# Properties

 Label 32.2.g.b Level $32$ Weight $2$ Character orbit 32.g Analytic conductor $0.256$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 32.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.255521286468$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1 - 2) q^{6} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b7 + b5 + b3 + b1) * q^3 + (b6 - b3 + b2) * q^4 + (-b7 - b6) * q^5 + (-2*b7 - b5 + b2 - b1 - 2) * q^6 + (b7 - 2*b5 + b4 - b3 + b2 - b1 - 2) * q^7 + (b7 - 2*b6 - b4 - b2) * q^8 + (-b7 + b6 - 2*b4 - 2*b1 - 1) * q^9 $$q - \beta_{2} q^{2} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1 - 2) q^{6} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{11} + (2 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{12} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{13} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{14} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{15} + ( - \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{17} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{18} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_1) q^{19} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{20} + (3 \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 1) q^{21} + (\beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{22} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{23} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{24} + (\beta_{7} + 3 \beta_{5} - 1) q^{25} + (3 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 - 4) q^{26} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{5} - 2 \beta_{3} + 1) q^{27} + ( - \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{28} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} - 2) q^{29} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{30} + ( - 2 \beta_{6} - 2 \beta_{5} + 4) q^{31} + ( - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{32} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} + 2 \beta_1 - 2) q^{33} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3}) q^{34} + (\beta_{7} + 3 \beta_{6} + \beta_{5} + 2 \beta_{2} + 1) q^{35} + (2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 - 4) q^{36} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2}) q^{37} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 + 2) q^{38} + ( - \beta_{7} + 4 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{39} + (3 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{40} + (\beta_{7} - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{41} + ( - 3 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{42} + (3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{43} + (\beta_{7} - 3 \beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{44} + (\beta_{6} + 2 \beta_{3} - 2 \beta_1 - 1) q^{45} + ( - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 + 2) q^{46} + ( - 2 \beta_{7} - 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 6) q^{48} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{49}+ \cdots + (\beta_{6} + \beta_{3} - 6 \beta_{2} + \beta_1 + 7) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b7 + b5 + b3 + b1) * q^3 + (b6 - b3 + b2) * q^4 + (-b7 - b6) * q^5 + (-2*b7 - b5 + b2 - b1 - 2) * q^6 + (b7 - 2*b5 + b4 - b3 + b2 - b1 - 2) * q^7 + (b7 - 2*b6 - b4 - b2) * q^8 + (-b7 + b6 - 2*b4 - 2*b1 - 1) * q^9 + (b7 + b6 + b4 + b3) * q^10 + (-b7 + b6 + b4 - b2 + 1) * q^11 + (2*b7 - b6 + b5 + 2*b4 + b3 + b1 + 2) * q^12 + (-b7 + b5 - 2*b3 + 2*b1) * q^13 + (-b6 + 3*b5 - 2*b4 + b3 + b1 + 2) * q^14 + (-b6 + b5 - b4 - b3 - b2 + b1 + 1) * q^15 + (-b7 + 3*b6 - b5 - b4 + b3 + b2 + b1) * q^16 + (2*b7 + 2*b6 - 2*b5 + 2*b4 + 2*b3) * q^17 + (b7 - b6 - 2*b5 + b4 - b3 - b2 + 2*b1 + 4) * q^18 + (-b7 + b5 - 2*b4 + b3 - b1) * q^19 + (-2*b7 - b6 + b5 - b3 - b1) * q^20 + (3*b7 - 3*b6 - b5 + 2*b4 - 2*b2 + 1) * q^21 + (b7 + b5 + b4 - 2*b3 - b1) * q^22 + (-b7 + 2*b6 - b4 + b3 + b2 - b1 - 2) * q^23 + (-3*b7 + b6 + b5 - b4 + b3 - b2 - 3*b1 - 2) * q^24 + (b7 + 3*b5 - 1) * q^25 + (3*b7 - b5 - b4 + 2*b2 - b1 - 4) * q^26 + (-3*b7 - 3*b6 - b5 - 2*b3 + 1) * q^27 + (-b7 + b6 - 5*b5 + b4 + b3 - b2 - b1 - 2) * q^28 + (-b7 + 2*b6 - b5 + 4*b2 - 2) * q^29 + (b7 + 2*b6 - 2*b5 - b4 + b2 - 2) * q^30 + (-2*b6 - 2*b5 + 4) * q^31 + (-4*b6 + 2*b4 - 2*b3 + 2*b1 - 2) * q^32 + (-b6 - b5 + 2*b2 + 2*b1 - 2) * q^33 + (-4*b7 - 2*b6 + 4*b5 - 2*b3) * q^34 + (b7 + 3*b6 + b5 + 2*b2 + 1) * q^35 + (2*b6 + 3*b5 - 2*b4 - b2 + b1 - 4) * q^36 + (3*b7 + 3*b6 + 2*b5 - 2*b4 + 4*b3 - 2*b2) * q^37 + (-3*b5 + 2*b4 - b2 + b1 + 2) * q^38 + (-b7 + 4*b5 + b4 + b3 - b2 - b1 + 2) * q^39 + (3*b7 + b6 - b5 + b4 + b3 - b2 - b1 + 2) * q^40 + (b7 - 4*b6 + 2*b4 - 2*b3 - 2*b2 + 2*b1 + 3) * q^41 + (-3*b7 + 5*b6 + 3*b5 - 3*b4 + b3 + b2 - b1) * q^42 + (3*b7 - 3*b6 - 2*b5 + b4 - b2 - 1) * q^43 + (b7 - 3*b4 - b2 - 2*b1 + 2) * q^44 + (b6 + 2*b3 - 2*b1 - 1) * q^45 + (-3*b6 - b5 + 2*b4 - b3 + b1 + 2) * q^46 + (-2*b7 - 2*b2 + 2*b1 + 2) * q^47 + (2*b7 - 2*b5 + 4*b4 - 2*b3 + 6) * q^48 + (-3*b7 - 2*b6 + 2*b5 - 2*b4 - 2*b3 - 2*b2 + 2*b1 + 2) * q^49 + (-b7 - 3*b5 - b4 + b2 - 3*b1) * q^50 + (4*b7 + 2*b6 - 4*b5 - 2) * q^51 + (-3*b7 - 2*b6 - 3*b4 + 2*b3 + b2 + 2*b1 + 2) * q^52 + (-3*b7 + 3*b6 - 2*b4 + 2*b2 - 4*b1 - 2) * q^53 + (5*b7 + 3*b6 + b5 + b4 + 3*b3 - b2 + b1) * q^54 + (-b7 + b4 + b3 + b2 + b1 - 2) * q^55 + (6*b5 + 2*b4 - 2*b3 + 2*b2 + 4*b1 + 2) * q^56 + (-3*b7 - 2*b5 - 2*b3 + 2*b2 + 1) * q^57 + (b7 - 6*b6 + b5 + b4 + 2*b3 - 2*b2 + b1) * q^58 + (3*b7 + 3*b6 + 2*b5 + b4 - 2*b3 + b2 - 3) * q^59 + (-b7 - 3*b6 + b5 - b4 - b3 + b2 + 3*b1) * q^60 + (-b7 - b5 - 2*b3 - 4*b2 - 2*b1 + 4) * q^61 + (2*b6 + 2*b5 + 2*b3 - 4*b2 + 2*b1) * q^62 + (5*b6 + 5*b5 - b4 + b3 + b2 + b1 - 5) * q^63 + (2*b7 + 4*b6 + 2*b5 - 2*b4 + 4*b3 + 4*b2 - 2*b1 - 4) * q^64 + (-b6 - b5 + 2*b4 - 2*b3 - 2*b2 - 2*b1) * q^65 + (-b6 + b5 + 3*b3 + 2*b2 + b1 - 4) * q^66 + (-3*b7 - 6*b6 - 3*b5 - 3*b3 - 3*b1 - 6) * q^67 + (6*b7 + 2*b6 - 4*b5 + 2*b4 + 2*b3 - 4*b1) * q^68 + (-b7 - b6 - 5*b5 + 2*b4 - 4*b3 + 2*b2 + 3) * q^69 + (-b7 - 5*b6 - b5 - b4 - b3 - 3*b2 - b1) * q^70 + (3*b7 - 2*b5 - 3*b4 + 3*b3 - 3*b2 + 3*b1) * q^71 + (-b6 - 5*b5 - 3*b3 + 6*b2 - b1 - 2) * q^72 + (-3*b7 + 2*b6 + 2*b3 + 2*b2 - 5) * q^73 + (-7*b7 - b6 - 4*b5 + b4 - 5*b3 + 2*b2) * q^74 + (-4*b7 + 4*b6 - b5 - 3*b4 + 3*b2 - 2*b1 - 4) * q^75 + (b6 + 5*b5 - b3 + b1 - 2) * q^76 + (b7 - b6 - b5 - 4*b4 + 2*b3 - 2*b1 + 1) * q^77 + (b6 - 3*b5 + 2*b4 - b3 - 2*b2 - 5*b1 + 2) * q^78 + (6*b7 - 2*b6 + 2*b5 + 4*b4 + 4*b3 + 2*b2 - 2*b1 - 2) * q^79 + (-4*b7 + 2*b5 - 2*b4 - 2*b3 - 2*b2 + 2) * q^80 + (-b7 - 3*b6 + 3*b5 + 2*b2 - 2*b1 - 2) * q^81 + (b7 + 6*b6 + 2*b5 - 3*b4 + 2*b3 + b2 - 2*b1 - 4) * q^82 + (-3*b7 - 4*b6 + 3*b5 + 6*b4 - 3*b3 + 3*b1 + 4) * q^83 + (2*b7 - 6*b6 - 6*b5 + 4*b4 - 4*b3 - 2*b2 + 2) * q^84 + (-2*b7 + 2*b6 + 4*b5 - 2*b4 + 2*b2 + 4*b1 + 2) * q^85 + (-3*b7 + 4*b6 + 3*b5 - 3*b4 + 2*b3 + 2*b2 + b1) * q^86 + (7*b7 - 2*b6 + 3*b4 - 3*b3 - 3*b2 + 3*b1 + 10) * q^87 + (-b7 + b6 - 3*b5 - b4 - b3 - 3*b2 + 3*b1 + 4) * q^88 + (3*b7 - 2*b5 - 2*b4 + 4*b3 - 4*b2 + 2*b1 + 1) * q^89 + (-2*b7 - b6 + 2*b4 - b3 - b2 + 4) * q^90 + (3*b7 + 3*b6 - 5*b5 + 6*b3 + 5) * q^91 + (b7 + 3*b6 + 3*b5 - b4 + 3*b3 - b2 - b1 - 2) * q^92 + (6*b7 - 2*b6 + 6*b5 + 4*b3 - 4*b2 + 4*b1 + 2) * q^93 + (2*b7 + 2*b6 + 2*b4 - 2*b3 + 2*b2 - 4) * q^94 + (-b6 - b5 - b4 + b3 - b2 - b1 - 1) * q^95 + (6*b5 - 4*b4 - 6*b2 - 2*b1) * q^96 + (4*b6 + 4*b5 - 2*b4 + 2*b3 + 4) * q^97 + (5*b7 + 4*b6 - 4*b5 + b4 + 2*b2 - 4) * q^98 + (b6 + b3 - 6*b2 + b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10})$$ 8 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 8 * q^6 - 8 * q^7 - 4 * q^8 $$8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8} + 4 q^{11} + 12 q^{12} - 8 q^{13} + 12 q^{14} + 20 q^{18} + 4 q^{19} + 4 q^{20} + 4 q^{22} - 8 q^{23} - 8 q^{24} - 8 q^{25} - 20 q^{26} + 8 q^{27} - 16 q^{28} - 12 q^{30} + 32 q^{31} - 24 q^{32} - 16 q^{33} + 16 q^{35} - 40 q^{36} - 8 q^{37} + 8 q^{38} + 16 q^{39} + 16 q^{40} + 8 q^{41} + 8 q^{42} - 12 q^{43} + 20 q^{44} + 12 q^{46} + 48 q^{48} + 16 q^{50} - 16 q^{51} + 12 q^{52} + 8 q^{53} - 8 q^{54} - 16 q^{55} + 8 q^{56} + 16 q^{57} - 12 q^{58} - 20 q^{59} - 8 q^{60} + 24 q^{61} - 24 q^{62} - 40 q^{63} - 8 q^{64} - 28 q^{66} - 36 q^{67} + 16 q^{68} + 32 q^{69} - 8 q^{70} - 24 q^{71} + 12 q^{72} - 32 q^{73} + 8 q^{74} - 12 q^{75} - 20 q^{76} + 16 q^{77} + 28 q^{78} + 8 q^{80} - 20 q^{82} + 20 q^{83} + 8 q^{84} + 8 q^{85} + 4 q^{86} + 56 q^{87} + 8 q^{88} - 16 q^{89} + 28 q^{90} + 40 q^{91} - 16 q^{92} - 16 q^{93} - 24 q^{94} - 8 q^{95} - 16 q^{96} + 32 q^{97} - 24 q^{98} + 28 q^{99}+O(q^{100})$$ 8 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 8 * q^6 - 8 * q^7 - 4 * q^8 + 4 * q^11 + 12 * q^12 - 8 * q^13 + 12 * q^14 + 20 * q^18 + 4 * q^19 + 4 * q^20 + 4 * q^22 - 8 * q^23 - 8 * q^24 - 8 * q^25 - 20 * q^26 + 8 * q^27 - 16 * q^28 - 12 * q^30 + 32 * q^31 - 24 * q^32 - 16 * q^33 + 16 * q^35 - 40 * q^36 - 8 * q^37 + 8 * q^38 + 16 * q^39 + 16 * q^40 + 8 * q^41 + 8 * q^42 - 12 * q^43 + 20 * q^44 + 12 * q^46 + 48 * q^48 + 16 * q^50 - 16 * q^51 + 12 * q^52 + 8 * q^53 - 8 * q^54 - 16 * q^55 + 8 * q^56 + 16 * q^57 - 12 * q^58 - 20 * q^59 - 8 * q^60 + 24 * q^61 - 24 * q^62 - 40 * q^63 - 8 * q^64 - 28 * q^66 - 36 * q^67 + 16 * q^68 + 32 * q^69 - 8 * q^70 - 24 * q^71 + 12 * q^72 - 32 * q^73 + 8 * q^74 - 12 * q^75 - 20 * q^76 + 16 * q^77 + 28 * q^78 + 8 * q^80 - 20 * q^82 + 20 * q^83 + 8 * q^84 + 8 * q^85 + 4 * q^86 + 56 * q^87 + 8 * q^88 - 16 * q^89 + 28 * q^90 + 40 * q^91 - 16 * q^92 - 16 * q^93 - 24 * q^94 - 8 * q^95 - 16 * q^96 + 32 * q^97 - 24 * q^98 + 28 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5$$ 2*v^7 - 7*v^6 + 24*v^5 - 42*v^4 + 59*v^3 - 48*v^2 + 24*v - 5 $$\beta_{2}$$ $$=$$ $$-2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8$$ -2*v^7 + 7*v^6 - 24*v^5 + 43*v^4 - 61*v^3 + 54*v^2 - 29*v + 8 $$\beta_{3}$$ $$=$$ $$-3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11$$ -3*v^7 + 10*v^6 - 35*v^5 + 60*v^4 - 87*v^3 + 73*v^2 - 42*v + 11 $$\beta_{4}$$ $$=$$ $$-3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13$$ -3*v^7 + 11*v^6 - 38*v^5 + 70*v^4 - 102*v^3 + 91*v^2 - 53*v + 13 $$\beta_{5}$$ $$=$$ $$5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19$$ 5*v^7 - 17*v^6 + 60*v^5 - 105*v^4 + 155*v^3 - 133*v^2 + 77*v - 19 $$\beta_{6}$$ $$=$$ $$-5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23$$ -5*v^7 + 18*v^6 - 63*v^5 + 115*v^4 - 170*v^3 + 152*v^2 - 89*v + 23 $$\beta_{7}$$ $$=$$ $$-8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31$$ -8*v^7 + 28*v^6 - 98*v^5 + 175*v^4 - 256*v^3 + 223*v^2 - 126*v + 31
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (-b7 + b6 - b5 - b4 - b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2$$ (-b7 + 3*b6 + b5 - 3*b4 + b3 + b2 - b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2$$ (5*b7 - b6 + 7*b5 - b4 + 5*b3 - 3*b2 + 3*b1 - 2) / 2 $$\nu^{4}$$ $$=$$ $$( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2$$ (11*b7 - 15*b6 + 3*b5 + 11*b4 - b3 - 5*b2 + 9*b1 + 14) / 2 $$\nu^{5}$$ $$=$$ $$( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2$$ (-13*b7 - 11*b6 - 29*b5 + 17*b4 - 23*b3 + 13*b2 - 3*b1 + 18) / 2 $$\nu^{6}$$ $$=$$ $$( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2$$ (-67*b7 + 59*b6 - 41*b5 - 29*b4 - 15*b3 + 37*b2 - 47*b1 - 48) / 2 $$\nu^{7}$$ $$=$$ $$( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2$$ (-7*b7 + 113*b6 + 97*b5 - 105*b4 + 91*b3 - 31*b2 - 39*b1 - 122) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 + 2.10607i 0.5 − 0.691860i 0.5 − 2.10607i 0.5 + 0.691860i 0.5 + 0.0297061i 0.5 − 1.44392i 0.5 − 0.0297061i 0.5 + 1.44392i
−1.26330 0.635665i −1.07947 2.60607i 1.19186 + 1.60607i 0.707107 + 0.292893i −0.292893 + 3.97844i 1.68554 + 1.68554i −0.484753 2.78658i −3.50504 + 3.50504i −0.707107 0.819496i
5.2 −0.443806 + 1.34277i 0.0794708 + 0.191860i −1.60607 1.19186i 0.707107 + 0.292893i −0.292893 + 0.0215628i −2.27133 2.27133i 2.31318 1.62764i 2.09083 2.09083i −0.707107 + 0.819496i
13.1 −1.26330 + 0.635665i −1.07947 + 2.60607i 1.19186 1.60607i 0.707107 0.292893i −0.292893 3.97844i 1.68554 1.68554i −0.484753 + 2.78658i −3.50504 3.50504i −0.707107 + 0.819496i
13.2 −0.443806 1.34277i 0.0794708 0.191860i −1.60607 + 1.19186i 0.707107 0.292893i −0.292893 0.0215628i −2.27133 + 2.27133i 2.31318 + 1.62764i 2.09083 + 2.09083i −0.707107 0.819496i
21.1 −1.40426 + 0.167452i 1.27882 0.529706i 1.94392 0.470294i −0.707107 + 1.70711i −1.70711 + 0.957989i −2.74912 2.74912i −2.65103 + 0.985930i −0.766519 + 0.766519i 0.707107 2.51564i
21.2 1.11137 0.874559i −2.27882 + 0.943920i 0.470294 1.94392i −0.707107 + 1.70711i −1.70711 + 3.04201i −0.665096 0.665096i −1.17740 2.57172i 2.18073 2.18073i 0.707107 + 2.51564i
29.1 −1.40426 0.167452i 1.27882 + 0.529706i 1.94392 + 0.470294i −0.707107 1.70711i −1.70711 0.957989i −2.74912 + 2.74912i −2.65103 0.985930i −0.766519 0.766519i 0.707107 + 2.51564i
29.2 1.11137 + 0.874559i −2.27882 0.943920i 0.470294 + 1.94392i −0.707107 1.70711i −1.70711 3.04201i −0.665096 + 0.665096i −1.17740 + 2.57172i 2.18073 + 2.18073i 0.707107 2.51564i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.2.g.b 8
3.b odd 2 1 288.2.v.b 8
4.b odd 2 1 128.2.g.b 8
5.b even 2 1 800.2.y.b 8
5.c odd 4 1 800.2.ba.c 8
5.c odd 4 1 800.2.ba.d 8
8.b even 2 1 256.2.g.d 8
8.d odd 2 1 256.2.g.c 8
12.b even 2 1 1152.2.v.b 8
16.e even 4 1 512.2.g.e 8
16.e even 4 1 512.2.g.h 8
16.f odd 4 1 512.2.g.f 8
16.f odd 4 1 512.2.g.g 8
32.g even 8 1 inner 32.2.g.b 8
32.g even 8 1 256.2.g.d 8
32.g even 8 1 512.2.g.e 8
32.g even 8 1 512.2.g.h 8
32.h odd 8 1 128.2.g.b 8
32.h odd 8 1 256.2.g.c 8
32.h odd 8 1 512.2.g.f 8
32.h odd 8 1 512.2.g.g 8
64.i even 16 2 4096.2.a.k 8
64.j odd 16 2 4096.2.a.q 8
96.o even 8 1 1152.2.v.b 8
96.p odd 8 1 288.2.v.b 8
160.v odd 8 1 800.2.ba.c 8
160.z even 8 1 800.2.y.b 8
160.bb odd 8 1 800.2.ba.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 1.a even 1 1 trivial
32.2.g.b 8 32.g even 8 1 inner
128.2.g.b 8 4.b odd 2 1
128.2.g.b 8 32.h odd 8 1
256.2.g.c 8 8.d odd 2 1
256.2.g.c 8 32.h odd 8 1
256.2.g.d 8 8.b even 2 1
256.2.g.d 8 32.g even 8 1
288.2.v.b 8 3.b odd 2 1
288.2.v.b 8 96.p odd 8 1
512.2.g.e 8 16.e even 4 1
512.2.g.e 8 32.g even 8 1
512.2.g.f 8 16.f odd 4 1
512.2.g.f 8 32.h odd 8 1
512.2.g.g 8 16.f odd 4 1
512.2.g.g 8 32.h odd 8 1
512.2.g.h 8 16.e even 4 1
512.2.g.h 8 32.g even 8 1
800.2.y.b 8 5.b even 2 1
800.2.y.b 8 160.z even 8 1
800.2.ba.c 8 5.c odd 4 1
800.2.ba.c 8 160.v odd 8 1
800.2.ba.d 8 5.c odd 4 1
800.2.ba.d 8 160.bb odd 8 1
1152.2.v.b 8 12.b even 2 1
1152.2.v.b 8 96.o even 8 1
4096.2.a.k 8 64.i even 16 2
4096.2.a.q 8 64.j odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} - 32T_{3}^{4} - 24T_{3}^{3} + 96T_{3}^{2} - 16T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{7} + 6 T^{6} + 4 T^{5} + \cdots + 16$$
$3$ $$T^{8} + 4 T^{7} + 8 T^{6} - 32 T^{4} + \cdots + 4$$
$5$ $$(T^{4} + 2 T^{2} - 4 T + 2)^{2}$$
$7$ $$T^{8} + 8 T^{7} + 32 T^{6} + 48 T^{5} + \cdots + 784$$
$11$ $$T^{8} - 4 T^{7} + 8 T^{6} - 64 T^{5} + \cdots + 4$$
$13$ $$T^{8} + 8 T^{7} + 36 T^{6} + \cdots + 6724$$
$17$ $$T^{8} + 64 T^{6} + 1056 T^{4} + \cdots + 256$$
$19$ $$T^{8} - 4 T^{7} - 8 T^{6} + 48 T^{5} + \cdots + 196$$
$23$ $$T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 16$$
$29$ $$T^{8} - 12 T^{6} + 168 T^{5} + \cdots + 188356$$
$31$ $$(T^{2} - 8 T + 8)^{4}$$
$37$ $$T^{8} + 8 T^{7} - 44 T^{6} + \cdots + 64516$$
$41$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 26896$$
$43$ $$T^{8} + 12 T^{7} + 56 T^{6} + \cdots + 31684$$
$47$ $$T^{8} + 64 T^{6} + 544 T^{4} + \cdots + 256$$
$53$ $$T^{8} - 8 T^{7} + 100 T^{6} + \cdots + 158404$$
$59$ $$T^{8} + 20 T^{7} + 136 T^{6} + \cdots + 643204$$
$61$ $$T^{8} - 24 T^{7} + 132 T^{6} + \cdots + 42436$$
$67$ $$T^{8} + 36 T^{7} + 504 T^{6} + \cdots + 1285956$$
$71$ $$T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 21196816$$
$73$ $$T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 38416$$
$79$ $$T^{8} + 512 T^{6} + \cdots + 99361024$$
$83$ $$T^{8} - 20 T^{7} + \cdots + 138250564$$
$89$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 17007376$$
$97$ $$(T^{4} - 16 T^{3} + 40 T^{2} + 288 T - 992)^{2}$$