ $$( - \nu^{11} + \nu^{10} - 12 \nu^{9} + 11 \nu^{8} - 50 \nu^{7} + 40 \nu^{6} - 89 \nu^{5} + 60 \nu^{4} + \cdots + 7 ) / 2$$ (-v^11 + v^10 - 12*v^9 + 11*v^8 - 50*v^7 + 40*v^6 - 89*v^5 + 60*v^4 - 61*v^3 + 38*v^2 - 5*v + 7) / 2 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 2 \nu^{10} + 13 \nu^{9} - 22 \nu^{8} + 60 \nu^{7} - 79 \nu^{6} + 120 \nu^{5} - 110 \nu^{4} + \cdots - 3 ) / 2$$ (v^11 - 2*v^10 + 13*v^9 - 22*v^8 + 60*v^7 - 79*v^6 + 120*v^5 - 110*v^4 + 98*v^3 - 51*v^2 + 23*v - 3) / 2 $$\beta_{9}$$ $$=$$ $$( - \nu^{11} - 2 \nu^{10} - 13 \nu^{9} - 22 \nu^{8} - 60 \nu^{7} - 79 \nu^{6} - 120 \nu^{5} - 110 \nu^{4} + \cdots - 3 ) / 2$$ (-v^11 - 2*v^10 - 13*v^9 - 22*v^8 - 60*v^7 - 79*v^6 - 120*v^5 - 110*v^4 - 98*v^3 - 51*v^2 - 23*v - 3) / 2 $$\beta_{10}$$ $$=$$ $$( -2\nu^{11} - 23\nu^{9} - 89\nu^{7} - 141\nu^{5} - 86\nu^{3} - 13\nu ) / 2$$ (-2*v^11 - 23*v^9 - 89*v^7 - 141*v^5 - 86*v^3 - 13*v) / 2 $$\beta_{11}$$ $$=$$ $$2\nu^{11} + 24\nu^{9} + 99\nu^{7} + 170\nu^{5} + 109\nu^{3} + 13\nu$$ 2*v^11 + 24*v^9 + 99*v^7 + 170*v^5 + 109*v^3 + 13*v
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 4 ) / 2$$ (-b5 + b4 - b3 - b2 + b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{11} + 2\beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4\beta_{3} - 4\beta_{2} ) / 2$$ (b11 + 2*b10 + b7 - b6 + b5 + b4 + 4*b3 - 4*b2) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 5\beta_{5} - 5\beta_{4} + 6\beta_{3} + 6\beta_{2} - 5\beta _1 + 14 ) / 2$$ (b9 + b8 + b7 + b6 + 5*b5 - 5*b4 + 6*b3 + 6*b2 - 5*b1 + 14) / 2 $$\nu^{5}$$ $$=$$ $$( - 8 \beta_{11} - 16 \beta_{10} - \beta_{9} + \beta_{8} - 6 \beta_{7} + 6 \beta_{6} + \cdots + 19 \beta_{2} ) / 2$$ (-8*b11 - 16*b10 - b9 + b8 - 6*b7 + 6*b6 - 7*b5 - 7*b4 - 19*b3 + 19*b2) / 2 $$\nu^{6}$$ $$=$$ $$( - 8 \beta_{9} - 8 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 25 \beta_{5} + 25 \beta_{4} - 35 \beta_{3} + \cdots - 62 ) / 2$$ (-8*b9 - 8*b8 - 6*b7 - 6*b6 - 25*b5 + 25*b4 - 35*b3 - 35*b2 + 25*b1 - 62) / 2 $$\nu^{7}$$ $$=$$ $$( 49 \beta_{11} + 102 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} + 31 \beta_{7} - 31 \beta_{6} + \cdots - 97 \beta_{2} ) / 2$$ (49*b11 + 102*b10 + 8*b9 - 8*b8 + 31*b7 - 31*b6 + 43*b5 + 43*b4 + 97*b3 - 97*b2) / 2 $$\nu^{8}$$ $$=$$ $$( 51 \beta_{9} + 51 \beta_{8} + 31 \beta_{7} + 31 \beta_{6} + 128 \beta_{5} - 128 \beta_{4} + 199 \beta_{3} + \cdots + 306 ) / 2$$ (51*b9 + 51*b8 + 31*b7 + 31*b6 + 128*b5 - 128*b4 + 199*b3 + 199*b2 - 130*b1 + 306) / 2 $$\nu^{9}$$ $$=$$ $$( - 279 \beta_{11} - 598 \beta_{10} - 51 \beta_{9} + 51 \beta_{8} - 159 \beta_{7} + 159 \beta_{6} + \cdots + 511 \beta_{2} ) / 2$$ (-279*b11 - 598*b10 - 51*b9 + 51*b8 - 159*b7 + 159*b6 - 250*b5 - 250*b4 - 511*b3 + 511*b2) / 2 $$\nu^{10}$$ $$=$$ $$( - 301 \beta_{9} - 301 \beta_{8} - 159 \beta_{7} - 159 \beta_{6} - 670 \beta_{5} + 670 \beta_{4} + \cdots - 1588 ) / 2$$ (-301*b9 - 301*b8 - 159*b7 - 159*b6 - 670*b5 + 670*b4 - 1111*b3 - 1111*b2 + 692*b1 - 1588) / 2 $$\nu^{11}$$ $$=$$ $$( 1549 \beta_{11} + 3378 \beta_{10} + 301 \beta_{9} - 301 \beta_{8} + 829 \beta_{7} + \cdots - 2734 \beta_{2} ) / 2$$ (1549*b11 + 3378*b10 + 301*b9 - 301*b8 + 829*b7 - 829*b6 + 1412*b5 + 1412*b4 + 2734*b3 - 2734*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$\beta_{10}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 2.33420i − 1.09958i − 0.274205i 0.594523i 1.37379i 1.73968i 2.33420i 1.09958i 0.274205i − 0.594523i − 1.37379i − 1.73968i
0 −2.33420 + 2.33420i 0 1.84263 + 1.26678i 0 1.63901 1.63901i 0 7.89698i 0
127.2 0 −1.09958 + 1.09958i 0 −0.504779 2.17835i 0 3.36850 3.36850i 0 0.581827i 0
127.3 0 −0.274205 + 0.274205i 0 2.23269 + 0.122814i 0 −2.49551 + 2.49551i 0 2.84962i 0
127.4 0 0.594523 0.594523i 0 −0.267807 + 2.21997i 0 −0.132476 + 0.132476i 0 2.29308i 0
127.5 0 1.37379 1.37379i 0 −2.14213 + 0.641320i 0 −0.287197 + 0.287197i 0 0.774596i 0
127.6 0 1.73968 1.73968i 0 0.839394 2.07254i 0 1.90768 1.90768i 0 3.05295i 0
383.1 0 −2.33420 2.33420i 0 1.84263 1.26678i 0 1.63901 + 1.63901i 0 7.89698i 0
383.2 0 −1.09958 1.09958i 0 −0.504779 + 2.17835i 0 3.36850 + 3.36850i 0 0.581827i 0
383.3 0 −0.274205 0.274205i 0 2.23269 0.122814i 0 −2.49551 2.49551i 0 2.84962i 0
383.4 0 0.594523 + 0.594523i 0 −0.267807 2.21997i 0 −0.132476 0.132476i 0 2.29308i 0
383.5 0 1.37379 + 1.37379i 0 −2.14213 0.641320i 0 −0.287197 0.287197i 0 0.774596i 0
383.6 0 1.73968 + 1.73968i 0 0.839394 + 2.07254i 0 1.90768 + 1.90768i 0 3.05295i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.o.v 12
4.b odd 2 1 1280.2.o.u 12
5.c odd 4 1 1280.2.o.s 12
8.b even 2 1 1280.2.o.t 12
8.d odd 2 1 1280.2.o.s 12
16.e even 4 1 640.2.n.a 12
16.e even 4 1 640.2.n.b yes 12
16.f odd 4 1 640.2.n.c yes 12
16.f odd 4 1 640.2.n.d yes 12
20.e even 4 1 1280.2.o.t 12
40.i odd 4 1 1280.2.o.u 12
40.k even 4 1 inner 1280.2.o.v 12
80.i odd 4 1 640.2.n.c yes 12
80.j even 4 1 640.2.n.b yes 12
80.s even 4 1 640.2.n.a 12
80.t odd 4 1 640.2.n.d yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.n.a 12 16.e even 4 1
640.2.n.a 12 80.s even 4 1
640.2.n.b yes 12 16.e even 4 1
640.2.n.b yes 12 80.j even 4 1
640.2.n.c yes 12 16.f odd 4 1
640.2.n.c yes 12 80.i odd 4 1
640.2.n.d yes 12 16.f odd 4 1
640.2.n.d yes 12 80.t odd 4 1
1280.2.o.s 12 5.c odd 4 1
1280.2.o.s 12 8.d odd 2 1
1280.2.o.t 12 8.b even 2 1
1280.2.o.t 12 20.e even 4 1
1280.2.o.u 12 4.b odd 2 1
1280.2.o.u 12 40.i odd 4 1
1280.2.o.v 12 1.a even 1 1 trivial
1280.2.o.v 12 40.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{12} - 8 T_{3}^{9} + 88 T_{3}^{8} - 80 T_{3}^{7} + 32 T_{3}^{6} + 32 T_{3}^{5} + 592 T_{3}^{4} + \cdots + 64$$ T3^12 - 8*T3^9 + 88*T3^8 - 80*T3^7 + 32*T3^6 + 32*T3^5 + 592*T3^4 - 384*T3^3 + 128*T3^2 + 128*T3 + 64 $$T_{7}^{12} - 8 T_{7}^{11} + 32 T_{7}^{10} - 40 T_{7}^{9} + 184 T_{7}^{8} - 1232 T_{7}^{7} + 4768 T_{7}^{6} + \cdots + 64$$ T7^12 - 8*T7^11 + 32*T7^10 - 40*T7^9 + 184*T7^8 - 1232*T7^7 + 4768*T7^6 - 6944*T7^5 + 3472*T7^4 + 5632*T7^3 + 3200*T7^2 + 640*T7 + 64 $$T_{13}^{12} - 4 T_{13}^{11} + 8 T_{13}^{10} - 8 T_{13}^{9} + 1036 T_{13}^{8} - 4000 T_{13}^{7} + \cdots + 937024$$ T13^12 - 4*T13^11 + 8*T13^10 - 8*T13^9 + 1036*T13^8 - 4000*T13^7 + 7744*T13^6 - 13376*T13^5 + 213808*T13^4 - 875072*T13^3 + 1874048*T13^2 - 1874048*T13 + 937024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 8 T^{9} + \cdots + 64$$
$5$ $$T^{12} - 4 T^{11} + \cdots + 15625$$
$7$ $$T^{12} - 8 T^{11} + \cdots + 64$$
$11$ $$(T^{6} + 4 T^{5} + \cdots + 448)^{2}$$
$13$ $$T^{12} - 4 T^{11} + \cdots + 937024$$
$17$ $$T^{12} - 4 T^{11} + \cdots + 506944$$
$19$ $$T^{12} + 120 T^{10} + \cdots + 200704$$
$23$ $$T^{12} + 16 T^{11} + \cdots + 25482304$$
$29$ $$(T^{6} - 8 T^{5} + \cdots + 512)^{2}$$
$31$ $$T^{12} + 192 T^{10} + \cdots + 32444416$$
$37$ $$T^{12} + 20 T^{11} + \cdots + 64$$
$41$ $$(T^{6} - 88 T^{4} + \cdots + 3136)^{2}$$
$43$ $$T^{12} - 16 T^{11} + \cdots + 3013696$$
$47$ $$T^{12} + \cdots + 476636224$$
$53$ $$T^{12} + \cdots + 742889536$$
$59$ $$T^{12} + 504 T^{10} + \cdots + 93392896$$
$61$ $$T^{12} + 352 T^{10} + \cdots + 58003456$$
$67$ $$T^{12} - 32 T^{11} + \cdots + 45914176$$
$71$ $$T^{12} + 256 T^{10} + \cdots + 4096$$
$73$ $$T^{12} + \cdots + 1575137344$$
$79$ $$(T^{6} - 16 T^{5} + \cdots - 4096)^{2}$$
$83$ $$T^{12} + \cdots + 18428605504$$
$89$ $$T^{12} + \cdots + 214228271104$$
$97$ $$T^{12} + \cdots + 128628387904$$