# Properties

 Label 1008.2.bt.d Level $1008$ Weight $2$ Character orbit 1008.bt Analytic conductor $8.049$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(17,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.bt (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: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401$$ x^16 + 10*x^14 + 61*x^12 + 266*x^10 + 852*x^8 + 1438*x^6 + 1933*x^4 + 3038*x^2 + 2401 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 504) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{10} + \beta_{4}) q^{5} + (\beta_{9} + \beta_{3}) q^{7}+O(q^{10})$$ q + (-b10 + b4) * q^5 + (b9 + b3) * q^7 $$q + ( - \beta_{10} + \beta_{4}) q^{5} + (\beta_{9} + \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{11}+ \cdots + (4 \beta_{9} + 4 \beta_{6} + 6 \beta_{5} + \cdots - 3) q^{97}+O(q^{100})$$ q + (-b10 + b4) * q^5 + (b9 + b3) * q^7 + (-b14 - b13 + b12 - b11 + b2) * q^11 + (-b9 - b8 - b7 - b6 + 1) * q^13 + (-b15 + b14 - b13 - b11) * q^17 + (2*b7 - b6 + b5 - 2*b1 - 3) * q^19 + (-b13 - 2*b12 + b11 - b2) * q^23 + (-b8 - b7 - 2*b5 + b1 + 3) * q^25 + (b15 + b14 + b11 - b10 + 2*b4 - b2) * q^29 + (b9 + b8 - b7 - b1 - 2) * q^31 + (2*b11 - 2*b10 + b4 - b2) * q^35 + (-2*b9 + 3*b8 - b6 + 2*b5 - 3*b3 - 3*b1 - 2) * q^37 + (-b15 - b13 - 4*b12 + b11 - b10 - b2) * q^41 + (b9 - b6 - 1) * q^43 + (-2*b13 + b12 - b11 + 2*b10 - 2*b4 + 2*b2) * q^47 + (2*b9 - 2*b8 + 2*b7 + 3*b5 - b3 - b1) * q^49 + (-b14 - b13 + 4*b12 - b11 - 4*b10 + 2*b4 + b2) * q^53 + (-b9 + b8 + b7 - b6 + 2*b5 - 2*b3 - 2*b1 - 2) * q^55 + (b12 - 2*b11 + 3*b4) * q^59 + (2*b8 - b7 - 3*b6 - 2*b5 + 2*b3 - b1 - 1) * q^61 + (b15 - b14 + b13 + b11 + 3*b10 + 3*b4 + b2) * q^65 + (-b8 - 3*b5 - b3 + 4) * q^67 + (-b11 - 2*b10 + 4*b4) * q^71 + (-3*b8 + b7 - 4*b5 + 2*b3 + b1 - 1) * q^73 + (b15 - 2*b13 + b12 + 3*b11 + 2*b10 - 3*b4 + 2*b2) * q^77 + (-2*b9 - 3*b8 - b7 - b6 + 3*b3 + 2*b1 + 4) * q^79 + (b15 - b14 - 4*b12 + 2*b11 - b10 + b2) * q^83 + (b9 - b8 + b7 - b6 + b3 - b1 - 3) * q^85 + (-2*b15 - 2*b14 - 2*b12 - 2*b11) * q^89 + (-b9 - b8 - 5*b7 - 3*b5 + 3*b3 + 3*b1 + 10) * q^91 + (2*b14 + 2*b13 + 5*b12 + 2*b11 - 4*b10 + 2*b4 - 2*b2) * q^95 + (4*b9 + 4*b6 + 6*b5 + 3*b3 + 3*b1 - 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{7}+O(q^{10})$$ 16 * q + 8 * q^7 $$16 q + 8 q^{7} - 12 q^{19} + 12 q^{25} - 24 q^{31} + 4 q^{37} - 8 q^{43} + 32 q^{49} + 28 q^{67} - 60 q^{73} + 32 q^{79} - 32 q^{85} + 84 q^{91}+O(q^{100})$$ 16 * q + 8 * q^7 - 12 * q^19 + 12 * q^25 - 24 * q^31 + 4 * q^37 - 8 * q^43 + 32 * q^49 + 28 * q^67 - 60 * q^73 + 32 * q^79 - 32 * q^85 + 84 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401$$ :

 $$\beta_{1}$$ $$=$$ $$( 34158 \nu^{14} + 852503 \nu^{12} + 4914956 \nu^{10} + 18664744 \nu^{8} + 59406470 \nu^{6} + \cdots + 139803027 ) / 392984900$$ (34158*v^14 + 852503*v^12 + 4914956*v^10 + 18664744*v^8 + 59406470*v^6 + 70780144*v^4 - 260640102*v^2 + 139803027) / 392984900 $$\beta_{2}$$ $$=$$ $$( 66287 \nu^{15} - 2125573 \nu^{13} - 20719966 \nu^{11} - 144889584 \nu^{9} + \cdots - 6339564777 \nu ) / 2750894300$$ (66287*v^15 - 2125573*v^13 - 20719966*v^11 - 144889584*v^9 - 625207850*v^7 - 2285038534*v^5 - 3180000603*v^3 - 6339564777*v) / 2750894300 $$\beta_{3}$$ $$=$$ $$( - 382234 \nu^{14} - 3667549 \nu^{12} - 21832848 \nu^{10} - 94498292 \nu^{8} - 297554910 \nu^{6} + \cdots - 1050178241 ) / 392984900$$ (-382234*v^14 - 3667549*v^12 - 21832848*v^10 - 94498292*v^8 - 297554910*v^6 - 530245552*v^4 - 674195374*v^2 - 1050178241) / 392984900 $$\beta_{4}$$ $$=$$ $$( - 88213 \nu^{15} - 1057753 \nu^{13} - 8202336 \nu^{11} - 41107094 \nu^{9} + \cdots - 905297057 \nu ) / 785969800$$ (-88213*v^15 - 1057753*v^13 - 8202336*v^11 - 41107094*v^9 - 157143590*v^7 - 407420164*v^5 - 856246593*v^3 - 905297057*v) / 785969800 $$\beta_{5}$$ $$=$$ $$( 623003 \nu^{14} + 5366258 \nu^{12} + 30938216 \nu^{10} + 125675214 \nu^{8} + 373946420 \nu^{6} + \cdots + 1273004222 ) / 392984900$$ (623003*v^14 + 5366258*v^12 + 30938216*v^10 + 125675214*v^8 + 373946420*v^6 + 445540384*v^4 + 843429583*v^2 + 1273004222) / 392984900 $$\beta_{6}$$ $$=$$ $$( 83051 \nu^{14} + 542488 \nu^{12} + 2797264 \nu^{10} + 9444519 \nu^{8} + 21581337 \nu^{6} + \cdots + 3256785 ) / 39298490$$ (83051*v^14 + 542488*v^12 + 2797264*v^10 + 9444519*v^8 + 21581337*v^6 - 16157524*v^4 + 47574110*v^2 + 3256785) / 39298490 $$\beta_{7}$$ $$=$$ $$( 17628 \nu^{14} + 144183 \nu^{12} + 817216 \nu^{10} + 3201064 \nu^{8} + 9190570 \nu^{6} + \cdots + 14486947 ) / 8020100$$ (17628*v^14 + 144183*v^12 + 817216*v^10 + 3201064*v^8 + 9190570*v^6 + 7363984*v^4 + 12646508*v^2 + 14486947) / 8020100 $$\beta_{8}$$ $$=$$ $$( 3705 \nu^{14} + 31546 \nu^{12} + 171760 \nu^{10} + 672790 \nu^{8} + 1902192 \nu^{6} + \cdots + 7562162 ) / 1604020$$ (3705*v^14 + 31546*v^12 + 171760*v^10 + 672790*v^8 + 1902192*v^6 + 1547740*v^4 + 2658005*v^2 + 7562162) / 1604020 $$\beta_{9}$$ $$=$$ $$( 1744691 \nu^{14} + 15899931 \nu^{12} + 90773542 \nu^{10} + 373226028 \nu^{8} + 1092704010 \nu^{6} + \cdots + 3124455159 ) / 392984900$$ (1744691*v^14 + 15899931*v^12 + 90773542*v^10 + 373226028*v^8 + 1092704010*v^6 + 1301735958*v^4 + 1516055181*v^2 + 3124455159) / 392984900 $$\beta_{10}$$ $$=$$ $$( - 154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} + \cdots - 238292243 \nu ) / 239208200$$ (-154017*v^15 - 1553547*v^13 - 9597064*v^11 - 41588666*v^9 - 133651610*v^7 - 226254436*v^5 - 303303017*v^3 - 238292243*v) / 239208200 $$\beta_{11}$$ $$=$$ $$( - 13169 \nu^{15} - 130759 \nu^{13} - 753868 \nu^{11} - 3141922 \nu^{9} - 9111910 \nu^{7} + \cdots - 21443331 \nu ) / 17355800$$ (-13169*v^15 - 130759*v^13 - 753868*v^11 - 3141922*v^9 - 9111910*v^7 - 10856432*v^5 - 5104409*v^3 - 21443331*v) / 17355800 $$\beta_{12}$$ $$=$$ $$( 7738601 \nu^{15} + 58163261 \nu^{13} + 313306372 \nu^{11} + 1167320938 \nu^{9} + \cdots + 9719272549 \nu ) / 5501788600$$ (7738601*v^15 + 58163261*v^13 + 313306372*v^11 + 1167320938*v^9 + 3102633990*v^7 + 1535163128*v^5 + 6928535561*v^3 + 9719272549*v) / 5501788600 $$\beta_{13}$$ $$=$$ $$( - 9919057 \nu^{15} - 78396587 \nu^{13} - 451981724 \nu^{11} - 1839736626 \nu^{9} + \cdots - 7354565183 \nu ) / 5501788600$$ (-9919057*v^15 - 78396587*v^13 - 451981724*v^11 - 1839736626*v^9 - 5463047630*v^7 - 6508976176*v^5 - 15803364417*v^3 - 7354565183*v) / 5501788600 $$\beta_{14}$$ $$=$$ $$( - 438892 \nu^{15} - 3881427 \nu^{13} - 22712419 \nu^{11} - 93951256 \nu^{9} + \cdots - 752948308 \nu ) / 125040650$$ (-438892*v^15 - 3881427*v^13 - 22712419*v^11 - 93951256*v^9 - 283749965*v^7 - 373105681*v^5 - 562359552*v^3 - 752948308*v) / 125040650 $$\beta_{15}$$ $$=$$ $$( 965736 \nu^{15} + 8386545 \nu^{13} + 48089618 \nu^{11} + 197972096 \nu^{9} + \cdots + 1747120823 \nu ) / 275089430$$ (965736*v^15 + 8386545*v^13 + 48089618*v^11 + 197972096*v^9 + 592040102*v^7 + 736276512*v^5 + 1335892868*v^3 + 1747120823*v) / 275089430
 $$\nu$$ $$=$$ $$( -\beta_{14} + \beta_{13} + 3\beta_{10} - 2\beta_{4} ) / 4$$ (-b14 + b13 + 3*b10 - 2*b4) / 4 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2\beta_{5} + \beta_{3} - 2$$ -b7 + 2*b5 + b3 - 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{15} + 5\beta_{14} - \beta_{13} + 2\beta_{12} - 4\beta_{11} - 3\beta_{10} - 8\beta_{4} + 2\beta_{2} ) / 4$$ (2*b15 + 5*b14 - b13 + 2*b12 - 4*b11 - 3*b10 - 8*b4 + 2*b2) / 4 $$\nu^{4}$$ $$=$$ $$-\beta_{8} - \beta_{7} + 2\beta_{6} - 3\beta_{5} - 5\beta_{3} + 2\beta _1 + 2$$ -b8 - b7 + 2*b6 - 3*b5 - 5*b3 + 2*b1 + 2 $$\nu^{5}$$ $$=$$ $$( -16\beta_{15} + 3\beta_{14} - 3\beta_{13} + 28\beta_{12} - 41\beta_{10} + 34\beta_{4} - 28\beta_{2} ) / 4$$ (-16*b15 + 3*b14 - 3*b13 + 28*b12 - 41*b10 + 34*b4 - 28*b2) / 4 $$\nu^{6}$$ $$=$$ $$-14\beta_{9} + 19\beta_{8} + 24\beta_{7} - 14\beta_{6} - 7\beta_{5} - 7\beta_{3} - 7\beta _1 - 14$$ -14*b9 + 19*b8 + 24*b7 - 14*b6 - 7*b5 - 7*b3 - 7*b1 - 14 $$\nu^{7}$$ $$=$$ $$( 98 \beta_{15} - 83 \beta_{14} + 91 \beta_{13} - 158 \beta_{12} + 256 \beta_{11} + 69 \beta_{10} + \cdots + 90 \beta_{2} ) / 4$$ (98*b15 - 83*b14 + 91*b13 - 158*b12 + 256*b11 + 69*b10 + 112*b4 + 90*b2) / 4 $$\nu^{8}$$ $$=$$ $$62\beta_{9} - 31\beta_{8} - 37\beta_{7} - 31\beta_{5} + 68\beta_{3} - 81\beta _1 + 31$$ 62*b9 - 31*b8 - 37*b7 - 31*b5 + 68*b3 - 81*b1 + 31 $$\nu^{9}$$ $$=$$ $$( - 112 \beta_{15} + 235 \beta_{14} - 535 \beta_{13} - 112 \beta_{12} - 1092 \beta_{11} + \cdots - 112 \beta_{2} ) / 4$$ (-112*b15 + 235*b14 - 535*b13 - 112*b12 - 1092*b11 + 759*b10 - 658*b4 - 112*b2) / 4 $$\nu^{10}$$ $$=$$ $$-404\beta_{8} - 99\beta_{7} + 198\beta_{6} + 454\beta_{5} + 32\beta_{3} + 503\beta _1 + 503$$ -404*b8 - 99*b7 + 198*b6 + 454*b5 + 32*b3 + 503*b1 + 503 $$\nu^{11}$$ $$=$$ $$( - 1526 \beta_{15} - 759 \beta_{14} + 759 \beta_{13} + 1942 \beta_{12} - 1671 \beta_{10} + \cdots + 614 \beta_{2} ) / 4$$ (-1526*b15 - 759*b14 + 759*b13 + 1942*b12 - 1671*b10 - 2132*b4 + 614*b2) / 4 $$\nu^{12}$$ $$=$$ $$-332\beta_{9} + 1742\beta_{8} - 788\beta_{7} - 332\beta_{6} - 166\beta_{5} - 166\beta_{3} - 166\beta _1 - 3969$$ -332*b9 + 1742*b8 - 788*b7 - 332*b6 - 166*b5 - 166*b3 - 166*b1 - 3969 $$\nu^{13}$$ $$=$$ $$( 7424 \beta_{15} + 7163 \beta_{14} + 3621 \beta_{13} + 1616 \beta_{12} + 5808 \beta_{11} + \cdots - 3360 \beta_{2} ) / 4$$ (7424*b15 + 7163*b14 + 3621*b13 + 1616*b12 + 5808*b11 - 14769*b10 + 15030*b4 - 3360*b2) / 4 $$\nu^{14}$$ $$=$$ $$-1244\beta_{9} + 622\beta_{8} + 6831\beta_{7} - 10592\beta_{5} - 7453\beta_{3} - 4438\beta _1 + 10592$$ -1244*b9 + 622*b8 + 6831*b7 - 10592*b5 - 7453*b3 - 4438*b1 + 10592 $$\nu^{15}$$ $$=$$ $$( - 15030 \beta_{15} - 29095 \beta_{14} - 13493 \beta_{13} - 15030 \beta_{12} + 22012 \beta_{11} + \cdots - 15030 \beta_{2} ) / 4$$ (-15030*b15 - 29095*b14 - 13493*b13 - 15030*b12 + 22012*b11 + 43553*b10 + 30632*b4 - 15030*b2) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.45333 + 1.51725i 1.01089 + 0.750919i 0.587308 + 2.01725i 0.144868 + 1.25092i −0.144868 − 1.25092i −0.587308 − 2.01725i −1.01089 − 0.750919i −1.45333 − 1.51725i 1.45333 − 1.51725i 1.01089 − 0.750919i 0.587308 − 2.01725i 0.144868 − 1.25092i −0.144868 + 1.25092i −0.587308 + 2.01725i −1.01089 + 0.750919i −1.45333 + 1.51725i
0 0 0 −1.45333 2.51725i 0 2.64571 + 0.0146827i 0 0 0
17.2 0 0 0 −1.01089 1.75092i 0 −0.561961 2.58538i 0 0 0
17.3 0 0 0 −0.587308 1.01725i 0 −2.35282 + 1.21006i 0 0 0
17.4 0 0 0 −0.144868 0.250919i 0 2.26907 + 1.36064i 0 0 0
17.5 0 0 0 0.144868 + 0.250919i 0 2.26907 + 1.36064i 0 0 0
17.6 0 0 0 0.587308 + 1.01725i 0 −2.35282 + 1.21006i 0 0 0
17.7 0 0 0 1.01089 + 1.75092i 0 −0.561961 2.58538i 0 0 0
17.8 0 0 0 1.45333 + 2.51725i 0 2.64571 + 0.0146827i 0 0 0
593.1 0 0 0 −1.45333 + 2.51725i 0 2.64571 0.0146827i 0 0 0
593.2 0 0 0 −1.01089 + 1.75092i 0 −0.561961 + 2.58538i 0 0 0
593.3 0 0 0 −0.587308 + 1.01725i 0 −2.35282 1.21006i 0 0 0
593.4 0 0 0 −0.144868 + 0.250919i 0 2.26907 1.36064i 0 0 0
593.5 0 0 0 0.144868 0.250919i 0 2.26907 1.36064i 0 0 0
593.6 0 0 0 0.587308 1.01725i 0 −2.35282 1.21006i 0 0 0
593.7 0 0 0 1.01089 1.75092i 0 −0.561961 + 2.58538i 0 0 0
593.8 0 0 0 1.45333 2.51725i 0 2.64571 0.0146827i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bt.d 16
3.b odd 2 1 inner 1008.2.bt.d 16
4.b odd 2 1 504.2.bl.a 16
7.c even 3 1 7056.2.k.h 16
7.d odd 6 1 inner 1008.2.bt.d 16
7.d odd 6 1 7056.2.k.h 16
12.b even 2 1 504.2.bl.a 16
21.g even 6 1 inner 1008.2.bt.d 16
21.g even 6 1 7056.2.k.h 16
21.h odd 6 1 7056.2.k.h 16
28.d even 2 1 3528.2.bl.a 16
28.f even 6 1 504.2.bl.a 16
28.f even 6 1 3528.2.k.b 16
28.g odd 6 1 3528.2.k.b 16
28.g odd 6 1 3528.2.bl.a 16
84.h odd 2 1 3528.2.bl.a 16
84.j odd 6 1 504.2.bl.a 16
84.j odd 6 1 3528.2.k.b 16
84.n even 6 1 3528.2.k.b 16
84.n even 6 1 3528.2.bl.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bl.a 16 4.b odd 2 1
504.2.bl.a 16 12.b even 2 1
504.2.bl.a 16 28.f even 6 1
504.2.bl.a 16 84.j odd 6 1
1008.2.bt.d 16 1.a even 1 1 trivial
1008.2.bt.d 16 3.b odd 2 1 inner
1008.2.bt.d 16 7.d odd 6 1 inner
1008.2.bt.d 16 21.g even 6 1 inner
3528.2.k.b 16 28.f even 6 1
3528.2.k.b 16 28.g odd 6 1
3528.2.k.b 16 84.j odd 6 1
3528.2.k.b 16 84.n even 6 1
3528.2.bl.a 16 28.d even 2 1
3528.2.bl.a 16 28.g odd 6 1
3528.2.bl.a 16 84.h odd 2 1
3528.2.bl.a 16 84.n even 6 1
7056.2.k.h 16 7.c even 3 1
7056.2.k.h 16 7.d odd 6 1
7056.2.k.h 16 21.g even 6 1
7056.2.k.h 16 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{16} + 14T_{5}^{14} + 143T_{5}^{12} + 638T_{5}^{10} + 2077T_{5}^{8} + 2644T_{5}^{6} + 2492T_{5}^{4} + 208T_{5}^{2} + 16$$ T5^16 + 14*T5^14 + 143*T5^12 + 638*T5^10 + 2077*T5^8 + 2644*T5^6 + 2492*T5^4 + 208*T5^2 + 16 $$T_{11}^{16} - 54 T_{11}^{14} + 2215 T_{11}^{12} - 33222 T_{11}^{10} + 364221 T_{11}^{8} - 1394988 T_{11}^{6} + \cdots + 4477456$$ T11^16 - 54*T11^14 + 2215*T11^12 - 33222*T11^10 + 364221*T11^8 - 1394988*T11^6 + 3880540*T11^4 - 4900656*T11^2 + 4477456

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 14 T^{14} + \cdots + 16$$
$7$ $$(T^{8} - 4 T^{7} + \cdots + 2401)^{2}$$
$11$ $$T^{16} - 54 T^{14} + \cdots + 4477456$$
$13$ $$(T^{8} + 82 T^{6} + \cdots + 24964)^{2}$$
$17$ $$T^{16} + \cdots + 157351936$$
$19$ $$(T^{8} + 6 T^{7} + \cdots + 111556)^{2}$$
$23$ $$T^{16} + \cdots + 2517630976$$
$29$ $$(T^{8} + 150 T^{6} + \cdots + 3136)^{2}$$
$31$ $$(T^{8} + 12 T^{7} + \cdots + 529)^{2}$$
$37$ $$(T^{8} - 2 T^{7} + \cdots + 586756)^{2}$$
$41$ $$(T^{8} - 124 T^{6} + \cdots + 160000)^{2}$$
$43$ $$(T^{4} + 2 T^{3} - 33 T^{2} + \cdots - 98)^{4}$$
$47$ $$T^{16} + \cdots + 100000000$$
$53$ $$T^{16} + \cdots + 9082363580416$$
$59$ $$T^{16} + \cdots + 64524128256$$
$61$ $$(T^{8} - 174 T^{6} + \cdots + 254016)^{2}$$
$67$ $$(T^{8} - 14 T^{7} + \cdots + 16)^{2}$$
$71$ $$(T^{8} + 176 T^{6} + \cdots + 26896)^{2}$$
$73$ $$(T^{8} + 30 T^{7} + \cdots + 446224)^{2}$$
$79$ $$(T^{8} - 16 T^{7} + \cdots + 11242609)^{2}$$
$83$ $$(T^{8} - 226 T^{6} + \cdots + 3844)^{2}$$
$89$ $$T^{16} + \cdots + 704536369954816$$
$97$ $$(T^{8} + 542 T^{6} + \cdots + 65480464)^{2}$$