# Properties

 Label 363.2.f.e Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.587785 + 0.809017i −0.587785 − 0.809017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
−1.53884 1.11803i −0.0877853 1.72982i 0.500000 + 1.53884i −1.53884 2.11803i −1.79892 + 2.76007i −0.690983 + 0.224514i −0.224514 + 0.690983i −2.98459 + 0.303706i 4.97980i
161.2 1.53884 + 1.11803i 1.08779 1.34786i 0.500000 + 1.53884i 1.53884 + 2.11803i 3.18088 0.857960i −0.690983 + 0.224514i 0.224514 0.690983i −0.633446 2.93236i 4.97980i
215.1 −0.363271 1.11803i −0.451057 + 1.67229i 0.500000 0.363271i −0.363271 0.118034i 2.03353 0.103198i −1.80902 2.48990i −2.48990 1.80902i −2.59310 1.50859i 0.449028i
215.2 0.363271 + 1.11803i 1.45106 0.945746i 0.500000 0.363271i 0.363271 + 0.118034i 1.58450 + 1.27877i −1.80902 2.48990i 2.48990 + 1.80902i 1.21113 2.74466i 0.449028i
233.1 −0.363271 + 1.11803i −0.451057 1.67229i 0.500000 + 0.363271i −0.363271 + 0.118034i 2.03353 + 0.103198i −1.80902 + 2.48990i −2.48990 + 1.80902i −2.59310 + 1.50859i 0.449028i
233.2 0.363271 1.11803i 1.45106 + 0.945746i 0.500000 + 0.363271i 0.363271 0.118034i 1.58450 1.27877i −1.80902 + 2.48990i 2.48990 1.80902i 1.21113 + 2.74466i 0.449028i
239.1 −1.53884 + 1.11803i −0.0877853 + 1.72982i 0.500000 1.53884i −1.53884 + 2.11803i −1.79892 2.76007i −0.690983 0.224514i −0.224514 0.690983i −2.98459 0.303706i 4.97980i
239.2 1.53884 1.11803i 1.08779 + 1.34786i 0.500000 1.53884i 1.53884 2.11803i 3.18088 + 0.857960i −0.690983 0.224514i 0.224514 + 0.690983i −0.633446 + 2.93236i 4.97980i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.2 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
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.e 8
3.b odd 2 1 inner 363.2.f.e 8
11.b odd 2 1 363.2.f.d 8
11.c even 5 1 33.2.f.a 8
11.c even 5 1 363.2.d.f 8
11.c even 5 1 363.2.f.b 8
11.c even 5 1 363.2.f.d 8
11.d odd 10 1 33.2.f.a 8
11.d odd 10 1 363.2.d.f 8
11.d odd 10 1 363.2.f.b 8
11.d odd 10 1 inner 363.2.f.e 8
33.d even 2 1 363.2.f.d 8
33.f even 10 1 33.2.f.a 8
33.f even 10 1 363.2.d.f 8
33.f even 10 1 363.2.f.b 8
33.f even 10 1 inner 363.2.f.e 8
33.h odd 10 1 33.2.f.a 8
33.h odd 10 1 363.2.d.f 8
33.h odd 10 1 363.2.f.b 8
33.h odd 10 1 363.2.f.d 8
44.g even 10 1 528.2.bn.c 8
44.h odd 10 1 528.2.bn.c 8
55.h odd 10 1 825.2.bi.b 8
55.j even 10 1 825.2.bi.b 8
55.k odd 20 1 825.2.bs.a 8
55.k odd 20 1 825.2.bs.d 8
55.l even 20 1 825.2.bs.a 8
55.l even 20 1 825.2.bs.d 8
99.m even 15 2 891.2.u.a 16
99.n odd 30 2 891.2.u.a 16
99.o odd 30 2 891.2.u.a 16
99.p even 30 2 891.2.u.a 16
132.n odd 10 1 528.2.bn.c 8
132.o even 10 1 528.2.bn.c 8
165.o odd 10 1 825.2.bi.b 8
165.r even 10 1 825.2.bi.b 8
165.u odd 20 1 825.2.bs.a 8
165.u odd 20 1 825.2.bs.d 8
165.v even 20 1 825.2.bs.a 8
165.v even 20 1 825.2.bs.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.f.a 8 11.c even 5 1
33.2.f.a 8 11.d odd 10 1
33.2.f.a 8 33.f even 10 1
33.2.f.a 8 33.h odd 10 1
363.2.d.f 8 11.c even 5 1
363.2.d.f 8 11.d odd 10 1
363.2.d.f 8 33.f even 10 1
363.2.d.f 8 33.h odd 10 1
363.2.f.b 8 11.c even 5 1
363.2.f.b 8 11.d odd 10 1
363.2.f.b 8 33.f even 10 1
363.2.f.b 8 33.h odd 10 1
363.2.f.d 8 11.b odd 2 1
363.2.f.d 8 11.c even 5 1
363.2.f.d 8 33.d even 2 1
363.2.f.d 8 33.h odd 10 1
363.2.f.e 8 1.a even 1 1 trivial
363.2.f.e 8 3.b odd 2 1 inner
363.2.f.e 8 11.d odd 10 1 inner
363.2.f.e 8 33.f even 10 1 inner
528.2.bn.c 8 44.g even 10 1
528.2.bn.c 8 44.h odd 10 1
528.2.bn.c 8 132.n odd 10 1
528.2.bn.c 8 132.o even 10 1
825.2.bi.b 8 55.h odd 10 1
825.2.bi.b 8 55.j even 10 1
825.2.bi.b 8 165.o odd 10 1
825.2.bi.b 8 165.r even 10 1
825.2.bs.a 8 55.k odd 20 1
825.2.bs.a 8 55.l even 20 1
825.2.bs.a 8 165.u odd 20 1
825.2.bs.a 8 165.v even 20 1
825.2.bs.d 8 55.k odd 20 1
825.2.bs.d 8 55.l even 20 1
825.2.bs.d 8 165.u odd 20 1
825.2.bs.d 8 165.v even 20 1
891.2.u.a 16 99.m even 15 2
891.2.u.a 16 99.n odd 30 2
891.2.u.a 16 99.o odd 30 2
891.2.u.a 16 99.p even 30 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{8} + 10T_{2}^{4} + 25T_{2}^{2} + 25$$ T2^8 + 10*T2^4 + 25*T2^2 + 25 $$T_{5}^{8} + 4T_{5}^{6} + 46T_{5}^{4} - 11T_{5}^{2} + 1$$ T5^8 + 4*T5^6 + 46*T5^4 - 11*T5^2 + 1 $$T_{7}^{4} + 5T_{7}^{3} + 15T_{7}^{2} + 15T_{7} + 5$$ T7^4 + 5*T7^3 + 15*T7^2 + 15*T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 10 T^{4} + 25 T^{2} + 25$$
$3$ $$T^{8} - 4 T^{7} + 13 T^{6} - 30 T^{5} + \cdots + 81$$
$5$ $$T^{8} + 4 T^{6} + 46 T^{4} - 11 T^{2} + \cdots + 1$$
$7$ $$(T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5)^{2}$$
$11$ $$T^{8}$$
$13$ $$(T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5)^{2}$$
$17$ $$T^{8} + 25 T^{6} + 250 T^{4} + \cdots + 15625$$
$19$ $$(T^{4} + 5 T^{3} + 125)^{2}$$
$23$ $$(T^{4} + 42 T^{2} + 121)^{2}$$
$29$ $$T^{8} + 20 T^{6} + 160 T^{4} + \cdots + 6400$$
$31$ $$(T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25)^{2}$$
$37$ $$(T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2}$$
$41$ $$T^{8} + 15 T^{6} + 85 T^{4} - 25 T^{2} + \cdots + 25$$
$43$ $$(T^{4} + 50 T^{2} + 125)^{2}$$
$47$ $$T^{8} - 44 T^{6} + 3966 T^{4} + \cdots + 13845841$$
$53$ $$T^{8} + 9 T^{6} + 486 T^{4} + \cdots + 6561$$
$59$ $$T^{8} - 11 T^{6} + 46 T^{4} + 4 T^{2} + \cdots + 1$$
$61$ $$(T^{4} + 10 T^{3} + 50 T^{2} + 125 T + 125)^{2}$$
$67$ $$(T^{2} + T - 61)^{4}$$
$71$ $$T^{8} + 20 T^{6} + 9150 T^{4} + \cdots + 9150625$$
$73$ $$(T^{4} + 40 T^{3} + 640 T^{2} + \cdots + 20480)^{2}$$
$79$ $$(T^{4} - 5 T^{3} + 35 T^{2} + 665 T + 1805)^{2}$$
$83$ $$T^{8} - 65 T^{6} + 37285 T^{4} + \cdots + 70644025$$
$89$ $$(T^{4} + 90 T^{2} + 25)^{2}$$
$97$ $$(T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2}$$