# Properties

 Label 784.2.x.h Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 112) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000 + 1.00000i −0.500000 1.86603i 2.00000i −0.866025 + 3.23205i 1.36603 2.36603i 0 −2.00000 + 2.00000i −0.633975 + 0.366025i −4.09808 + 2.36603i
373.1 1.00000 + 1.00000i −0.500000 0.133975i 2.00000i 0.866025 0.232051i −0.366025 0.633975i 0 −2.00000 + 2.00000i −2.36603 1.36603i 1.09808 + 0.633975i
557.1 1.00000 1.00000i −0.500000 + 0.133975i 2.00000i 0.866025 + 0.232051i −0.366025 + 0.633975i 0 −2.00000 2.00000i −2.36603 + 1.36603i 1.09808 0.633975i
765.1 1.00000 1.00000i −0.500000 + 1.86603i 2.00000i −0.866025 3.23205i 1.36603 + 2.36603i 0 −2.00000 2.00000i −0.633975 0.366025i −4.09808 2.36603i
 $$n$$: 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
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.h 4
7.b odd 2 1 112.2.w.b yes 4
7.c even 3 1 784.2.m.d 4
7.c even 3 1 784.2.x.a 4
7.d odd 6 1 112.2.w.a 4
7.d odd 6 1 784.2.m.e 4
16.e even 4 1 784.2.x.a 4
28.d even 2 1 448.2.ba.a 4
28.f even 6 1 448.2.ba.b 4
56.e even 2 1 896.2.ba.c 4
56.h odd 2 1 896.2.ba.b 4
56.j odd 6 1 896.2.ba.d 4
56.m even 6 1 896.2.ba.a 4
112.j even 4 1 448.2.ba.b 4
112.j even 4 1 896.2.ba.a 4
112.l odd 4 1 112.2.w.a 4
112.l odd 4 1 896.2.ba.d 4
112.v even 12 1 448.2.ba.a 4
112.v even 12 1 896.2.ba.c 4
112.w even 12 1 784.2.m.d 4
112.w even 12 1 inner 784.2.x.h 4
112.x odd 12 1 112.2.w.b yes 4
112.x odd 12 1 784.2.m.e 4
112.x odd 12 1 896.2.ba.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.a 4 7.d odd 6 1
112.2.w.a 4 112.l odd 4 1
112.2.w.b yes 4 7.b odd 2 1
112.2.w.b yes 4 112.x odd 12 1
448.2.ba.a 4 28.d even 2 1
448.2.ba.a 4 112.v even 12 1
448.2.ba.b 4 28.f even 6 1
448.2.ba.b 4 112.j even 4 1
784.2.m.d 4 7.c even 3 1
784.2.m.d 4 112.w even 12 1
784.2.m.e 4 7.d odd 6 1
784.2.m.e 4 112.x odd 12 1
784.2.x.a 4 7.c even 3 1
784.2.x.a 4 16.e even 4 1
784.2.x.h 4 1.a even 1 1 trivial
784.2.x.h 4 112.w even 12 1 inner
896.2.ba.a 4 56.m even 6 1
896.2.ba.a 4 112.j even 4 1
896.2.ba.b 4 56.h odd 2 1
896.2.ba.b 4 112.x odd 12 1
896.2.ba.c 4 56.e even 2 1
896.2.ba.c 4 112.v even 12 1
896.2.ba.d 4 56.j odd 6 1
896.2.ba.d 4 112.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 1$$ T3^4 + 2*T3^3 + 5*T3^2 + 4*T3 + 1 $$T_{5}^{4} + 9T_{5}^{2} - 18T_{5} + 9$$ T5^4 + 9*T5^2 - 18*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{2}$$
$3$ $$T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1$$
$5$ $$T^{4} + 9 T^{2} - 18 T + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169$$
$13$ $$T^{4} + 8 T^{3} + 32 T^{2} + 16 T + 4$$
$17$ $$T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9$$
$19$ $$T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169$$
$23$ $$T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121$$
$29$ $$T^{4} + 8 T^{3} + 32 T^{2} + 16 T + 4$$
$31$ $$T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1$$
$37$ $$T^{4} + 8 T^{3} + 137 T^{2} + \cdots + 169$$
$41$ $$T^{4} + 104T^{2} + 1936$$
$43$ $$T^{4} - 12 T^{3} + 72 T^{2} + 72 T + 36$$
$47$ $$T^{4} - 12 T^{3} + 111 T^{2} + \cdots + 1089$$
$53$ $$T^{4} - 20 T^{3} + 101 T^{2} + \cdots + 2209$$
$59$ $$T^{4} - 26 T^{3} + 365 T^{2} + \cdots + 14641$$
$61$ $$T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1$$
$67$ $$T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$T^{4} + 18 T^{3} + 131 T^{2} + \cdots + 529$$
$79$ $$T^{4} - 16 T^{3} + 267 T^{2} + \cdots + 121$$
$83$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 676$$
$89$ $$(T^{2} - 9 T + 27)^{2}$$
$97$ $$(T^{2} + 8 T - 32)^{2}$$