# Properties

 Label 507.2.f.e Level $507$ Weight $2$ Character orbit 507.f Analytic conductor $4.048$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( -2 + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{6} + ( -1 - \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( -2 + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{6} + ( -1 - \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{9} + ( -1 + \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{10} + ( 1 - \beta_{3} - 2 \beta_{7} ) q^{11} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{12} + ( 1 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{14} + ( -2 + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{15} + ( 1 - 2 \beta_{2} + 2 \beta_{5} ) q^{16} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{18} -2 \beta_{5} q^{19} + ( -4 - 3 \beta_{1} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{20} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{21} + ( -1 + 3 \beta_{2} - 3 \beta_{5} ) q^{22} + ( 1 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{24} + ( -1 + \beta_{2} + \beta_{5} ) q^{25} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{27} + ( 3 - 3 \beta_{3} - 2 \beta_{5} ) q^{28} + ( 1 - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( -1 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{30} + ( -3 + 3 \beta_{3} + 4 \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{32} + ( -2 - \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{33} + ( 5 - \beta_{2} + 4 \beta_{3} ) q^{34} + ( 1 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{35} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{36} + ( -2 + 5 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( -3 - 4 \beta_{2} + 4 \beta_{5} ) q^{40} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{41} + ( 3 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{42} + ( 3 - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{5} ) q^{43} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{44} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{45} + ( -4 - 4 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 6 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{48} -5 \beta_{3} q^{49} + ( -1 + \beta_{3} + 2 \beta_{7} ) q^{50} + ( 1 + \beta_{1} + 4 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{51} + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{53} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} ) q^{54} + ( -1 + 3 \beta_{2} - 3 \beta_{5} ) q^{55} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -2 + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{57} + ( 9 - 9 \beta_{3} - \beta_{5} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( -1 + \beta_{1} - 6 \beta_{2} - 9 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{60} -7 q^{61} + ( -3 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{62} + ( -1 + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{7} ) q^{63} + ( -5 + 5 \beta_{2} + 7 \beta_{3} + 5 \beta_{5} ) q^{64} + ( -9 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{66} + ( -4 + 4 \beta_{3} + 6 \beta_{5} ) q^{67} + ( -3 + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{68} + ( 5 - 5 \beta_{3} - 2 \beta_{5} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{71} + ( 10 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{72} + ( 2 + 3 \beta_{2} + 5 \beta_{3} ) q^{73} + ( -8 + 3 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{74} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{77} + 2 q^{79} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{80} + ( -3 + 6 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{81} + ( -2 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{82} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{83} + ( -2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{84} + ( 5 - \beta_{2} + 4 \beta_{3} ) q^{85} + ( 9 + 6 \beta_{1} - 3 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} ) q^{86} + ( 6 + 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{87} + ( 1 - \beta_{2} + 8 \beta_{3} - \beta_{5} ) q^{88} + ( 5 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{7} ) q^{89} + ( 6 + 2 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} - 6 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{90} + ( 4 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{93} + ( -16 - 4 \beta_{2} + 4 \beta_{5} ) q^{94} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -2 + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{96} + ( -7 + 7 \beta_{3} + 6 \beta_{5} ) q^{97} + ( 5 + 5 \beta_{1} - 5 \beta_{4} - 5 \beta_{5} ) q^{98} + ( -9 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} - 16q^{6} - 8q^{7} - 8q^{9} + O(q^{10})$$ $$8q + 4q^{3} - 16q^{6} - 8q^{7} - 8q^{9} - 16q^{15} + 8q^{16} - 4q^{18} - 8q^{19} - 4q^{21} - 8q^{22} - 12q^{24} + 4q^{27} + 16q^{28} - 8q^{31} - 4q^{33} + 36q^{34} + 4q^{37} - 24q^{40} + 32q^{42} - 4q^{45} + 28q^{48} + 8q^{54} - 8q^{55} - 16q^{57} + 68q^{58} - 44q^{60} - 56q^{61} + 8q^{63} - 40q^{66} - 8q^{67} + 32q^{70} + 36q^{72} + 28q^{73} + 8q^{76} + 16q^{79} - 8q^{81} - 4q^{84} + 36q^{85} + 68q^{87} + 20q^{93} - 128q^{94} - 16q^{96} - 32q^{97} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} - \nu^{6} - 25 \nu^{5} - 46 \nu^{4} - 5 \nu^{3} - 132 \nu^{2} + 28 \nu - 55$$$$)/37$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 41 \nu - 29$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 7 \nu^{6} + 47 \nu^{5} - 100 \nu^{4} + 261 \nu^{3} - 295 \nu^{2} + 344 \nu - 126$$$$)/37$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 36 \nu^{6} - 136 \nu^{5} + 361 \nu^{4} - 634 \nu^{3} + 793 \nu^{2} - 601 \nu + 278$$$$)/37$$ $$\beta_{6}$$ $$=$$ $$($$$$-18 \nu^{7} + 63 \nu^{6} - 238 \nu^{5} + 419 \nu^{4} - 684 \nu^{3} + 546 \nu^{2} - 395 \nu + 61$$$$)/37$$ $$\beta_{7}$$ $$=$$ $$($$$$-18 \nu^{7} + 63 \nu^{6} - 238 \nu^{5} + 456 \nu^{4} - 758 \nu^{3} + 805 \nu^{2} - 617 \nu + 283$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + 3 \beta_{1} - 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{6} - 3 \beta_{5} - 12 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} - 8 \beta_{1} + 10$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} - \beta_{6} - 2 \beta_{5} + 11 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} - 25 \beta_{1} + 19$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$5 \beta_{7} + 12 \beta_{6} + 2 \beta_{5} + 32 \beta_{4} - 42 \beta_{3} + 7 \beta_{1} - 11$$ $$\nu^{7}$$ $$=$$ $$($$$$-46 \beta_{7} + 38 \beta_{6} + 17 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 140 \beta_{1} - 96$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 0.5 − 2.19293i 0.5 − 1.56488i 0.5 + 0.564882i 0.5 + 1.19293i 0.5 + 2.19293i 0.5 + 1.56488i 0.5 − 0.564882i 0.5 − 1.19293i
−1.69293 1.69293i −0.366025 1.69293i 3.73205i −1.69293 1.69293i −2.24637 + 3.48568i −1.00000 1.00000i 2.93225 2.93225i −2.73205 + 1.23931i 5.73205i
239.2 −1.06488 1.06488i 1.36603 1.06488i 0.267949i −1.06488 1.06488i −2.58863 0.320682i −1.00000 1.00000i −1.84443 + 1.84443i 0.732051 2.90931i 2.26795i
239.3 1.06488 + 1.06488i 1.36603 + 1.06488i 0.267949i 1.06488 + 1.06488i 0.320682 + 2.58863i −1.00000 1.00000i 1.84443 1.84443i 0.732051 + 2.90931i 2.26795i
239.4 1.69293 + 1.69293i −0.366025 + 1.69293i 3.73205i 1.69293 + 1.69293i −3.48568 + 2.24637i −1.00000 1.00000i −2.93225 + 2.93225i −2.73205 1.23931i 5.73205i
437.1 −1.69293 + 1.69293i −0.366025 + 1.69293i 3.73205i −1.69293 + 1.69293i −2.24637 3.48568i −1.00000 + 1.00000i 2.93225 + 2.93225i −2.73205 1.23931i 5.73205i
437.2 −1.06488 + 1.06488i 1.36603 + 1.06488i 0.267949i −1.06488 + 1.06488i −2.58863 + 0.320682i −1.00000 + 1.00000i −1.84443 1.84443i 0.732051 + 2.90931i 2.26795i
437.3 1.06488 1.06488i 1.36603 1.06488i 0.267949i 1.06488 1.06488i 0.320682 2.58863i −1.00000 + 1.00000i 1.84443 + 1.84443i 0.732051 2.90931i 2.26795i
437.4 1.69293 1.69293i −0.366025 1.69293i 3.73205i 1.69293 1.69293i −3.48568 2.24637i −1.00000 + 1.00000i −2.93225 2.93225i −2.73205 + 1.23931i 5.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 437.4 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
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.e 8
3.b odd 2 1 inner 507.2.f.e 8
13.b even 2 1 507.2.f.f 8
13.c even 3 1 507.2.k.d 8
13.c even 3 1 507.2.k.f 8
13.d odd 4 1 inner 507.2.f.e 8
13.d odd 4 1 507.2.f.f 8
13.e even 6 1 39.2.k.b 8
13.e even 6 1 507.2.k.e 8
13.f odd 12 1 39.2.k.b 8
13.f odd 12 1 507.2.k.d 8
13.f odd 12 1 507.2.k.e 8
13.f odd 12 1 507.2.k.f 8
39.d odd 2 1 507.2.f.f 8
39.f even 4 1 inner 507.2.f.e 8
39.f even 4 1 507.2.f.f 8
39.h odd 6 1 39.2.k.b 8
39.h odd 6 1 507.2.k.e 8
39.i odd 6 1 507.2.k.d 8
39.i odd 6 1 507.2.k.f 8
39.k even 12 1 39.2.k.b 8
39.k even 12 1 507.2.k.d 8
39.k even 12 1 507.2.k.e 8
39.k even 12 1 507.2.k.f 8
52.i odd 6 1 624.2.cn.c 8
52.l even 12 1 624.2.cn.c 8
65.l even 6 1 975.2.bo.d 8
65.o even 12 1 975.2.bp.e 8
65.r odd 12 1 975.2.bp.e 8
65.r odd 12 1 975.2.bp.f 8
65.s odd 12 1 975.2.bo.d 8
65.t even 12 1 975.2.bp.f 8
156.r even 6 1 624.2.cn.c 8
156.v odd 12 1 624.2.cn.c 8
195.y odd 6 1 975.2.bo.d 8
195.bc odd 12 1 975.2.bp.f 8
195.bf even 12 1 975.2.bp.e 8
195.bf even 12 1 975.2.bp.f 8
195.bh even 12 1 975.2.bo.d 8
195.bn odd 12 1 975.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 13.e even 6 1
39.2.k.b 8 13.f odd 12 1
39.2.k.b 8 39.h odd 6 1
39.2.k.b 8 39.k even 12 1
507.2.f.e 8 1.a even 1 1 trivial
507.2.f.e 8 3.b odd 2 1 inner
507.2.f.e 8 13.d odd 4 1 inner
507.2.f.e 8 39.f even 4 1 inner
507.2.f.f 8 13.b even 2 1
507.2.f.f 8 13.d odd 4 1
507.2.f.f 8 39.d odd 2 1
507.2.f.f 8 39.f even 4 1
507.2.k.d 8 13.c even 3 1
507.2.k.d 8 13.f odd 12 1
507.2.k.d 8 39.i odd 6 1
507.2.k.d 8 39.k even 12 1
507.2.k.e 8 13.e even 6 1
507.2.k.e 8 13.f odd 12 1
507.2.k.e 8 39.h odd 6 1
507.2.k.e 8 39.k even 12 1
507.2.k.f 8 13.c even 3 1
507.2.k.f 8 13.f odd 12 1
507.2.k.f 8 39.i odd 6 1
507.2.k.f 8 39.k even 12 1
624.2.cn.c 8 52.i odd 6 1
624.2.cn.c 8 52.l even 12 1
624.2.cn.c 8 156.r even 6 1
624.2.cn.c 8 156.v odd 12 1
975.2.bo.d 8 65.l even 6 1
975.2.bo.d 8 65.s odd 12 1
975.2.bo.d 8 195.y odd 6 1
975.2.bo.d 8 195.bh even 12 1
975.2.bp.e 8 65.o even 12 1
975.2.bp.e 8 65.r odd 12 1
975.2.bp.e 8 195.bf even 12 1
975.2.bp.e 8 195.bn odd 12 1
975.2.bp.f 8 65.r odd 12 1
975.2.bp.f 8 65.t even 12 1
975.2.bp.f 8 195.bc odd 12 1
975.2.bp.f 8 195.bf even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{8} + 38 T_{2}^{4} + 169$$ $$T_{5}^{8} + 38 T_{5}^{4} + 169$$ $$T_{7}^{2} + 2 T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 + 38 T^{4} + T^{8}$$
$3$ $$( 9 - 6 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$5$ $$169 + 38 T^{4} + T^{8}$$
$7$ $$( 2 + 2 T + T^{2} )^{4}$$
$11$ $$2704 + 296 T^{4} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$( 117 - 30 T^{2} + T^{4} )^{2}$$
$19$ $$( 16 - 16 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$( 1573 + 82 T^{2} + T^{4} )^{2}$$
$31$ $$( 484 - 88 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 1369 + 74 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$41$ $$169 + 998 T^{4} + T^{8}$$
$43$ $$( 324 + 72 T^{2} + T^{4} )^{2}$$
$47$ $$11075584 + 9728 T^{4} + T^{8}$$
$53$ $$( 13 + 22 T^{2} + T^{4} )^{2}$$
$59$ $$43264 + 608 T^{4} + T^{8}$$
$61$ $$( 7 + T )^{8}$$
$67$ $$( 2704 - 208 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$71$ $$43264 + 608 T^{4} + T^{8}$$
$73$ $$( 121 - 154 T + 98 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$79$ $$( -2 + T )^{8}$$
$83$ $$2704 + 296 T^{4} + T^{8}$$
$89$ $$77228944 + 17768 T^{4} + T^{8}$$
$97$ $$( 484 - 352 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2}$$