# Properties

 Label 24.3.b.a Level $24$ Weight $3$ Character orbit 24.b Analytic conductor $0.654$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 24.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.653952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.4752.1 Defining polynomial: $$x^{4} + 3x^{2} - 6x + 6$$ x^4 + 3*x^2 - 6*x + 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2} - 2) q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} + \beta_1 - 2) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{8} + 3 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 + b2 * q^3 + (-b3 - b2 - 2) * q^4 + 2*b3 * q^5 + (b3 + b1 - 2) * q^6 + (-2*b3 + 2*b2 - 4*b1 + 2) * q^7 + (-2*b3 + 4*b2 - 2*b1) * q^8 + 3 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2} - 2) q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} + \beta_1 - 2) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{8} + 3 q^{9} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{10} - 8 q^{11} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{12} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{13} + (6 \beta_{2} + 2 \beta_1 + 8) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{15} + (2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 4) q^{16} + ( - 8 \beta_{2} - 2) q^{17} + ( - 3 \beta_{2} + 3 \beta_1) q^{18} + ( - 4 \beta_{2} + 8) q^{19} + ( - 2 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 20) q^{20} + ( - 2 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 4) q^{21} + (8 \beta_{2} - 8 \beta_1) q^{22} + (4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{23} + (2 \beta_{2} - 6 \beta_1 + 12) q^{24} + (16 \beta_{2} - 11) q^{25} + ( - 4 \beta_{3} - 4 \beta_{2} - 24) q^{26} + 3 \beta_{2} q^{27} + (6 \beta_{3} - 10 \beta_{2} + 16 \beta_1 - 20) q^{28} + ( - 6 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 8) q^{29} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 16) q^{30} + (10 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 6) q^{31} + (8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 24) q^{32} - 8 \beta_{2} q^{33} + ( - 8 \beta_{3} + 2 \beta_{2} - 10 \beta_1 + 16) q^{34} + (4 \beta_{2} + 24) q^{35} + ( - 3 \beta_{3} - 3 \beta_{2} - 6) q^{36} + ( - 4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{37} + ( - 4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 8) q^{38} + (8 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{39} + ( - 8 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 24) q^{40} + (24 \beta_{2} + 10) q^{41} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 20) q^{42} + ( - 12 \beta_{2} + 8) q^{43} + (8 \beta_{3} + 8 \beta_{2} + 16) q^{44} + 6 \beta_{3} q^{45} + ( - 12 \beta_{2} - 4 \beta_1 - 16) q^{46} + ( - 12 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{47} + (2 \beta_{3} - 6 \beta_{2} + 8 \beta_1 + 20) q^{48} + ( - 32 \beta_{2} - 11) q^{49} + (16 \beta_{3} + 11 \beta_{2} + 5 \beta_1 - 32) q^{50} + ( - 2 \beta_{2} - 24) q^{51} + ( - 8 \beta_{3} + 32 \beta_{2} - 24 \beta_1) q^{52} + ( - 2 \beta_{3} - 8 \beta_{2} + 16 \beta_1 - 8) q^{53} + (3 \beta_{3} + 3 \beta_1 - 6) q^{54} - 16 \beta_{3} q^{55} + ( - 4 \beta_{3} - 8 \beta_{2} - 20 \beta_1 - 32) q^{56} + (8 \beta_{2} - 12) q^{57} + (2 \beta_{3} + 20 \beta_{2} + 6 \beta_1 + 36) q^{58} + (12 \beta_{2} - 32) q^{59} + ( - 2 \beta_{3} + 22 \beta_{2} - 8 \beta_1 - 20) q^{60} + (12 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 4) q^{61} + (16 \beta_{3} - 14 \beta_{2} - 10 \beta_1 + 56) q^{62} + ( - 6 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 6) q^{63} + (4 \beta_{3} + 4 \beta_{2} - 32 \beta_1 + 8) q^{64} + ( - 40 \beta_{2} + 24) q^{65} + ( - 8 \beta_{3} - 8 \beta_1 + 16) q^{66} + (12 \beta_{2} - 64) q^{67} + ( - 6 \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 20) q^{68} + (4 \beta_{3} - 8 \beta_{2} + 16 \beta_1 - 8) q^{69} + (4 \beta_{3} - 24 \beta_{2} + 28 \beta_1 - 8) q^{70} + ( - 4 \beta_{3} + 20 \beta_{2} - 40 \beta_1 + 20) q^{71} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{72} + (32 \beta_{2} + 50) q^{73} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 32) q^{74} + ( - 11 \beta_{2} + 48) q^{75} + ( - 12 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 8) q^{76} + (16 \beta_{3} - 16 \beta_{2} + 32 \beta_1 - 16) q^{77} + (4 \beta_{3} - 20 \beta_{2} - 8 \beta_1 - 8) q^{78} + (2 \beta_{3} - 18 \beta_{2} + 36 \beta_1 - 18) q^{79} + ( - 20 \beta_{3} - 20 \beta_{2} + 32 \beta_1 - 40) q^{80} + 9 q^{81} + (24 \beta_{3} - 10 \beta_{2} + 34 \beta_1 - 48) q^{82} + ( - 16 \beta_{2} + 40) q^{83} + (10 \beta_{3} - 26 \beta_{2} + 28 \beta_1 - 20) q^{84} + (12 \beta_{3} + 16 \beta_{2} - 32 \beta_1 + 16) q^{85} + ( - 12 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 24) q^{86} + ( - 10 \beta_{3} + 14 \beta_{2} - 28 \beta_1 + 14) q^{87} + (16 \beta_{3} - 32 \beta_{2} + 16 \beta_1) q^{88} + (48 \beta_{2} - 50) q^{89} + (6 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 12) q^{90} + (56 \beta_{2} + 72) q^{91} + ( - 12 \beta_{3} + 20 \beta_{2} - 32 \beta_1 + 40) q^{92} + ( - 22 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{93} + ( - 16 \beta_{3} + 20 \beta_{2} + 12 \beta_1 - 48) q^{94} + (24 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 8) q^{95} + ( - 4 \beta_{3} - 32 \beta_{2} + 20 \beta_1 - 16) q^{96} + ( - 48 \beta_{2} + 14) q^{97} + ( - 32 \beta_{3} + 11 \beta_{2} - 43 \beta_1 + 64) q^{98} - 24 q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 + b2 * q^3 + (-b3 - b2 - 2) * q^4 + 2*b3 * q^5 + (b3 + b1 - 2) * q^6 + (-2*b3 + 2*b2 - 4*b1 + 2) * q^7 + (-2*b3 + 4*b2 - 2*b1) * q^8 + 3 * q^9 + (2*b3 - 4*b2 - 2*b1 + 4) * q^10 - 8 * q^11 + (b3 - b2 - 2*b1 - 2) * q^12 + (-4*b2 + 8*b1 - 4) * q^13 + (6*b2 + 2*b1 + 8) * q^14 + (-2*b3 - 2*b2 + 4*b1 - 2) * q^15 + (2*b3 + 6*b2 + 4*b1 - 4) * q^16 + (-8*b2 - 2) * q^17 + (-3*b2 + 3*b1) * q^18 + (-4*b2 + 8) * q^19 + (-2*b3 - 6*b2 - 4*b1 + 20) * q^20 + (-2*b3 + 4*b2 - 8*b1 + 4) * q^21 + (8*b2 - 8*b1) * q^22 + (4*b3 - 4*b2 + 8*b1 - 4) * q^23 + (2*b2 - 6*b1 + 12) * q^24 + (16*b2 - 11) * q^25 + (-4*b3 - 4*b2 - 24) * q^26 + 3*b2 * q^27 + (6*b3 - 10*b2 + 16*b1 - 20) * q^28 + (-6*b3 + 8*b2 - 16*b1 + 8) * q^29 + (-4*b3 + 2*b2 + 2*b1 - 16) * q^30 + (10*b3 + 6*b2 - 12*b1 + 6) * q^31 + (8*b3 - 4*b2 + 4*b1 - 24) * q^32 - 8*b2 * q^33 + (-8*b3 + 2*b2 - 10*b1 + 16) * q^34 + (4*b2 + 24) * q^35 + (-3*b3 - 3*b2 - 6) * q^36 + (-4*b3 - 4*b2 + 8*b1 - 4) * q^37 + (-4*b3 - 8*b2 + 4*b1 + 8) * q^38 + (8*b3 - 4*b2 + 8*b1 - 4) * q^39 + (-8*b3 - 12*b2 + 12*b1 + 24) * q^40 + (24*b2 + 10) * q^41 + (2*b3 + 8*b2 + 2*b1 + 20) * q^42 + (-12*b2 + 8) * q^43 + (8*b3 + 8*b2 + 16) * q^44 + 6*b3 * q^45 + (-12*b2 - 4*b1 - 16) * q^46 + (-12*b3 - 4*b2 + 8*b1 - 4) * q^47 + (2*b3 - 6*b2 + 8*b1 + 20) * q^48 + (-32*b2 - 11) * q^49 + (16*b3 + 11*b2 + 5*b1 - 32) * q^50 + (-2*b2 - 24) * q^51 + (-8*b3 + 32*b2 - 24*b1) * q^52 + (-2*b3 - 8*b2 + 16*b1 - 8) * q^53 + (3*b3 + 3*b1 - 6) * q^54 - 16*b3 * q^55 + (-4*b3 - 8*b2 - 20*b1 - 32) * q^56 + (8*b2 - 12) * q^57 + (2*b3 + 20*b2 + 6*b1 + 36) * q^58 + (12*b2 - 32) * q^59 + (-2*b3 + 22*b2 - 8*b1 - 20) * q^60 + (12*b3 + 4*b2 - 8*b1 + 4) * q^61 + (16*b3 - 14*b2 - 10*b1 + 56) * q^62 + (-6*b3 + 6*b2 - 12*b1 + 6) * q^63 + (4*b3 + 4*b2 - 32*b1 + 8) * q^64 + (-40*b2 + 24) * q^65 + (-8*b3 - 8*b1 + 16) * q^66 + (12*b2 - 64) * q^67 + (-6*b3 + 10*b2 + 16*b1 + 20) * q^68 + (4*b3 - 8*b2 + 16*b1 - 8) * q^69 + (4*b3 - 24*b2 + 28*b1 - 8) * q^70 + (-4*b3 + 20*b2 - 40*b1 + 20) * q^71 + (-6*b3 + 12*b2 - 6*b1) * q^72 + (32*b2 + 50) * q^73 + (-8*b3 + 4*b2 + 4*b1 - 32) * q^74 + (-11*b2 + 48) * q^75 + (-12*b3 - 4*b2 + 8*b1 - 8) * q^76 + (16*b3 - 16*b2 + 32*b1 - 16) * q^77 + (4*b3 - 20*b2 - 8*b1 - 8) * q^78 + (2*b3 - 18*b2 + 36*b1 - 18) * q^79 + (-20*b3 - 20*b2 + 32*b1 - 40) * q^80 + 9 * q^81 + (24*b3 - 10*b2 + 34*b1 - 48) * q^82 + (-16*b2 + 40) * q^83 + (10*b3 - 26*b2 + 28*b1 - 20) * q^84 + (12*b3 + 16*b2 - 32*b1 + 16) * q^85 + (-12*b3 - 8*b2 - 4*b1 + 24) * q^86 + (-10*b3 + 14*b2 - 28*b1 + 14) * q^87 + (16*b3 - 32*b2 + 16*b1) * q^88 + (48*b2 - 50) * q^89 + (6*b3 - 12*b2 - 6*b1 + 12) * q^90 + (56*b2 + 72) * q^91 + (-12*b3 + 20*b2 - 32*b1 + 40) * q^92 + (-22*b3 - 4*b2 + 8*b1 - 4) * q^93 + (-16*b3 + 20*b2 + 12*b1 - 48) * q^94 + (24*b3 + 8*b2 - 16*b1 + 8) * q^95 + (-4*b3 - 32*b2 + 20*b1 - 16) * q^96 + (-48*b2 + 14) * q^97 + (-32*b3 + 11*b2 - 43*b1 + 64) * q^98 - 24 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 8 q^{4} - 6 q^{6} - 4 q^{8} + 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 8 * q^4 - 6 * q^6 - 4 * q^8 + 12 * q^9 $$4 q + 2 q^{2} - 8 q^{4} - 6 q^{6} - 4 q^{8} + 12 q^{9} + 12 q^{10} - 32 q^{11} - 12 q^{12} + 36 q^{14} - 8 q^{16} - 8 q^{17} + 6 q^{18} + 32 q^{19} + 72 q^{20} - 16 q^{22} + 36 q^{24} - 44 q^{25} - 96 q^{26} - 48 q^{28} - 60 q^{30} - 88 q^{32} + 44 q^{34} + 96 q^{35} - 24 q^{36} + 40 q^{38} + 120 q^{40} + 40 q^{41} + 84 q^{42} + 32 q^{43} + 64 q^{44} - 72 q^{46} + 96 q^{48} - 44 q^{49} - 118 q^{50} - 96 q^{51} - 48 q^{52} - 18 q^{54} - 168 q^{56} - 48 q^{57} + 156 q^{58} - 128 q^{59} - 96 q^{60} + 204 q^{62} - 32 q^{64} + 96 q^{65} + 48 q^{66} - 256 q^{67} + 112 q^{68} + 24 q^{70} - 12 q^{72} + 200 q^{73} - 120 q^{74} + 192 q^{75} - 16 q^{76} - 48 q^{78} - 96 q^{80} + 36 q^{81} - 124 q^{82} + 160 q^{83} - 24 q^{84} + 88 q^{86} + 32 q^{88} - 200 q^{89} + 36 q^{90} + 288 q^{91} + 96 q^{92} - 168 q^{94} - 24 q^{96} + 56 q^{97} + 170 q^{98} - 96 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 8 * q^4 - 6 * q^6 - 4 * q^8 + 12 * q^9 + 12 * q^10 - 32 * q^11 - 12 * q^12 + 36 * q^14 - 8 * q^16 - 8 * q^17 + 6 * q^18 + 32 * q^19 + 72 * q^20 - 16 * q^22 + 36 * q^24 - 44 * q^25 - 96 * q^26 - 48 * q^28 - 60 * q^30 - 88 * q^32 + 44 * q^34 + 96 * q^35 - 24 * q^36 + 40 * q^38 + 120 * q^40 + 40 * q^41 + 84 * q^42 + 32 * q^43 + 64 * q^44 - 72 * q^46 + 96 * q^48 - 44 * q^49 - 118 * q^50 - 96 * q^51 - 48 * q^52 - 18 * q^54 - 168 * q^56 - 48 * q^57 + 156 * q^58 - 128 * q^59 - 96 * q^60 + 204 * q^62 - 32 * q^64 + 96 * q^65 + 48 * q^66 - 256 * q^67 + 112 * q^68 + 24 * q^70 - 12 * q^72 + 200 * q^73 - 120 * q^74 + 192 * q^75 - 16 * q^76 - 48 * q^78 - 96 * q^80 + 36 * q^81 - 124 * q^82 + 160 * q^83 - 24 * q^84 + 88 * q^86 + 32 * q^88 - 200 * q^89 + 36 * q^90 + 288 * q^91 + 96 * q^92 - 168 * q^94 - 24 * q^96 + 56 * q^97 + 170 * q^98 - 96 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} - 6x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 3\nu^{2} + 4\nu + 2 ) / 4$$ (v^3 + 3*v^2 + 4*v + 2) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 6 ) / 4$$ (-v^3 + v^2 + 6) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} + 8\nu - 6 ) / 4$$ (v^3 - v^2 + 8*v - 6) / 4
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2$$ (-b3 + b2 + 2*b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} - 7\beta_{2} + 2\beta _1 + 8 ) / 2$$ (-b3 - 7*b2 + 2*b1 + 8) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.866025 − 0.719687i 0.866025 + 0.719687i −0.866025 + 1.99551i −0.866025 − 1.99551i
−0.366025 1.96622i 1.73205 −3.73205 + 1.43937i 2.87875i −0.633975 3.40559i 10.7436i 4.19615 + 6.81119i 3.00000 −5.66025 + 1.05369i
19.2 −0.366025 + 1.96622i 1.73205 −3.73205 1.43937i 2.87875i −0.633975 + 3.40559i 10.7436i 4.19615 6.81119i 3.00000 −5.66025 1.05369i
19.3 1.36603 1.46081i −1.73205 −0.267949 3.99102i 7.98203i −2.36603 + 2.53020i 2.13878i −6.19615 5.06040i 3.00000 11.6603 + 10.9037i
19.4 1.36603 + 1.46081i −1.73205 −0.267949 + 3.99102i 7.98203i −2.36603 2.53020i 2.13878i −6.19615 + 5.06040i 3.00000 11.6603 10.9037i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.3.b.a 4
3.b odd 2 1 72.3.b.b 4
4.b odd 2 1 96.3.b.a 4
5.b even 2 1 600.3.g.a 4
5.c odd 4 2 600.3.p.a 8
8.b even 2 1 96.3.b.a 4
8.d odd 2 1 inner 24.3.b.a 4
12.b even 2 1 288.3.b.b 4
16.e even 4 2 768.3.g.h 8
16.f odd 4 2 768.3.g.h 8
20.d odd 2 1 2400.3.g.a 4
20.e even 4 2 2400.3.p.a 8
24.f even 2 1 72.3.b.b 4
24.h odd 2 1 288.3.b.b 4
40.e odd 2 1 600.3.g.a 4
40.f even 2 1 2400.3.g.a 4
40.i odd 4 2 2400.3.p.a 8
40.k even 4 2 600.3.p.a 8
48.i odd 4 2 2304.3.g.z 8
48.k even 4 2 2304.3.g.z 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 1.a even 1 1 trivial
24.3.b.a 4 8.d odd 2 1 inner
72.3.b.b 4 3.b odd 2 1
72.3.b.b 4 24.f even 2 1
96.3.b.a 4 4.b odd 2 1
96.3.b.a 4 8.b even 2 1
288.3.b.b 4 12.b even 2 1
288.3.b.b 4 24.h odd 2 1
600.3.g.a 4 5.b even 2 1
600.3.g.a 4 40.e odd 2 1
600.3.p.a 8 5.c odd 4 2
600.3.p.a 8 40.k even 4 2
768.3.g.h 8 16.e even 4 2
768.3.g.h 8 16.f odd 4 2
2304.3.g.z 8 48.i odd 4 2
2304.3.g.z 8 48.k even 4 2
2400.3.g.a 4 20.d odd 2 1
2400.3.g.a 4 40.f even 2 1
2400.3.p.a 8 20.e even 4 2
2400.3.p.a 8 40.i odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 6 T^{2} - 8 T + 16$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4} + 72T^{2} + 528$$
$7$ $$T^{4} + 120T^{2} + 528$$
$11$ $$(T + 8)^{4}$$
$13$ $$T^{4} + 384 T^{2} + 33792$$
$17$ $$(T^{2} + 4 T - 188)^{2}$$
$19$ $$(T^{2} - 16 T + 16)^{2}$$
$23$ $$T^{4} + 480T^{2} + 8448$$
$29$ $$T^{4} + 1608T^{2} + 528$$
$31$ $$T^{4} + 3384 T^{2} + 279312$$
$37$ $$T^{4} + 864 T^{2} + 76032$$
$41$ $$(T^{2} - 20 T - 1628)^{2}$$
$43$ $$(T^{2} - 16 T - 368)^{2}$$
$47$ $$T^{4} + 3552 T^{2} + 8448$$
$53$ $$T^{4} + 1800 T^{2} + 803088$$
$59$ $$(T^{2} + 64 T + 592)^{2}$$
$61$ $$T^{4} + 3552 T^{2} + 8448$$
$67$ $$(T^{2} + 128 T + 3664)^{2}$$
$71$ $$T^{4} + 8928 T^{2} + \cdots + 12849408$$
$73$ $$(T^{2} - 100 T - 572)^{2}$$
$79$ $$T^{4} + 7416 T^{2} + \cdots + 10797072$$
$83$ $$(T^{2} - 80 T + 832)^{2}$$
$89$ $$(T^{2} + 100 T - 4412)^{2}$$
$97$ $$(T^{2} - 28 T - 6716)^{2}$$