# Properties

 Label 99.2.m.a Level $99$ Weight $2$ Character orbit 99.m Analytic conductor $0.791$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.m (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} +O(q^{10})$$ q + (-z^7 + z^6 + z^5 - z^4 + z^3 - 2*z^2 + 1) * q^2 + (-2*z^6 - z) * q^3 + (-3*z^7 - 3*z) * q^4 + (2*z^5 - 2*z + 2) * q^5 + (2*z^7 + z^6 - 4*z^5 + 2*z^4 - z^3 + 4*z - 3) * q^6 + (z^7 - z^6 + z^4 - z^3 + z^2 - 1) * q^7 + (4*z^7 - 5*z^6 + 4*z^2 - 4) * q^8 - 3*z^2 * q^9 $$q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} - 2) q^{10} + ( - 2 \zeta_{15}^{7} + 4 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - \zeta_{15}^{2} + 2 \zeta_{15} + 2) q^{11} + (3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 3 \zeta_{15} + 3) q^{12} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4) q^{13} + (\zeta_{15}^{7} - \zeta_{15}^{5} + 2 \zeta_{15} - 1) q^{14} + (4 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{2} + 2 \zeta_{15}) q^{15} + (5 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 8 \zeta_{15}^{5} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{3} + 8 \zeta_{15} - 5) q^{16} + (3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} + 3) q^{17} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{4} - 3 \zeta_{15}) q^{18} + \zeta_{15}^{6} q^{19} + (6 \zeta_{15}^{7} - 6 \zeta_{15}^{6} - 6 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 12 \zeta_{15}^{2} + \cdots - 6) q^{20} + \cdots + ( - 6 \zeta_{15}^{7} + 6 \zeta_{15}^{5} - 9 \zeta_{15}^{4} - 12 \zeta_{15} + 6) q^{99} +O(q^{100})$$ q + (-z^7 + z^6 + z^5 - z^4 + z^3 - 2*z^2 + 1) * q^2 + (-2*z^6 - z) * q^3 + (-3*z^7 - 3*z) * q^4 + (2*z^5 - 2*z + 2) * q^5 + (2*z^7 + z^6 - 4*z^5 + 2*z^4 - z^3 + 4*z - 3) * q^6 + (z^7 - z^6 + z^4 - z^3 + z^2 - 1) * q^7 + (4*z^7 - 5*z^6 + 4*z^2 - 4) * q^8 - 3*z^2 * q^9 + (-4*z^7 + 4*z^3 - 4*z^2 - 2) * q^10 + (-2*z^7 + 4*z^6 - 2*z^4 + 2*z^3 - z^2 + 2*z + 2) * q^11 + (3*z^7 + 3*z^5 - 3*z^4 - 3*z^3 + 3*z^2 - 3*z + 3) * q^12 + (-4*z^7 + 4*z^5 - 4*z^4 + 4) * q^13 + (z^7 - z^5 + 2*z - 1) * q^14 + (4*z^7 - 2*z^6 + 2*z^2 + 2*z) * q^15 + (5*z^7 + 3*z^6 - 8*z^5 + 5*z^4 - 5*z^3 + 8*z - 5) * q^16 + (3*z^6 + 3*z^3 + 3) * q^17 + (-3*z^7 + 3*z^4 - 3*z) * q^18 + z^6 * q^19 + (6*z^7 - 6*z^6 - 6*z^5 + 6*z^4 - 6*z^3 + 12*z^2 - 6) * q^20 + (2*z^5 + 1) * q^21 + (-3*z^7 + 6*z^5 - 3*z^4 - 7*z + 6) * q^22 + (-3*z^5 - z^4 - z - 3) * q^23 + (-9*z^7 + 8*z^6 + 4*z^5 - 4*z^4 + 8*z^3 - 10*z^2 + 4) * q^24 + (-3*z^6 + 4*z^5 + 4*z^2 - 3*z) * q^25 + (-4*z^7 + 4*z^6 - 4*z^2) * q^26 + (6*z^7 - 6*z^5 + 6*z^4 - 3*z^3 + 6*z - 6) * q^27 + (3*z^6 + 3) * q^28 + (3*z^7 + 3*z^4 - 3*z^3 + 3*z - 3) * q^29 + (-4*z^7 + 12*z^6 - 4*z^5 - 8*z^2 + 6*z + 4) * q^30 + (-z^5 + z^4 - 1) * q^31 + (9*z^5 - 3*z^4 - 3*z + 9) * q^32 + (-4*z^7 - 2*z^5 - 2*z^4 + z^3 + 2*z^2 - 2*z) * q^33 + (-3*z^7 + 3*z^5 - 3*z^4 + 3*z^3 - 3*z^2 - 3*z + 3) * q^34 + (2*z^7 - 2*z^6 - 2*z^3 + 2*z^2) * q^35 + (9*z^7 - 9*z^6 + 9*z^2 - 9) * q^36 + (3*z^7 - 3*z^6 + 2*z^3 + 3*z^2 + 2) * q^37 + (-z^7 - z^6 + 2*z^5 - z^4 + z^3 - 2*z + 1) * q^38 + (-4*z^7 - 4*z^6 - 4*z^4 - 4*z^3 - 4) * q^39 + (10*z^7 - 8*z^4 + 10*z) * q^40 + (-4*z^7 + 4*z^5 - 9*z + 4) * q^41 + (-3*z^7 + z^6 - z^5 + z^4 + z^3 - 2*z^2 - 1) * q^42 + (3*z^7 - 3*z^5 + 3*z^4 - 3*z^3 + 3*z^2 + 3*z - 3) * q^43 + (-9*z^7 - 3*z^6 + 6*z^3 - 9*z^2 - 3) * q^44 + (-6*z^7 + 6*z^3 - 6*z^2) * q^45 + (5*z^7 - 2*z^6 - 2*z^3 + 5*z^2) * q^46 + (z^6 - 8*z^5 - 8*z^2 + z) * q^47 + (2*z^7 - 8*z^6 + 10*z^4 - 5*z^3 + 8*z^2 - 6*z - 5) * q^48 + (6*z^7 - 6*z^5 + 6*z^4 + 6*z - 6) * q^49 + (3*z^7 - 10*z^5 + 3*z^4 + 10*z - 10) * q^50 + (-3*z^7 - 3*z^4 + 6*z^3 - 3*z + 6) * q^51 + (-12*z^6 - 12*z) * q^52 + (4*z^7 - 5*z^6 - 5*z^3 + 4*z^2) * q^53 + (3*z^7 + 6*z^5 - 3*z^4 - 3*z^3 + 3*z^2 - 3*z + 9) * q^54 + (-6*z^7 + 8*z^6 + 2*z^3 - 6*z^2) * q^55 + (-4*z^7 + 5*z^5 - 4*z^4 + 4*z^3 - 4*z^2 - 4*z + 4) * q^56 + (z^7 + 2*z^2) * q^57 + (-3*z^7 + 3*z^5 - 3*z + 3) * q^58 + (z^7 + z^4 + z) * q^59 + (-12*z^7 - 6*z^6 + 24*z^5 - 12*z^4 + 6*z^3 - 24*z + 18) * q^60 + (9*z^7 - 3*z^6 - 6*z^5 + 9*z^4 - 9*z^3 + 6*z - 9) * q^61 + (3*z^7 - 3*z^6 - z^3 + 3*z^2 - 1) * q^62 + 3*z * q^63 + (-5*z^7 + 6*z^6 + 6*z^3 - 5*z^2) * q^64 + 8*z^5 * q^65 + (11*z^7 - 6*z^6 - 3*z^4 - 3*z^3 + 7*z^2 + 3*z - 3) * q^66 + (-6*z^5 - 6) * q^67 - 9*z^7 * q^68 + (2*z^7 + 3*z^6 - z^5 + z^2 - 3*z - 2) * q^69 + (-2*z^7 + 6*z^6 - 2*z^4 + 2*z^3 - 6*z^2 + 4*z + 2) * q^70 + (5*z^6 - 6*z^3 + 5) * q^71 + (3*z^7 + 12*z^6 - 15*z^5 + 3*z^4 - 3*z^3 + 15*z - 3) * q^72 + (-5*z^7 + 5*z^6 + 2*z^3 - 5*z^2 + 2) * q^73 + (11*z^6 - 8*z^5 - 8*z^2 + 11*z) * q^74 + (-5*z^7 + 4*z^6 + 8*z^5 - 8*z^4 + 4*z^3 - 3*z^2 + 8) * q^75 + (-3*z^5 + 3*z^4 + 3*z - 3) * q^76 + (2*z^7 - 4*z^5 + 2*z^4 + z - 4) * q^77 + (8*z^7 - 4*z^5 + 4*z^4 - 8*z^3 + 8*z^2 + 4*z - 4) * q^78 + (-9*z^7 + 9*z^6 + 9*z^5 - 9*z^4 + 9*z^3 - 4*z^2 + 9) * q^79 + (-16*z^7 + 22*z^6 - 16*z^2 + 16) * q^80 + 9*z^4 * q^81 + (-13*z^6 + 9*z^3 - 13) * q^82 + (-11*z^7 - 2*z^6 + 13*z^5 - 11*z^4 + 11*z^3 - 13*z + 11) * q^83 + (3*z^7 - 6*z^6 + 6*z^2 - 3*z) * q^84 - 6*z * q^85 + (3*z^7 + 6*z^5 - 6*z^4 + 6) * q^86 + (3*z^7 + 6*z^4 - 3*z^3 + 3*z^2 + 6*z - 3) * q^87 + (15*z^7 - 12*z^6 - 13*z^5 + 15*z^4 - 15*z^3 + 18*z^2 + 3*z - 15) * q^88 + (-2*z^7 + 2*z^3 - 2*z^2 - 8) * q^89 + (12*z^7 - 12*z^6 - 12*z^5 + 12*z^4 - 12*z^3 + 18*z^2 - 12) * q^90 + (4*z^7 + 4*z^2 - 4) * q^91 + (3*z^7 + 6*z^6 + 3*z^4 - 3*z^3 - 6*z^2 + 9*z - 3) * q^92 + (z^6 + z^5 - z + 2) * q^93 + (-z^7 + 10*z^5 - z^4 - 10*z + 10) * q^94 + (-2*z^7 - 2*z) * q^95 + (6*z^7 - 9*z^6 - 3*z^5 + 3*z^2 + 9*z - 6) * q^96 + 6*z^2 * q^97 + (6*z^7 - 6*z^3 + 6*z^2 + 6) * q^98 + (-6*z^7 + 6*z^5 - 9*z^4 - 12*z + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9}+O(q^{10})$$ 8 * q - 4 * q^2 + 3 * q^3 - 6 * q^4 + 6 * q^5 - q^7 - 14 * q^8 - 3 * q^9 $$8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9} - 32 q^{10} + q^{11} + 18 q^{12} + 8 q^{13} - q^{14} + 12 q^{15} + 14 q^{16} + 12 q^{17} - 3 q^{18} - 2 q^{19} + 24 q^{20} + 11 q^{22} - 14 q^{23} - 39 q^{24} - 9 q^{25} - 16 q^{26} + 18 q^{28} - 9 q^{29} + 18 q^{30} - 3 q^{31} + 30 q^{32} - 6 q^{34} + 12 q^{35} - 36 q^{36} + 24 q^{37} - 4 q^{38} - 24 q^{39} + 12 q^{40} + 3 q^{41} - 12 q^{42} + 6 q^{43} - 48 q^{44} - 24 q^{45} + 18 q^{46} + 23 q^{47} - 6 q^{49} - 24 q^{50} + 27 q^{51} + 12 q^{52} + 28 q^{53} + 54 q^{54} - 32 q^{55} - 12 q^{56} + 3 q^{57} + 6 q^{58} + 3 q^{59} + 6 q^{62} + 3 q^{63} - 34 q^{64} - 32 q^{65} + 12 q^{66} - 24 q^{67} - 9 q^{68} - 18 q^{69} - 6 q^{70} + 42 q^{71} + 39 q^{72} - 8 q^{73} + 13 q^{74} - 6 q^{76} - 11 q^{77} + 24 q^{78} - 22 q^{79} + 52 q^{80} + 9 q^{81} - 96 q^{82} - 17 q^{83} + 18 q^{84} - 6 q^{85} + 21 q^{86} + 37 q^{88} - 72 q^{89} + 42 q^{90} - 24 q^{91} - 21 q^{92} + 9 q^{93} + 28 q^{94} - 4 q^{95} + 6 q^{97} + 72 q^{98} - 3 q^{99}+O(q^{100})$$ 8 * q - 4 * q^2 + 3 * q^3 - 6 * q^4 + 6 * q^5 - q^7 - 14 * q^8 - 3 * q^9 - 32 * q^10 + q^11 + 18 * q^12 + 8 * q^13 - q^14 + 12 * q^15 + 14 * q^16 + 12 * q^17 - 3 * q^18 - 2 * q^19 + 24 * q^20 + 11 * q^22 - 14 * q^23 - 39 * q^24 - 9 * q^25 - 16 * q^26 + 18 * q^28 - 9 * q^29 + 18 * q^30 - 3 * q^31 + 30 * q^32 - 6 * q^34 + 12 * q^35 - 36 * q^36 + 24 * q^37 - 4 * q^38 - 24 * q^39 + 12 * q^40 + 3 * q^41 - 12 * q^42 + 6 * q^43 - 48 * q^44 - 24 * q^45 + 18 * q^46 + 23 * q^47 - 6 * q^49 - 24 * q^50 + 27 * q^51 + 12 * q^52 + 28 * q^53 + 54 * q^54 - 32 * q^55 - 12 * q^56 + 3 * q^57 + 6 * q^58 + 3 * q^59 + 6 * q^62 + 3 * q^63 - 34 * q^64 - 32 * q^65 + 12 * q^66 - 24 * q^67 - 9 * q^68 - 18 * q^69 - 6 * q^70 + 42 * q^71 + 39 * q^72 - 8 * q^73 + 13 * q^74 - 6 * q^76 - 11 * q^77 + 24 * q^78 - 22 * q^79 + 52 * q^80 + 9 * q^81 - 96 * q^82 - 17 * q^83 + 18 * q^84 - 6 * q^85 + 21 * q^86 + 37 * q^88 - 72 * q^89 + 42 * q^90 - 24 * q^91 - 21 * q^92 + 9 * q^93 + 28 * q^94 - 4 * q^95 + 6 * q^97 + 72 * q^98 - 3 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$$ $$-1 - \zeta_{15}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.669131 + 0.743145i −0.104528 + 0.994522i 0.669131 − 0.743145i −0.104528 − 0.994522i 0.913545 − 0.406737i −0.978148 − 0.207912i −0.978148 + 0.207912i 0.913545 + 0.406737i
0.273659 2.60369i −1.28716 + 1.15897i −4.74803 1.00922i −0.338261 3.21834i 2.66535 + 3.66854i −0.669131 + 0.743145i −2.30902 + 7.10642i 0.313585 2.98357i −8.47214
16.1 0.373619 + 0.0794152i 1.72256 + 0.181049i −1.69381 0.754131i 1.20906 0.256993i 0.629204 + 0.204441i 0.104528 + 0.994522i −1.19098 0.865300i 2.93444 + 0.623735i 0.472136
25.1 0.273659 + 2.60369i −1.28716 1.15897i −4.74803 + 1.00922i −0.338261 + 3.21834i 2.66535 3.66854i −0.669131 0.743145i −2.30902 7.10642i 0.313585 + 2.98357i −8.47214
31.1 0.373619 0.0794152i 1.72256 0.181049i −1.69381 + 0.754131i 1.20906 + 0.256993i 0.629204 0.204441i 0.104528 0.994522i −1.19098 + 0.865300i 2.93444 0.623735i 0.472136
49.1 −0.255585 + 0.283856i 0.704489 + 1.58231i 0.193806 + 1.84395i −0.827091 0.918578i −0.629204 0.204441i −0.913545 0.406737i −1.19098 0.865300i −2.00739 + 2.22943i 0.472136
58.1 −2.39169 1.06485i 0.360114 1.69420i 3.24803 + 3.60730i 2.95630 1.31623i −2.66535 + 3.66854i 0.978148 0.207912i −2.30902 7.10642i −2.74064 1.22021i −8.47214
70.1 −2.39169 + 1.06485i 0.360114 + 1.69420i 3.24803 3.60730i 2.95630 + 1.31623i −2.66535 3.66854i 0.978148 + 0.207912i −2.30902 + 7.10642i −2.74064 + 1.22021i −8.47214
97.1 −0.255585 0.283856i 0.704489 1.58231i 0.193806 1.84395i −0.827091 + 0.918578i −0.629204 + 0.204441i −0.913545 + 0.406737i −1.19098 + 0.865300i −2.00739 2.22943i 0.472136
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.m.a 8
3.b odd 2 1 297.2.n.a 8
9.c even 3 1 inner 99.2.m.a 8
9.c even 3 1 891.2.f.b 4
9.d odd 6 1 297.2.n.a 8
9.d odd 6 1 891.2.f.a 4
11.c even 5 1 inner 99.2.m.a 8
11.c even 5 1 1089.2.e.g 4
11.d odd 10 1 1089.2.e.d 4
33.h odd 10 1 297.2.n.a 8
99.m even 15 1 inner 99.2.m.a 8
99.m even 15 1 891.2.f.b 4
99.m even 15 1 1089.2.e.g 4
99.m even 15 1 9801.2.a.n 2
99.n odd 30 1 297.2.n.a 8
99.n odd 30 1 891.2.f.a 4
99.n odd 30 1 9801.2.a.bb 2
99.o odd 30 1 1089.2.e.d 4
99.o odd 30 1 9801.2.a.bc 2
99.p even 30 1 9801.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.a 8 1.a even 1 1 trivial
99.2.m.a 8 9.c even 3 1 inner
99.2.m.a 8 11.c even 5 1 inner
99.2.m.a 8 99.m even 15 1 inner
297.2.n.a 8 3.b odd 2 1
297.2.n.a 8 9.d odd 6 1
297.2.n.a 8 33.h odd 10 1
297.2.n.a 8 99.n odd 30 1
891.2.f.a 4 9.d odd 6 1
891.2.f.a 4 99.n odd 30 1
891.2.f.b 4 9.c even 3 1
891.2.f.b 4 99.m even 15 1
1089.2.e.d 4 11.d odd 10 1
1089.2.e.d 4 99.o odd 30 1
1089.2.e.g 4 11.c even 5 1
1089.2.e.g 4 99.m even 15 1
9801.2.a.m 2 99.p even 30 1
9801.2.a.n 2 99.m even 15 1
9801.2.a.bb 2 99.n odd 30 1
9801.2.a.bc 2 99.o odd 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 4T_{2}^{7} + 10T_{2}^{6} + 26T_{2}^{5} + 39T_{2}^{4} - 14T_{2}^{3} - 5T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{7} + 10 T^{6} + 26 T^{5} + \cdots + 1$$
$3$ $$T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + \cdots + 81$$
$5$ $$T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256$$
$7$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$11$ $$T^{8} - T^{7} - 20 T^{6} + T^{5} + \cdots + 14641$$
$13$ $$T^{8} - 8 T^{7} - 128 T^{5} + \cdots + 65536$$
$17$ $$(T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81)^{2}$$
$19$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$23$ $$(T^{4} + 7 T^{3} + 38 T^{2} + 77 T + 121)^{2}$$
$29$ $$T^{8} + 9 T^{7} + 45 T^{6} + \cdots + 6561$$
$31$ $$T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1$$
$37$ $$(T^{4} - 12 T^{3} + 94 T^{2} - 403 T + 961)^{2}$$
$41$ $$T^{8} - 3 T^{7} - 100 T^{6} + \cdots + 707281$$
$43$ $$(T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2}$$
$47$ $$T^{8} - 23 T^{7} + 280 T^{6} + \cdots + 25411681$$
$53$ $$(T^{4} - 14 T^{3} + 96 T^{2} - 319 T + 841)^{2}$$
$59$ $$T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1$$
$61$ $$T^{8} - 90 T^{6} + 1350 T^{5} + \cdots + 4100625$$
$67$ $$(T^{2} + 6 T + 36)^{4}$$
$71$ $$(T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2}$$
$73$ $$(T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1)^{2}$$
$79$ $$T^{8} + 22 T^{7} + \cdots + 104060401$$
$83$ $$T^{8} + 17 T^{7} + \cdots + 373301041$$
$89$ $$(T^{2} + 18 T + 76)^{4}$$
$97$ $$T^{8} - 6 T^{7} + 216 T^{5} + \cdots + 1679616$$