# Properties

 Label 845.2.l.c Level $845$ Weight $2$ Character orbit 845.l Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{5} - \beta_{3}) q^{7} + (\beta_{5} + 2 \beta_{3}) q^{8} + (2 \beta_{4} - 2) q^{9}+O(q^{10})$$ q + (b5 + b1) * q^2 + (-b5 - b3) * q^3 + (b6 + b4 + b2) * q^4 + (b7 + b6 - b5 + b4 - 2*b3 - b1) * q^5 + (b6 - b2 + 1) * q^6 + (-2*b5 - b3) * q^7 + (b5 + 2*b3) * q^8 + (2*b4 - 2) * q^9 $$q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{5} - \beta_{3}) q^{7} + (\beta_{5} + 2 \beta_{3}) q^{8} + (2 \beta_{4} - 2) q^{9} + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{10} + (\beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{7} + \beta_{5} + \beta_1) q^{12} + (\beta_{6} - \beta_{4} - 2 \beta_{2} + 2) q^{14} + (\beta_{5} + \beta_{2} + \beta_1 + 1) q^{15} + (2 \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{16} + ( - 2 \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + 4 \beta_1) q^{17} + (2 \beta_{7} - 2 \beta_1) q^{18} + (\beta_{4} + 1) q^{19} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3) q^{20} + ( - 2 \beta_{4} + 1) q^{21} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{22} + (4 \beta_{7} - \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{4} - 2) q^{24} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} - 1) q^{25} + 5 \beta_{5} q^{27} + (2 \beta_{5} + \beta_{3} + 3 \beta_1) q^{28} + ( - 4 \beta_{6} + \beta_{4} + 2 \beta_{2} - 2) q^{29} + (\beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{30} + ( - 6 \beta_{6} - 2 \beta_{4} - 2) q^{31} + (\beta_{7} + 7 \beta_{5} + 3 \beta_{3}) q^{32} + (2 \beta_{7} - \beta_{3}) q^{33} + (3 \beta_{6} + 7 \beta_{4} - 2) q^{34} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{35} + ( - 4 \beta_{6} + 2 \beta_{2} - 2) q^{36} + (3 \beta_{5} + 3 \beta_{3} + 6 \beta_1) q^{37} + (\beta_{7} + 2 \beta_{5} + \beta_1) q^{38} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{40} + ( - \beta_{4} - 2 \beta_{2}) q^{41} + ( - 2 \beta_{7} - \beta_{5} + \beta_1) q^{42} + ( - 2 \beta_{7} + 2 \beta_{5} + 9 \beta_{3} + 4 \beta_1) q^{43} + ( - \beta_{6} - 7 \beta_{4} + 3) q^{44} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{45} + ( - 3 \beta_{6} - 2 \beta_{4} + 3 \beta_{2} - 5) q^{46} + ( - 4 \beta_{7} + 4 \beta_1) q^{47} + ( - \beta_{7} + \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{48} + 4 \beta_{4} q^{49} + ( - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{50} + (2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} + 1) q^{51} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{53} + (5 \beta_{4} + 5 \beta_{2} - 5) q^{54} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{55} + (3 \beta_{4} - 3) q^{56} + ( - \beta_{5} - 2 \beta_{3}) q^{57} + ( - 3 \beta_{7} - 7 \beta_{5} - 2 \beta_{3}) q^{58} + ( - 4 \beta_{6} + 7 \beta_{4} + 4 \beta_{2} + 3) q^{59} + (2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{60} + ( - 2 \beta_{6} - 7 \beta_{4} - 2 \beta_{2} + 5) q^{61} + ( - 8 \beta_{7} - 10 \beta_{5} - 6 \beta_{3} + 4 \beta_1) q^{62} + (2 \beta_{5} - 2 \beta_{3}) q^{63} + ( - 2 \beta_{6} + 2 \beta_{4} + 4 \beta_{2} - 9) q^{64} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2} - 4) q^{66} + ( - 7 \beta_{5} + \beta_{3} - 6 \beta_1) q^{67} + (2 \beta_{7} + 10 \beta_{5} + 9 \beta_{3} - \beta_1) q^{68} + (2 \beta_{6} + \beta_{4} + 2 \beta_{2} + 1) q^{69} + (3 \beta_{6} + \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 1) q^{70} + (4 \beta_{6} - 3 \beta_{4} - 4 \beta_{2} + 1) q^{71} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{72} + (3 \beta_{6} + 12 \beta_{4} + 3 \beta_{2} - 9) q^{74} + (4 \beta_{7} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{75} + (2 \beta_{4} + 3 \beta_{2} - 1) q^{76} + (2 \beta_{7} + \beta_{5} + 2 \beta_1) q^{77} + 6 q^{79} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{80} - \beta_{4} q^{81} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{82} + ( - 2 \beta_{7} - 4 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{83} + (3 \beta_{6} - \beta_{4} - 3 \beta_{2} + 2) q^{84} + (3 \beta_{7} + 3 \beta_{6} + 7 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + \cdots - 2) q^{85}+ \cdots + ( - 4 \beta_{6} - 2) q^{99}+O(q^{100})$$ q + (b5 + b1) * q^2 + (-b5 - b3) * q^3 + (b6 + b4 + b2) * q^4 + (b7 + b6 - b5 + b4 - 2*b3 - b1) * q^5 + (b6 - b2 + 1) * q^6 + (-2*b5 - b3) * q^7 + (b5 + 2*b3) * q^8 + (2*b4 - 2) * q^9 + (2*b7 + 2*b5 - b4 + b3 - b1) * q^10 + (b4 + 2*b2) * q^11 + (b7 + b5 + b1) * q^12 + (b6 - b4 - 2*b2 + 2) * q^14 + (b5 + b2 + b1 + 1) * q^15 + (2*b6 - b4 - b2 + 1) * q^16 + (-2*b7 + 2*b5 + 3*b3 + 4*b1) * q^17 + (2*b7 - 2*b1) * q^18 + (b4 + 1) * q^19 + (b7 - 2*b6 - 3*b5 - b4 - 2*b3 + 2*b2 - 3) * q^20 + (-2*b4 + 1) * q^21 + (b7 - b5 + 2*b3 - 2*b1) * q^22 + (4*b7 - b5 - 3*b3 - 2*b1) * q^23 + (b4 - 2) * q^24 + (-2*b6 - 2*b5 + 2*b4 + 4*b2 - 1) * q^25 + 5*b5 * q^27 + (2*b5 + b3 + 3*b1) * q^28 + (-4*b6 + b4 + 2*b2 - 2) * q^29 + (b6 + 3*b4 + b3 + b2 - 2) * q^30 + (-6*b6 - 2*b4 - 2) * q^31 + (b7 + 7*b5 + 3*b3) * q^32 + (2*b7 - b3) * q^33 + (3*b6 + 7*b4 - 2) * q^34 + (-b7 + b6 + b5 - b4 + b2 + 2*b1 + 2) * q^35 + (-4*b6 + 2*b2 - 2) * q^36 + (3*b5 + 3*b3 + 6*b1) * q^37 + (b7 + 2*b5 + b1) * q^38 + (-b7 + b6 - 2*b5 - b4 - 2*b2 - b1 - 1) * q^40 + (-b4 - 2*b2) * q^41 + (-2*b7 - b5 + b1) * q^42 + (-2*b7 + 2*b5 + 9*b3 + 4*b1) * q^43 + (-b6 - 7*b4 + 3) * q^44 + (-2*b7 - 2*b6 + 2*b5 + 2*b3 + 2*b2 - 2) * q^45 + (-3*b6 - 2*b4 + 3*b2 - 5) * q^46 + (-4*b7 + 4*b1) * q^47 + (-b7 + b5 + 2*b3 + 2*b1) * q^48 + 4*b4 * q^49 + (-5*b5 - 2*b4 + 2*b3 - 2*b2 - 3*b1 + 2) * q^50 + (2*b6 - 2*b4 - 4*b2 + 1) * q^51 + (-2*b7 + 2*b5 - 2*b1) * q^53 + (5*b4 + 5*b2 - 5) * q^54 + (2*b7 - 2*b6 - 4*b5 - 3*b4 - 5*b3 + b2 - b1 - 1) * q^55 + (3*b4 - 3) * q^56 + (-b5 - 2*b3) * q^57 + (-3*b7 - 7*b5 - 2*b3) * q^58 + (-4*b6 + 7*b4 + 4*b2 + 3) * q^59 + (2*b7 + b6 + b5 - 3*b4 + 2*b3 - 2*b1 + 2) * q^60 + (-2*b6 - 7*b4 - 2*b2 + 5) * q^61 + (-8*b7 - 10*b5 - 6*b3 + 4*b1) * q^62 + (2*b5 - 2*b3) * q^63 + (-2*b6 + 2*b4 + 4*b2 - 9) * q^64 + (-b6 + b4 + 2*b2 - 4) * q^66 + (-7*b5 + b3 - 6*b1) * q^67 + (2*b7 + 10*b5 + 9*b3 - b1) * q^68 + (2*b6 + b4 + 2*b2 + 1) * q^69 + (3*b6 + b5 + 5*b4 + 2*b3 - 1) * q^70 + (4*b6 - 3*b4 - 4*b2 + 1) * q^71 + (-4*b5 - 2*b3) * q^72 + (3*b6 + 12*b4 + 3*b2 - 9) * q^74 + (4*b7 - b5 - 2*b4 - 3*b3 - 2*b1) * q^75 + (2*b4 + 3*b2 - 1) * q^76 + (2*b7 + b5 + 2*b1) * q^77 + 6 * q^79 + (-b5 + 2*b4 + 3*b3 + b2 + 2*b1 - 3) * q^80 - b4 * q^81 + (-b7 + b5 - 2*b3 + 2*b1) * q^82 + (-2*b7 - 4*b5 - 8*b3 + 2*b1) * q^83 + (3*b6 - b4 - 3*b2 + 2) * q^84 + (3*b7 + 3*b6 + 7*b5 - 5*b4 + 2*b3 - 3*b2 - 2) * q^85 + (-3*b6 + b4 - 2) * q^86 + (2*b7 - 2*b5 - 3*b3 - 4*b1) * q^87 + (-4*b7 + b5 + 3*b3 + 2*b1) * q^88 + (3*b4 - 2*b2 - 8) * q^89 + (-2*b7 - 6*b5 - 2*b1 + 2) * q^90 + (-b7 - 11*b5 - b1) * q^92 + (-4*b5 - 2*b3 - 6*b1) * q^93 + (8*b6 + 8*b4 - 4*b2 + 4) * q^94 + (b7 + b6 - b5 + 2*b4 - 3*b3 + b2 - 2*b1 - 1) * q^95 + (b6 + 7*b4 - 3) * q^96 + (6*b7 - 4*b5 - 5*b3) * q^97 + (4*b7 + 4*b5) * q^98 + (-4*b6 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10})$$ 8 * q - 2 * q^4 + 6 * q^6 - 8 * q^9 $$8 q - 2 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{10} + 12 q^{14} + 6 q^{15} - 2 q^{16} + 12 q^{19} - 24 q^{20} - 12 q^{24} - 10 q^{30} + 6 q^{35} - 4 q^{36} - 12 q^{40} - 12 q^{45} - 42 q^{46} + 16 q^{49} + 12 q^{50} - 30 q^{54} - 14 q^{55} - 12 q^{56} + 60 q^{59} + 24 q^{61} - 64 q^{64} - 28 q^{66} - 12 q^{71} - 42 q^{74} - 8 q^{75} - 6 q^{76} + 48 q^{79} - 18 q^{80} - 4 q^{81} + 6 q^{84} - 42 q^{85} - 48 q^{89} + 16 q^{90} + 40 q^{94} - 6 q^{95}+O(q^{100})$$ 8 * q - 2 * q^4 + 6 * q^6 - 8 * q^9 - 4 * q^10 + 12 * q^14 + 6 * q^15 - 2 * q^16 + 12 * q^19 - 24 * q^20 - 12 * q^24 - 10 * q^30 + 6 * q^35 - 4 * q^36 - 12 * q^40 - 12 * q^45 - 42 * q^46 + 16 * q^49 + 12 * q^50 - 30 * q^54 - 14 * q^55 - 12 * q^56 + 60 * q^59 + 24 * q^61 - 64 * q^64 - 28 * q^66 - 12 * q^71 - 42 * q^74 - 8 * q^75 - 6 * q^76 + 48 * q^79 - 18 * q^80 - 4 * q^81 + 6 * q^84 - 42 * q^85 - 48 * q^89 + 16 * q^90 + 40 * q^94 - 6 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20$$ (-v^6 + 5*v^4 - 5*v^2 - 12) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40$$ (-v^7 + 5*v^5 - 5*v^3 - 12*v) / 40 $$\beta_{4}$$ $$=$$ $$( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20$$ (3*v^6 + 5*v^4 + 15*v^2 + 36) / 20 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40$$ (-3*v^7 - 5*v^5 + 5*v^3 - 16*v) / 40 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 7 ) / 5$$ (-v^6 - 7) / 5 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8$$ (v^7 + 3*v^5 + 5*v^3 + 12*v) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{2} - 1$$ b6 + b4 - b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1$$ b7 + 2*b5 - b3 - b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{2}$$ b4 + 3*b2 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 5\beta_{3}$$ b7 + 5*b3 $$\nu^{6}$$ $$=$$ $$-5\beta_{6} - 7$$ -5*b6 - 7 $$\nu^{7}$$ $$=$$ $$-10\beta_{5} - 10\beta_{3} - 7\beta_1$$ -10*b5 - 10*b3 - 7*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$1 - \beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
654.1
 −1.09445 + 0.895644i −0.228425 + 1.39564i 0.228425 − 1.39564i 1.09445 − 0.895644i −1.09445 − 0.895644i −0.228425 − 1.39564i 0.228425 + 1.39564i 1.09445 + 0.895644i
−1.09445 + 1.89564i −0.866025 0.500000i −1.39564 2.41733i 0.456850 2.18890i 1.89564 1.09445i −0.866025 1.50000i 1.73205 −1.00000 1.73205i 3.64938 + 3.26167i
654.2 −0.228425 + 0.395644i 0.866025 + 0.500000i 0.895644 + 1.55130i 2.18890 + 0.456850i −0.395644 + 0.228425i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i −0.680750 + 0.761669i
654.3 0.228425 0.395644i −0.866025 0.500000i 0.895644 + 1.55130i −2.18890 + 0.456850i −0.395644 + 0.228425i −0.866025 1.50000i 1.73205 −1.00000 1.73205i −0.319250 + 0.970381i
654.4 1.09445 1.89564i 0.866025 + 0.500000i −1.39564 2.41733i −0.456850 2.18890i 1.89564 1.09445i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i −4.64938 1.52962i
699.1 −1.09445 1.89564i −0.866025 + 0.500000i −1.39564 + 2.41733i 0.456850 + 2.18890i 1.89564 + 1.09445i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i 3.64938 3.26167i
699.2 −0.228425 0.395644i 0.866025 0.500000i 0.895644 1.55130i 2.18890 0.456850i −0.395644 0.228425i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i −0.680750 0.761669i
699.3 0.228425 + 0.395644i −0.866025 + 0.500000i 0.895644 1.55130i −2.18890 0.456850i −0.395644 0.228425i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −0.319250 0.970381i
699.4 1.09445 + 1.89564i 0.866025 0.500000i −1.39564 + 2.41733i −0.456850 + 2.18890i 1.89564 + 1.09445i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i −4.64938 + 1.52962i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.c 8
5.b even 2 1 inner 845.2.l.c 8
13.b even 2 1 65.2.l.a 8
13.c even 3 1 65.2.l.a 8
13.c even 3 1 845.2.d.c 8
13.d odd 4 1 845.2.n.c 8
13.d odd 4 1 845.2.n.d 8
13.e even 6 1 845.2.d.c 8
13.e even 6 1 inner 845.2.l.c 8
13.f odd 12 2 845.2.b.f 8
13.f odd 12 1 845.2.n.c 8
13.f odd 12 1 845.2.n.d 8
39.d odd 2 1 585.2.bf.a 8
39.i odd 6 1 585.2.bf.a 8
52.b odd 2 1 1040.2.df.b 8
52.j odd 6 1 1040.2.df.b 8
65.d even 2 1 65.2.l.a 8
65.g odd 4 1 845.2.n.c 8
65.g odd 4 1 845.2.n.d 8
65.h odd 4 1 325.2.n.b 4
65.h odd 4 1 325.2.n.c 4
65.l even 6 1 845.2.d.c 8
65.l even 6 1 inner 845.2.l.c 8
65.n even 6 1 65.2.l.a 8
65.n even 6 1 845.2.d.c 8
65.o even 12 1 4225.2.a.bj 4
65.o even 12 1 4225.2.a.bk 4
65.q odd 12 1 325.2.n.b 4
65.q odd 12 1 325.2.n.c 4
65.s odd 12 2 845.2.b.f 8
65.s odd 12 1 845.2.n.c 8
65.s odd 12 1 845.2.n.d 8
65.t even 12 1 4225.2.a.bj 4
65.t even 12 1 4225.2.a.bk 4
195.e odd 2 1 585.2.bf.a 8
195.x odd 6 1 585.2.bf.a 8
260.g odd 2 1 1040.2.df.b 8
260.v odd 6 1 1040.2.df.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 13.b even 2 1
65.2.l.a 8 13.c even 3 1
65.2.l.a 8 65.d even 2 1
65.2.l.a 8 65.n even 6 1
325.2.n.b 4 65.h odd 4 1
325.2.n.b 4 65.q odd 12 1
325.2.n.c 4 65.h odd 4 1
325.2.n.c 4 65.q odd 12 1
585.2.bf.a 8 39.d odd 2 1
585.2.bf.a 8 39.i odd 6 1
585.2.bf.a 8 195.e odd 2 1
585.2.bf.a 8 195.x odd 6 1
845.2.b.f 8 13.f odd 12 2
845.2.b.f 8 65.s odd 12 2
845.2.d.c 8 13.c even 3 1
845.2.d.c 8 13.e even 6 1
845.2.d.c 8 65.l even 6 1
845.2.d.c 8 65.n even 6 1
845.2.l.c 8 1.a even 1 1 trivial
845.2.l.c 8 5.b even 2 1 inner
845.2.l.c 8 13.e even 6 1 inner
845.2.l.c 8 65.l even 6 1 inner
845.2.n.c 8 13.d odd 4 1
845.2.n.c 8 13.f odd 12 1
845.2.n.c 8 65.g odd 4 1
845.2.n.c 8 65.s odd 12 1
845.2.n.d 8 13.d odd 4 1
845.2.n.d 8 13.f odd 12 1
845.2.n.d 8 65.g odd 4 1
845.2.n.d 8 65.s odd 12 1
1040.2.df.b 8 52.b odd 2 1
1040.2.df.b 8 52.j odd 6 1
1040.2.df.b 8 260.g odd 2 1
1040.2.df.b 8 260.v odd 6 1
4225.2.a.bj 4 65.o even 12 1
4225.2.a.bj 4 65.t even 12 1
4225.2.a.bk 4 65.o even 12 1
4225.2.a.bk 4 65.t even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 5T_{2}^{6} + 24T_{2}^{4} + 5T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 5 T^{6} + 24 T^{4} + 5 T^{2} + \cdots + 1$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8} - 34T^{4} + 625$$
$7$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$11$ $$(T^{4} - 7 T^{2} + 49)^{2}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} - 21 T^{2} + 441)^{2}$$
$19$ $$(T^{2} - 3 T + 3)^{4}$$
$23$ $$(T^{4} - 21 T^{2} + 441)^{2}$$
$29$ $$(T^{4} + 21 T^{2} + 441)^{2}$$
$31$ $$(T^{4} + 132 T^{2} + 3600)^{2}$$
$37$ $$(T^{4} + 63 T^{2} + 3969)^{2}$$
$41$ $$(T^{4} - 7 T^{2} + 49)^{2}$$
$43$ $$T^{8} - 114 T^{6} + 12771 T^{4} + \cdots + 50625$$
$47$ $$(T^{4} - 80 T^{2} + 256)^{2}$$
$53$ $$(T^{4} + 60 T^{2} + 144)^{2}$$
$59$ $$(T^{4} - 30 T^{3} + 347 T^{2} - 1410 T + 2209)^{2}$$
$61$ $$(T^{4} - 12 T^{3} + 129 T^{2} - 180 T + 225)^{2}$$
$67$ $$T^{8} + 222 T^{6} + 49059 T^{4} + \cdots + 50625$$
$71$ $$(T^{4} + 6 T^{3} - 13 T^{2} - 150 T + 625)^{2}$$
$73$ $$T^{8}$$
$79$ $$(T - 6)^{8}$$
$83$ $$(T^{4} - 164 T^{2} + 4624)^{2}$$
$89$ $$(T^{4} + 24 T^{3} + 233 T^{2} + 984 T + 1681)^{2}$$
$97$ $$T^{8} + 150 T^{6} + 19899 T^{4} + \cdots + 6765201$$