# Properties

 Label 31.2.g.a Level $31$ Weight $2$ Character orbit 31.g Analytic conductor $0.248$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 31.g (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.247536246266$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{15})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1$$ x^16 + 19*x^14 + 140*x^12 + 511*x^10 + 979*x^8 + 956*x^6 + 410*x^4 + 44*x^2 + 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 basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{2}+ \cdots + (\beta_{13} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{9}+O(q^{10})$$ q + (b15 - b14 + b13 - b10 - b8 - b7 + b6 + b5 - b4 + b3 - b1 - 1) * q^2 + (-b15 + b14 + b11 - b8 - b7 + b6 - b4 + b3 - 1) * q^3 + (b12 - b11 + b10 + 3*b8 + b7 - 2*b6 - b5 - b3) * q^4 + (b14 - 2*b13 - b11 + b9 + b8 + b7 - b6 + b4 - b3 + 1) * q^5 + (-b14 - 2*b12 - b11 + b10 - b9 - b8 + 2*b7 - b5 + 5*b4 - 2*b3 + 2*b1 - 1) * q^6 + (-b14 - b9 + b8 + b7 - b6 - b4 + 1) * q^7 + (-b15 + b14 - b13 + b12 + 3*b11 - b8 - b7 - 2*b4 - b3 + b1 + 3) * q^8 + (b13 - b8 + b7 - b6 + b4 - b3 + b1 - 1) * q^9 $$q + (\beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{2}+ \cdots + (2 \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + 2 \beta_{10} - \beta_{9} + 4 \beta_{8} + \beta_{7} + \cdots + 1) q^{99}+O(q^{100})$$ q + (b15 - b14 + b13 - b10 - b8 - b7 + b6 + b5 - b4 + b3 - b1 - 1) * q^2 + (-b15 + b14 + b11 - b8 - b7 + b6 - b4 + b3 - 1) * q^3 + (b12 - b11 + b10 + 3*b8 + b7 - 2*b6 - b5 - b3) * q^4 + (b14 - 2*b13 - b11 + b9 + b8 + b7 - b6 + b4 - b3 + 1) * q^5 + (-b14 - 2*b12 - b11 + b10 - b9 - b8 + 2*b7 - b5 + 5*b4 - 2*b3 + 2*b1 - 1) * q^6 + (-b14 - b9 + b8 + b7 - b6 - b4 + 1) * q^7 + (-b15 + b14 - b13 + b12 + 3*b11 - b8 - b7 - 2*b4 - b3 + b1 + 3) * q^8 + (b13 - b8 + b7 - b6 + b4 - b3 + b1 - 1) * q^9 + (-b15 + 2*b14 - 3*b12 + 2*b11 + b10 + b9 - 2*b8 - b7 + 3*b6 - b5 + 3*b4 + 2*b3 + 2*b1 - 3) * q^10 + (-b15 + b14 + 4*b12 + b11 - b10 + b9 - 2*b7 - 2*b4 + b3 - b1) * q^11 + (3*b15 - b14 - b13 - 2*b12 - 2*b11 + b10 + b8 + b7 + b6 + b5 - b4 + 2*b3 - b2 - 3*b1 + 1) * q^12 + (b15 - 2*b12 - 2*b11 + b10 - b8 + b6 + b4 + b3 + b2 + b1 - 2) * q^13 + (-b15 - b14 + b13 + 2*b12 - 2*b11 - 3*b10 + 2*b8 - 2*b7 - b6 + b5 - 5*b4 - b3 - b1 + 1) * q^14 + (-2*b14 + 2*b13 + 4*b12 - 2*b10 + 4*b8 - 2*b7 - 2*b6 - 3*b4 + 2*b2 - 5*b1 + 2) * q^15 + (b15 - 3*b14 + 3*b13 + 2*b12 - 3*b10 - b9 - 3*b8 + b6 + 3*b5 + b4 + b2 - 3*b1 - 1) * q^16 + (2*b15 + b13 - 4*b12 - b11 + b10 - 2*b9 - b8 + b6 + b4 + b3 - 2*b2 + 2*b1 - 4) * q^17 + (-b13 + 2*b12 + 3*b8 + b5 - 4*b4 + b3 - 2*b2 - 3*b1 + 3) * q^18 + (-b14 - b13 - b12 - b11 + b10 + 2*b8 + b7 - b6 + b5 - 2*b4 - b3 - 3*b2 - b1 + 4) * q^19 + (b14 + b13 - b12 - 4*b8 - b6 + b3 + 3*b1) * q^20 + (2*b14 - b13 - 2*b12 + 3*b11 + 2*b10 + 2*b9 - 4*b8 - b7 + 5*b6 - b5 + 4*b4 + b3 + b2 + 3*b1 - 1) * q^21 + (-2*b15 + 4*b14 - 2*b13 - b12 + b11 + b10 + b9 - b8 - b7 - 3*b5 + 2*b4 + b3 + 2*b1 - 2) * q^22 + (-b14 + 2*b11 - 2*b8 + b7 + 2*b6 - b5 + 5*b4 - b3 + 4*b2 + 3*b1 - 1) * q^23 + (-4*b15 + 5*b14 - 2*b13 + b12 + 4*b11 + 3*b9 - b7 - 3*b6 - 4*b5 + 4*b2 + 3*b1 - 1) * q^24 + (b15 - 3*b14 + b13 + b12 + 3*b11 - 2*b9 - 2*b8 + 2*b7 + 3*b6 + 2*b4 - 3*b3 + b2 + 1) * q^25 + (-b15 + b14 - 3*b13 + 2*b11 + 2*b10 + b9 + b8 + 2*b7 - 3*b6 + b5 - b3 + b1 + 5) * q^26 + (-b14 - b12 - 2*b11 - 2*b10 + 2*b8 - b7 - 2*b6 + 2*b5 - b4 - 3*b2) * q^27 + (b15 - b14 + b12 - 5*b11 + b10 - b9 + 3*b8 + 2*b7 - 2*b6 + 2*b4 - 2*b3 - 2*b2 - 2*b1 - 2) * q^28 + (b15 - 3*b14 + b13 - 3*b12 - 4*b11 + b10 - 2*b9 + b8 + 3*b7 + b6 + b5 + 2*b4 - b3 - 2*b2 - b1 - 1) * q^29 + (b15 - 2*b14 + 2*b12 - 6*b11 + b10 - 4*b9 + 5*b8 + 3*b7 - 5*b6 + b5 - 2*b4 - 3*b3 - 4*b2 + b1) * q^30 + (-b15 + 2*b14 - b9 + 2*b8 - 4*b6 - 3*b5 - 3*b3 - b1) * q^31 + (b15 + 2*b12 - 4*b11 + b10 + 8*b8 + b7 + 2*b6 - b5 - 2*b4 + b3 - 3*b1 - 1) * q^32 + (b15 - 2*b14 + b13 - 2*b12 - 2*b11 - b10 + b8 + 3*b7 - 4*b6 - 2*b3) * q^33 + (-b15 + b14 - 2*b13 + 3*b12 + 6*b11 - b10 + 3*b9 - 4*b8 - 2*b7 - 2*b4 + 5*b2 - 2*b1 + 2) * q^34 + (2*b14 - b13 - 4*b12 + 2*b11 + b10 + b9 - 4*b8 - b7 + 3*b4 + 3*b3 - 2*b2 - 4) * q^35 + (-2*b15 + b14 + b13 - b12 + 2*b11 + b10 - b9 - 3*b8 - 2*b7 + 6*b6 - b5 + 3*b4 + b3 + 4*b2 + 5*b1 - 3) * q^36 + (-b15 + 5*b14 - 2*b13 + b12 + b11 - b10 + 4*b9 - 2*b8 - 5*b7 + b6 + b5 - 6*b4 + 6*b3 - 1) * q^37 + (-2*b14 + 3*b13 + 3*b12 + 2*b11 - 2*b10 + b9 - b8 - 2*b7 + 3*b6 + 2*b4 + 2*b2 - b1 - 1) * q^38 + (b14 + b13 + 2*b12 - b8 - b7 + 3*b6 + b5 - 3*b4 + 2*b3 - 2*b2 - b1) * q^39 + (2*b15 - 4*b14 - b13 - b11 - b10 - b8 - 2*b7 + 4*b6 + 4*b5 - b4 + 2*b3 - 2*b2 + 1) * q^40 + (-b15 - b14 + 3*b12 + b11 - 2*b10 + b9 + 4*b8 - b7 - 2*b6 - b5 - b4 - b3 + b2 - 2*b1 + 1) * q^41 + (3*b15 - b14 + 2*b13 - 3*b12 + 5*b11 + b10 - 2*b9 - 6*b8 + 2*b7 + b6 + 6*b4 + 4*b2 + 2*b1 + 1) * q^42 + (-3*b15 + 4*b14 + 2*b13 + b11 - 2*b10 + 2*b9 - b8 - 2*b7 - 2*b6 - 2*b5 - 5*b4 + b3 - 2*b2) * q^43 + (b15 - b14 + b13 - 4*b12 + 3*b11 + b10 - b9 - 5*b8 + 3*b6 + 2*b4 + 1) * q^44 + (b15 - b14 + b13 - 3*b11 + 2*b10 - 2*b9 - 2*b8 + 3*b6 + b5 - 3*b4 + 2*b3 + 4) * q^45 + (3*b15 - b14 + b13 - 2*b12 - 6*b11 - b10 - 3*b9 + 3*b8 + 2*b7 - 3*b6 + 3*b5 - 6*b4 - 3*b2 - 2*b1 + 3) * q^46 + (b15 - 4*b13 - b12 - b11 + 4*b10 - b9 + 4*b8 + 3*b7 - 2*b6 - b5 + 2*b4 - 2*b3 - 2*b2 + 4) * q^47 + (-b15 - b14 - 6*b12 + b11 - 2*b10 - 5*b8 - 2*b6 + b5 + 4*b4 - 2*b3 - 4*b2 + 3*b1 - 1) * q^48 + (-b15 + 4*b12 + 2*b11 - 2*b10 + b8 - b7 + b5 + 2*b4 - b3 + b2 + 2*b1) * q^49 + (2*b14 + 3*b13 + 2*b12 - 3*b11 - b10 + b9 + 4*b8 - b7 - 2*b6 - 3*b5 - 4*b4 + 2*b3 - 5*b1 - 2) * q^50 + (-b15 - 2*b14 - 3*b13 + 2*b12 - b11 + 2*b9 + 4*b8 - b4 - b3 - b2 - 2*b1 + 2) * q^51 + (-b15 - 5*b14 + 3*b13 + 4*b12 + b11 - 6*b10 - b9 - 2*b7 - b6 + 2*b5 - b4 - 2*b3 - 4*b1 + 2) * q^52 + (b15 - 2*b14 + b13 + 5*b12 - 2*b11 + 3*b8 + b7 - 5*b6 + 2*b5 - 2*b4 - b3 + 4*b2 - 4*b1 + 4) * q^53 + (3*b15 - 2*b14 + b13 + 2*b12 - 4*b11 + 4*b6 + b5 + 2*b4 + b2 - b1 - 4) * q^54 + (5*b15 - 5*b14 + b13 - 4*b12 - b11 + b10 - 4*b9 - b8 + 2*b7 + 7*b6 + 5*b5 + 5*b4 + 5*b2 + 3*b1 - 1) * q^55 + (-3*b15 + 6*b14 - 3*b13 - 4*b12 - 2*b11 + 3*b10 + 3*b9 + 3*b8 - 4*b6 - 3*b5 - 5*b4 + 3*b3 - 3*b2 + b1 - 1) * q^56 + (-b15 + b14 + 2*b13 + 3*b12 + 6*b11 - b10 - b9 - 5*b8 - 3*b7 + 5*b6 - 2*b5 + 4*b4 + b3 + 8*b2 + 2*b1 - 5) * q^57 + (4*b14 - 3*b13 - 2*b12 - 2*b11 + 5*b10 + 3*b9 + 6*b8 + 2*b7 - b6 - 4*b5 - b4 + b3 - 2*b2 + 2*b1 + 1) * q^58 + (-b15 + b14 + b13 + b12 - b11 + b10 - 3*b9 - 3*b8 + b7 + 2*b6 - b5 + b4 + 2*b2 + 3*b1 - 2) * q^59 + (-5*b15 + 8*b14 - 5*b13 + 7*b11 - 2*b10 + 7*b9 - 5*b7 - 2*b6 - 3*b5 - 4*b4 + 2*b3 - 5*b2 - 2*b1 - 3) * q^60 + (-2*b15 + 4*b14 + b12 + 4*b11 - 2*b10 + 5*b9 - 5*b8 - 3*b7 - b6 - 2*b5 - b4 + 3*b3 + b1 - 6) * q^61 + (b15 - 2*b14 - 5*b12 + b11 + b9 - 8*b8 + 2*b6 + 3*b5 + 2*b4 + 3*b3 - 5*b2 - 4*b1 - 3) * q^62 + (b12 - 2*b11 - b9 + b7 - 4*b6 + b5 - b4 - 2*b3 - b2 - 2) * q^63 + (-2*b15 + 4*b14 - 2*b13 - 3*b12 + 3*b11 + 6*b10 - 4*b9 + b8 + b6 - 3*b5 + 4*b4 - 2*b3 - b2 + 9*b1 - 1) * q^64 + (-b13 - 3*b12 - b11 + b10 - 3*b9 + 5*b8 + 4*b7 - 3*b6 - 3*b5 + 4*b4 - 5*b3 - b2 + b1 - 1) * q^65 + (-3*b14 + b13 + 6*b12 - 5*b11 - b10 - b9 + 10*b8 - 3*b6 + 4*b5 - 11*b4 - b3 - 3*b2 - 4*b1 + 6) * q^66 + (3*b15 - 6*b14 + 4*b13 - b12 - 2*b11 - b9 - 2*b8 + 2*b7 + 3*b6 + 3*b5 + b4 + b3 + 2*b2 - b1 + 1) * q^67 + (b15 - 6*b14 + 4*b13 + 3*b12 - b11 - 2*b10 - 4*b9 - 2*b8 + 3*b7 + b6 + 2*b5 + 7*b4 - 4*b3 + 3*b2 + 2*b1) * q^68 + (4*b14 - 3*b13 - 3*b11 + b10 + b9 + 2*b8 + b7 - 4*b6 - 5*b4 - 5*b2 - 3*b1 + 5) * q^69 + (2*b15 - 4*b14 + b13 - 2*b12 + 4*b11 - 6*b8 + 4*b7 + 3*b6 - 2*b5 + 9*b4 - b3 + 8*b2 + 4*b1 - 1) * q^70 + (-b15 + 2*b14 - 2*b12 + 3*b11 - b9 + b6 - 3*b5 - b4 + b1 - 2) * q^71 + (3*b15 - 3*b14 - b13 - 3*b12 - 3*b11 - 3*b9 + 2*b8 + 4*b7 - b6 + 2*b5 + 4*b4 - 4*b3 - 2*b2 + 3) * q^72 + (-3*b15 - 2*b14 - 5*b13 + 6*b12 - 3*b11 + 11*b8 - 6*b6 - 3*b4 - 5*b3 - 3*b2 - 3*b1 + 8) * q^73 + (-3*b15 - 2*b14 + b12 + 6*b11 - 2*b9 - 5*b8 + b6 + 11*b4 - 5*b3 + 7*b2 + 8*b1 - 1) * q^74 + (-3*b15 + 5*b14 - 2*b13 + 2*b12 + b11 - 4*b10 + 5*b9 - 3*b8 - b7 - 5*b6 - 2*b5 + b4 - b3 + 4*b2 + b1) * q^75 + (3*b15 + b14 - 2*b13 - 4*b12 - 7*b11 + 4*b10 - 2*b9 + 6*b8 + 6*b7 - 4*b6 - b5 + b4 - 2*b3 - 3*b2 + 2*b1) * q^76 + (-4*b15 + 4*b14 - 4*b13 - 5*b12 - 2*b11 + 4*b10 + 4*b9 - b8 + 2*b7 - b6 - 2*b5 + 5*b4 - 6*b2 + 3*b1 + 1) * q^77 + (-3*b15 + 6*b14 - 2*b13 + 6*b11 + 2*b10 + 3*b9 - b8 - 4*b7 + 2*b6 - 6*b5 + 2*b4 + 2*b3 + 3*b2 + 3*b1 - 4) * q^78 + (-4*b15 + 4*b14 - 4*b13 + 4*b12 + b11 + 2*b10 + 8*b9 + 2*b8 - 6*b6 - 4*b5 - b4 + 2*b3 - 2*b2 - 2*b1 + 5) * q^79 + (2*b15 + b14 + b13 + 2*b10 + b9 - b8 + 3*b6 - 2*b5 + b3 + 4*b2 - 3*b1 + 2) * q^80 + (3*b15 - b14 - 4*b12 + 5*b11 - b10 - 2*b9 - 7*b8 - b7 + 4*b6 + 2*b4 + 2*b3 + 3*b2 + b1 - 2) * q^81 + (b15 + b13 - b12 - 7*b11 + b10 - 4*b9 + 3*b8 + 2*b7 - 3*b6 - b4 - 3*b3 - 3*b2 + 3*b1 - 1) * q^82 + (b15 + 3*b14 - b13 - 3*b12 + b11 + 4*b10 - 2*b9 + b8 + 3*b7 + 8*b6 + 2*b4 + 2*b3 + 3*b2 + 7*b1 - 1) * q^83 + (4*b15 - 8*b14 + 4*b13 + 4*b12 - 2*b11 - 6*b10 - b9 + 2*b8 - 2*b7 + 2*b6 + 7*b5 - 9*b4 + 2*b3 + b2 - 9*b1 + 5) * q^84 + (-3*b15 + 5*b14 + 2*b13 - b12 - b11 + 3*b8 - 3*b7 - 2*b6 + 2*b5 - 9*b4 + 6*b3 - 9*b2 - 4*b1 + 1) * q^85 + (4*b15 - 6*b14 + 3*b13 - 2*b12 - 3*b11 + b10 - 3*b9 - 2*b8 - b7 + 3*b6 + 4*b5 - 4*b2 - b1 - 3) * q^86 + (2*b15 - 2*b14 + 4*b13 + 4*b12 + 4*b11 - 4*b10 - b9 - 8*b8 - 4*b7 + 7*b6 + 4*b5 + 2*b3 + 3*b2 - 2) * q^87 + (2*b15 - 3*b14 + 5*b13 + 4*b12 + b11 - 3*b10 - b9 - b8 - 2*b7 - b6 - b5 - 2*b4 + b3 - 6*b1 - 4) * q^88 + (-4*b14 + 5*b13 - 2*b12 - 3*b10 - 5*b9 - 2*b8 - 2*b7 + b6 + 4*b5 - 3*b4 + b3 - 4*b2 - 2*b1 - 3) * q^89 + (-b15 + b14 - b13 + 2*b12 + 5*b11 - 3*b10 + 6*b9 - 2*b8 - 6*b7 + 2*b5 - 6*b4 + 5*b3 - 5*b1 + 1) * q^90 + (6*b15 - 4*b14 + 6*b13 - b12 - 2*b11 - 3*b10 - 3*b9 - 5*b8 - 4*b7 + 5*b6 + 3*b5 - 3*b4 + 2*b3 + 3*b2 - b1 - 5) * q^91 + (-4*b15 + 2*b14 + 8*b12 + 4*b11 - 4*b10 + 6*b9 + b8 - 8*b7 + b6 - 8*b4 + 4*b3 + 2*b2 - 6*b1 - 1) * q^92 + (-3*b14 - 2*b13 - 4*b11 + b10 - b9 + 9*b8 - b7 + b6 + 6*b5 - 10*b4 + 3*b3 - 3*b2 - 4*b1 + 9) * q^93 + (-5*b15 + 4*b14 - b12 + 12*b11 - 5*b10 + 5*b9 - 10*b8 - 6*b7 + 4*b6 - 2*b5 + b4 + 3*b3 + 5*b2 + b1) * q^94 + (-3*b15 + 3*b14 - 3*b13 + b12 - b11 - 3*b10 + 6*b9 - b8 - 5*b7 - 3*b6 + b5 - 6*b4 + 5*b3 - 4*b2 - b1 - 1) * q^95 + (8*b15 - 8*b14 + 8*b13 - 2*b12 - 7*b11 - 4*b9 - 7*b8 + 4*b7 + 4*b6 + 4*b5 + 4*b4 + 4*b3 + 3*b2 - 2*b1 - 6) * q^96 + (-2*b14 + 3*b13 + 7*b12 + 2*b11 - 3*b10 - 3*b9 + b7 - 4*b5 + b4 - 5*b3 + 9*b2) * q^97 + (2*b15 + 2*b14 - b12 - 4*b11 - 2*b10 - 2*b9 - b8 - 2*b7 + 4*b6 - 2*b4 + 2*b3 - 2*b2 - 2*b1 - 5) * q^98 + (2*b15 - b14 - b13 - 3*b12 + 2*b10 - b9 + 4*b8 + b7 - b6 - 2*b5 + 2*b4 - 3*b3 - 2*b2 - 3*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9}+O(q^{10})$$ 16 * q - 6 * q^2 - 12 * q^3 - 14 * q^4 - 3 * q^5 + 11 * q^6 + 2 * q^7 + 17 * q^8 - 10 * q^9 $$16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9} - 2 q^{10} - 7 q^{11} + 5 q^{12} - 7 q^{13} - 6 q^{14} + 14 q^{15} - 2 q^{16} - 6 q^{17} - 3 q^{18} + 16 q^{19} + 37 q^{20} + 9 q^{21} + 9 q^{22} + q^{23} - 20 q^{24} - 13 q^{25} + 9 q^{26} + 9 q^{27} - 30 q^{28} - 14 q^{29} - 22 q^{30} + 15 q^{31} - 42 q^{32} - 13 q^{33} - 32 q^{34} - 9 q^{35} + q^{36} - 8 q^{37} + 8 q^{38} - 3 q^{39} - q^{40} - 8 q^{41} + 69 q^{42} + 23 q^{43} + 39 q^{44} + 65 q^{45} + 34 q^{46} + 14 q^{47} + 34 q^{48} + 2 q^{49} + 3 q^{50} - 42 q^{51} + 29 q^{52} + 6 q^{53} - 46 q^{54} - 7 q^{55} - 30 q^{56} - 17 q^{57} - 15 q^{58} + 4 q^{59} - 75 q^{60} - 60 q^{61} - 25 q^{62} - 46 q^{63} + 23 q^{64} - 12 q^{65} - 30 q^{66} + 13 q^{67} + 30 q^{68} + 38 q^{69} + 12 q^{70} - 14 q^{71} + 37 q^{72} + 2 q^{73} + 13 q^{74} + 13 q^{75} - 12 q^{76} + 18 q^{77} - 15 q^{78} + 18 q^{79} + 36 q^{80} + 23 q^{81} + 14 q^{82} - 16 q^{83} + 8 q^{84} + 37 q^{85} - 26 q^{86} + 15 q^{87} - 17 q^{88} + q^{89} - 23 q^{90} + 8 q^{91} - 64 q^{92} + 17 q^{93} + 44 q^{94} - 22 q^{95} + 8 q^{96} + 3 q^{97} - 10 q^{98} + 6 q^{99}+O(q^{100})$$ 16 * q - 6 * q^2 - 12 * q^3 - 14 * q^4 - 3 * q^5 + 11 * q^6 + 2 * q^7 + 17 * q^8 - 10 * q^9 - 2 * q^10 - 7 * q^11 + 5 * q^12 - 7 * q^13 - 6 * q^14 + 14 * q^15 - 2 * q^16 - 6 * q^17 - 3 * q^18 + 16 * q^19 + 37 * q^20 + 9 * q^21 + 9 * q^22 + q^23 - 20 * q^24 - 13 * q^25 + 9 * q^26 + 9 * q^27 - 30 * q^28 - 14 * q^29 - 22 * q^30 + 15 * q^31 - 42 * q^32 - 13 * q^33 - 32 * q^34 - 9 * q^35 + q^36 - 8 * q^37 + 8 * q^38 - 3 * q^39 - q^40 - 8 * q^41 + 69 * q^42 + 23 * q^43 + 39 * q^44 + 65 * q^45 + 34 * q^46 + 14 * q^47 + 34 * q^48 + 2 * q^49 + 3 * q^50 - 42 * q^51 + 29 * q^52 + 6 * q^53 - 46 * q^54 - 7 * q^55 - 30 * q^56 - 17 * q^57 - 15 * q^58 + 4 * q^59 - 75 * q^60 - 60 * q^61 - 25 * q^62 - 46 * q^63 + 23 * q^64 - 12 * q^65 - 30 * q^66 + 13 * q^67 + 30 * q^68 + 38 * q^69 + 12 * q^70 - 14 * q^71 + 37 * q^72 + 2 * q^73 + 13 * q^74 + 13 * q^75 - 12 * q^76 + 18 * q^77 - 15 * q^78 + 18 * q^79 + 36 * q^80 + 23 * q^81 + 14 * q^82 - 16 * q^83 + 8 * q^84 + 37 * q^85 - 26 * q^86 + 15 * q^87 - 17 * q^88 + q^89 - 23 * q^90 + 8 * q^91 - 64 * q^92 + 17 * q^93 + 44 * q^94 - 22 * q^95 + 8 * q^96 + 3 * q^97 - 10 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} - 641 \nu^{8} + 5467 \nu^{7} + 691 \nu^{6} + 6431 \nu^{5} + 3382 \nu^{4} + 2409 \nu^{3} + 2839 \nu^{2} + \cdots + 255 ) / 186$$ (2*v^15 - 4*v^14 + 48*v^13 - 65*v^12 + 458*v^11 - 358*v^10 + 2196*v^9 - 641*v^8 + 5467*v^7 + 691*v^6 + 6431*v^5 + 3382*v^4 + 2409*v^3 + 2839*v^2 - 360*v + 255) / 186 $$\beta_{2}$$ $$=$$ $$( 2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + 641 \nu^{8} + 5467 \nu^{7} - 691 \nu^{6} + 6431 \nu^{5} - 3382 \nu^{4} + 2409 \nu^{3} - 2839 \nu^{2} + \cdots - 255 ) / 186$$ (2*v^15 + 4*v^14 + 48*v^13 + 65*v^12 + 458*v^11 + 358*v^10 + 2196*v^9 + 641*v^8 + 5467*v^7 - 691*v^6 + 6431*v^5 - 3382*v^4 + 2409*v^3 - 2839*v^2 - 360*v - 255) / 186 $$\beta_{3}$$ $$=$$ $$( 6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} - 7260 \nu^{8} + 16835 \nu^{7} - 16144 \nu^{6} + 21308 \nu^{5} - 17926 \nu^{4} + \cdots - 463 ) / 186$$ (6*v^15 - 10*v^14 + 144*v^13 - 209*v^12 + 1374*v^11 - 1732*v^10 + 6619*v^9 - 7260*v^8 + 16835*v^7 - 16144*v^6 + 21308*v^5 - 17926*v^4 + 10699*v^3 - 7953*v^2 + 439*v - 463) / 186 $$\beta_{4}$$ $$=$$ $$( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + 1125 \nu + 93 ) / 186$$ (17*v^15 + 315*v^13 + 2250*v^11 + 7940*v^9 + 14865*v^7 + 14844*v^5 + 7255*v^3 + 1125*v + 93) / 186 $$\beta_{5}$$ $$=$$ $$( 6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + 7260 \nu^{8} + 16835 \nu^{7} + 16144 \nu^{6} + 21308 \nu^{5} + 17926 \nu^{4} + \cdots + 463 ) / 186$$ (6*v^15 + 10*v^14 + 144*v^13 + 209*v^12 + 1374*v^11 + 1732*v^10 + 6619*v^9 + 7260*v^8 + 16835*v^7 + 16144*v^6 + 21308*v^5 + 17926*v^4 + 10699*v^3 + 7953*v^2 + 439*v + 463) / 186 $$\beta_{6}$$ $$=$$ $$( - 6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + 12516 \nu^{8} - 16835 \nu^{7} + 21730 \nu^{6} - 21308 \nu^{5} + 18145 \nu^{4} + \cdots + 354 ) / 186$$ (-6*v^15 + 28*v^14 - 144*v^13 + 517*v^12 - 1374*v^11 + 3653*v^10 - 6619*v^9 + 12516*v^8 - 16835*v^7 + 21730*v^6 - 21308*v^5 + 18145*v^4 - 10699*v^3 + 6074*v^2 - 532*v + 354) / 186 $$\beta_{7}$$ $$=$$ $$( 8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + 10969 \nu^{8} - 1382 \nu^{7} + 14550 \nu^{6} - 6857 \nu^{5} + 6617 \nu^{4} - 6329 \nu^{3} + \cdots - 143 ) / 186$$ (8*v^15 + 33*v^14 + 130*v^13 + 575*v^12 + 716*v^11 + 3713*v^10 + 1282*v^9 + 10969*v^8 - 1382*v^7 + 14550*v^6 - 6857*v^5 + 6617*v^4 - 6329*v^3 - 412*v^2 - 1347*v - 143) / 186 $$\beta_{8}$$ $$=$$ $$( 6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + 12516 \nu^{8} + 16835 \nu^{7} + 21730 \nu^{6} + 21308 \nu^{5} + 18145 \nu^{4} + \cdots + 354 ) / 186$$ (6*v^15 + 28*v^14 + 144*v^13 + 517*v^12 + 1374*v^11 + 3653*v^10 + 6619*v^9 + 12516*v^8 + 16835*v^7 + 21730*v^6 + 21308*v^5 + 18145*v^4 + 10699*v^3 + 6074*v^2 + 532*v + 354) / 186 $$\beta_{9}$$ $$=$$ $$( - 36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} - 21362 \nu^{9} + 16025 \nu^{8} - 44466 \nu^{7} + 26249 \nu^{6} - 47899 \nu^{5} + 19734 \nu^{4} + \cdots + 538 ) / 186$$ (-36*v^15 + 38*v^14 - 709*v^13 + 695*v^12 - 5485*v^11 + 4827*v^10 - 21362*v^9 + 16025*v^8 - 44466*v^7 + 26249*v^6 - 47899*v^5 + 19734*v^4 - 22747*v^3 + 5719*v^2 - 2510*v + 538) / 186 $$\beta_{10}$$ $$=$$ $$( - 53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + 1770 \nu^{8} - 35213 \nu^{7} + 2430 \nu^{6} - 26132 \nu^{5} + 1337 \nu^{4} - 7000 \nu^{3} + \cdots + 61 ) / 186$$ (-53*v^15 + 5*v^14 - 962*v^13 + 89*v^12 - 6619*v^11 + 587*v^10 - 21738*v^9 + 1770*v^8 - 35213*v^7 + 2430*v^6 - 26132*v^5 + 1337*v^4 - 7000*v^3 + 303*v^2 - 225*v + 61) / 186 $$\beta_{11}$$ $$=$$ $$( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} - 9222 \nu^{8} + 59919 \nu^{7} - 13483 \nu^{6} + 62350 \nu^{5} - 7987 \nu^{4} + \cdots + 36 ) / 186$$ (50*v^15 - 25*v^14 + 983*v^13 - 445*v^12 + 7575*v^11 - 2966*v^10 + 29263*v^9 - 9222*v^8 + 59919*v^7 - 13483*v^6 + 62350*v^5 - 7987*v^4 + 27117*v^3 - 926*v^2 + 1788*v + 36) / 186 $$\beta_{12}$$ $$=$$ $$( 57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + 8581 \nu^{8} + 46581 \nu^{7} + 14174 \nu^{6} + 41009 \nu^{5} + 11369 \nu^{4} + \cdots + 126 ) / 186$$ (57*v^15 + 21*v^14 + 1058*v^13 + 380*v^12 + 7535*v^11 + 2608*v^10 + 26161*v^9 + 8581*v^8 + 46581*v^7 + 14174*v^6 + 41009*v^5 + 11369*v^4 + 15383*v^3 + 3765*v^2 + 1489*v + 126) / 186 $$\beta_{13}$$ $$=$$ $$( 27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + 21738 \nu^{8} + 31009 \nu^{7} + 35213 \nu^{6} + 32677 \nu^{5} + 26132 \nu^{4} + \cdots + 597 ) / 186$$ (27*v^15 + 53*v^14 + 524*v^13 + 962*v^12 + 3982*v^11 + 6619*v^10 + 15200*v^9 + 21738*v^8 + 31009*v^7 + 35213*v^6 + 32677*v^5 + 26132*v^4 + 14464*v^3 + 7093*v^2 + 658*v + 597) / 186 $$\beta_{14}$$ $$=$$ $$( 36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + 27451 \nu^{8} + 44466 \nu^{7} + 44177 \nu^{6} + 47899 \nu^{5} + 32530 \nu^{4} + \cdots + 470 ) / 186$$ (36*v^15 + 68*v^14 + 709*v^13 + 1229*v^12 + 5485*v^11 + 8411*v^10 + 21362*v^9 + 27451*v^8 + 44466*v^7 + 44177*v^6 + 47899*v^5 + 32530*v^4 + 22747*v^3 + 8467*v^2 + 2510*v + 470) / 186 $$\beta_{15}$$ $$=$$ $$( 135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + 67940 \nu^{9} + 12098 \nu^{8} + 128230 \nu^{7} + 17671 \nu^{6} + 121504 \nu^{5} + 11336 \nu^{4} + \cdots + 359 ) / 186$$ (135*v^15 + 34*v^14 + 2558*v^13 + 599*v^12 + 18763*v^11 + 3942*v^10 + 67940*v^9 + 12098*v^8 + 128230*v^7 + 17671*v^6 + 121504*v^5 + 11336*v^4 + 48574*v^3 + 2730*v^2 + 3724*v + 359) / 186
 $$\nu$$ $$=$$ $$\beta_{8} - \beta_{6} - \beta_{5} - \beta_{3}$$ b8 - b6 - b5 - b3 $$\nu^{2}$$ $$=$$ $$\beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{4} + \beta _1 - 3$$ b14 - b12 + b11 + b9 - 2*b8 + b4 + b1 - 3 $$\nu^{3}$$ $$=$$ $$- \beta_{15} + \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 6 \beta_{6} + 3 \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1$$ -b15 + b14 + b12 + 2*b11 + b10 - 6*b8 + 6*b6 + 3*b5 + b4 + 4*b3 + 2*b2 + 2*b1 $$\nu^{4}$$ $$=$$ $$2 \beta_{15} - 8 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 7 \beta_{9} + 12 \beta_{8} + \beta_{7} + 4 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{2} - 6 \beta _1 + 14$$ 2*b15 - 8*b14 + 3*b12 - 6*b11 + 2*b10 - 7*b9 + 12*b8 + b7 + 4*b6 + b5 - 3*b4 + b2 - 6*b1 + 14 $$\nu^{5}$$ $$=$$ $$7 \beta_{15} - 6 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} - 14 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + 37 \beta_{8} + \beta_{7} - 37 \beta_{6} - 13 \beta_{5} - 4 \beta_{4} - 21 \beta_{3} - 14 \beta_{2} - 15 \beta _1 + 2$$ 7*b15 - 6*b14 - 4*b13 - 6*b12 - 14*b11 - 7*b10 + 2*b9 + 37*b8 + b7 - 37*b6 - 13*b5 - 4*b4 - 21*b3 - 14*b2 - 15*b1 + 2 $$\nu^{6}$$ $$=$$ $$- 19 \beta_{15} + 55 \beta_{14} - 8 \beta_{12} + 38 \beta_{11} - 19 \beta_{10} + 47 \beta_{9} - 72 \beta_{8} - 11 \beta_{7} - 34 \beta_{6} - 11 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - 11 \beta_{2} + 38 \beta _1 - 77$$ -19*b15 + 55*b14 - 8*b12 + 38*b11 - 19*b10 + 47*b9 - 72*b8 - 11*b7 - 34*b6 - 11*b5 + 8*b4 + 3*b3 - 11*b2 + 38*b1 - 77 $$\nu^{7}$$ $$=$$ $$- 42 \beta_{15} + 35 \beta_{14} + 44 \beta_{13} + 29 \beta_{12} + 86 \beta_{11} + 42 \beta_{10} - 22 \beta_{9} - 235 \beta_{8} - 15 \beta_{7} + 235 \beta_{6} + 68 \beta_{5} + 7 \beta_{4} + 125 \beta_{3} + 85 \beta_{2} + 98 \beta _1 - 26$$ -42*b15 + 35*b14 + 44*b13 + 29*b12 + 86*b11 + 42*b10 - 22*b9 - 235*b8 - 15*b7 + 235*b6 + 68*b5 + 7*b4 + 125*b3 + 85*b2 + 98*b1 - 26 $$\nu^{8}$$ $$=$$ $$145 \beta_{15} - 365 \beta_{14} + 14 \beta_{12} - 248 \beta_{11} + 145 \beta_{10} - 309 \beta_{9} + 440 \beta_{8} + 89 \beta_{7} + 234 \beta_{6} + 92 \beta_{5} - 14 \beta_{4} - 36 \beta_{3} + 86 \beta_{2} - 245 \beta _1 + 455$$ 145*b15 - 365*b14 + 14*b12 - 248*b11 + 145*b10 - 309*b9 + 440*b8 + 89*b7 + 234*b6 + 92*b5 - 14*b4 - 36*b3 + 86*b2 - 245*b1 + 455 $$\nu^{9}$$ $$=$$ $$245 \beta_{15} - 212 \beta_{14} - 356 \beta_{13} - 128 \beta_{12} - 518 \beta_{11} - 245 \beta_{10} + 178 \beta_{9} + 1508 \beta_{8} + 145 \beta_{7} - 1508 \beta_{6} - 391 \beta_{5} + 50 \beta_{4} - 781 \beta_{3} - 513 \beta_{2} + \cdots + 234$$ 245*b15 - 212*b14 - 356*b13 - 128*b12 - 518*b11 - 245*b10 + 178*b9 + 1508*b8 + 145*b7 - 1508*b6 - 391*b5 + 50*b4 - 781*b3 - 513*b2 - 630*b1 + 234 $$\nu^{10}$$ $$=$$ $$- 1022 \beta_{15} + 2385 \beta_{14} + 34 \beta_{12} + 1625 \beta_{11} - 1022 \beta_{10} + 2000 \beta_{9} - 2723 \beta_{8} - 637 \beta_{7} - 1517 \beta_{6} - 688 \beta_{5} - 34 \beta_{4} + 303 \beta_{3} - 601 \beta_{2} + \cdots - 2780$$ -1022*b15 + 2385*b14 + 34*b12 + 1625*b11 - 1022*b10 + 2000*b9 - 2723*b8 - 637*b7 - 1517*b6 - 688*b5 - 34*b4 + 303*b3 - 601*b2 + 1589*b1 - 2780 $$\nu^{11}$$ $$=$$ $$- 1432 \beta_{15} + 1322 \beta_{14} + 2578 \beta_{13} + 518 \beta_{12} + 3129 \beta_{11} + 1432 \beta_{10} - 1289 \beta_{9} - 9702 \beta_{8} - 1179 \beta_{7} + 9702 \beta_{6} + 2357 \beta_{5} - 756 \beta_{4} + \cdots - 1831$$ -1432*b15 + 1322*b14 + 2578*b13 + 518*b12 + 3129*b11 + 1432*b10 - 1289*b9 - 9702*b8 - 1179*b7 + 9702*b6 + 2357*b5 - 756*b4 + 4968*b3 + 3142*b2 + 4056*b1 - 1831 $$\nu^{12}$$ $$=$$ $$6922 \beta_{15} - 15445 \beta_{14} - 619 \beta_{12} - 10601 \beta_{11} + 6922 \beta_{10} - 12821 \beta_{9} + 16992 \beta_{8} + 4298 \beta_{7} + 9634 \beta_{6} + 4853 \beta_{5} + 619 \beta_{4} - 2229 \beta_{3} + \cdots + 17280$$ 6922*b15 - 15445*b14 - 619*b12 - 10601*b11 + 6922*b10 - 12821*b9 + 16992*b8 + 4298*b7 + 9634*b6 + 4853*b5 + 619*b4 - 2229*b3 + 4010*b2 - 10313*b1 + 17280 $$\nu^{13}$$ $$=$$ $$8460 \beta_{15} - 8372 \beta_{14} - 17726 \beta_{13} - 1817 \beta_{12} - 19052 \beta_{11} - 8460 \beta_{10} + 8863 \beta_{9} + 62396 \beta_{8} + 8775 \beta_{7} - 62396 \beta_{6} - 14559 \beta_{5} + \cdots + 13340$$ 8460*b15 - 8372*b14 - 17726*b13 - 1817*b12 - 19052*b11 - 8460*b10 + 8863*b9 + 62396*b8 + 8775*b7 - 62396*b6 - 14559*b5 + 6779*b4 - 31794*b3 - 19510*b2 - 26153*b1 + 13340 $$\nu^{14}$$ $$=$$ $$- 45841 \beta_{15} + 99462 \beta_{14} + 5217 \beta_{12} + 68772 \beta_{11} - 45841 \beta_{10} + 81769 \beta_{9} - 106633 \beta_{8} - 28148 \beta_{7} - 60771 \beta_{6} - 33083 \beta_{5} + \cdots - 108434$$ -45841*b15 + 99462*b14 + 5217*b12 + 68772*b11 - 45841*b10 + 81769*b9 - 106633*b8 - 28148*b7 - 60771*b6 - 33083*b5 - 5217*b4 + 15390*b3 - 26186*b2 + 66810*b1 - 108434 $$\nu^{15}$$ $$=$$ $$- 50619 \beta_{15} + 53382 \beta_{14} + 118538 \beta_{13} + 4347 \beta_{12} + 116998 \beta_{11} + 50619 \beta_{10} - 59269 \beta_{9} - 400724 \beta_{8} - 62032 \beta_{7} + 400724 \beta_{6} + \cdots - 93153$$ -50619*b15 + 53382*b14 + 118538*b13 + 4347*b12 + 116998*b11 + 50619*b10 - 59269*b9 - 400724*b8 - 62032*b7 + 400724*b6 + 91111*b5 - 51949*b4 + 203762*b3 + 122303*b2 + 168575*b1 - 93153

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 − 2.16544i 0.176392i 2.16544i − 0.176392i 1.03739i − 2.52368i − 1.14660i 0.333129i − 1.42343i 1.83925i 1.42343i − 1.83925i 1.14660i − 0.333129i − 1.03739i 2.52368i
−0.571745 1.75965i −0.488442 + 0.103822i −1.15144 + 0.836573i −0.603681 + 1.04561i 0.461954 + 0.800128i 3.41030 + 1.51837i −0.863288 0.627215i −2.51284 + 1.11879i 2.18505 + 0.464447i
7.2 0.380762 + 1.17187i −2.02963 + 0.431412i 0.389745 0.283166i 0.772811 1.33855i −1.27836 2.21419i −3.47491 1.54713i 2.47393 + 1.79742i 1.19265 0.531003i 1.86286 + 0.395962i
9.1 −0.571745 + 1.75965i −0.488442 0.103822i −1.15144 0.836573i −0.603681 1.04561i 0.461954 0.800128i 3.41030 1.51837i −0.863288 + 0.627215i −2.51284 1.11879i 2.18505 0.464447i
9.2 0.380762 1.17187i −2.02963 0.431412i 0.389745 + 0.283166i 0.772811 + 1.33855i −1.27836 + 2.21419i −3.47491 + 1.54713i 2.47393 1.79742i 1.19265 + 0.531003i 1.86286 0.395962i
10.1 −1.02470 0.744490i −0.155153 1.47618i −0.122284 0.376353i 1.90016 + 3.29117i −0.940018 + 1.62816i −2.14115 0.455117i −0.937688 + 2.88591i 0.779397 0.165666i 0.503147 4.78712i
10.2 −0.284315 0.206567i 0.302431 + 2.87744i −0.579869 1.78465i −1.48661 2.57489i 0.508398 0.880572i 1.05848 + 0.224987i −0.420982 + 1.29565i −5.25377 + 1.11672i −0.109221 + 1.03917i
14.1 −0.831304 + 2.55849i 0.949606 1.05464i −4.23677 3.07819i −0.304192 + 0.526876i 1.90889 + 3.30629i 0.180508 1.71742i 7.04481 5.11835i 0.103062 + 0.980572i −1.09513 1.21627i
14.2 0.640321 1.97070i −1.43153 + 1.58988i −1.85563 1.34820i −1.17396 + 2.03335i 2.21654 + 3.83916i 0.384094 3.65441i −0.492333 + 0.357701i −0.164841 1.56836i 3.25543 + 3.61552i
18.1 −1.86683 + 1.35633i −2.32289 + 1.03422i 1.02738 3.16196i 1.24923 + 2.16373i 2.93370 5.08132i 1.07187 + 1.19043i 0.944583 + 2.90713i 2.31884 2.57533i −5.26683 2.34494i
18.2 0.557811 0.405274i −0.824384 + 0.367040i −0.471127 + 1.44998i −1.85376 3.21080i −0.311099 + 0.538840i 0.510810 + 0.567312i 0.750969 + 2.31124i −1.46250 + 1.62427i −2.33530 1.03974i
19.1 −1.86683 1.35633i −2.32289 1.03422i 1.02738 + 3.16196i 1.24923 2.16373i 2.93370 + 5.08132i 1.07187 1.19043i 0.944583 2.90713i 2.31884 + 2.57533i −5.26683 + 2.34494i
19.2 0.557811 + 0.405274i −0.824384 0.367040i −0.471127 1.44998i −1.85376 + 3.21080i −0.311099 0.538840i 0.510810 0.567312i 0.750969 2.31124i −1.46250 1.62427i −2.33530 + 1.03974i
20.1 −0.831304 2.55849i 0.949606 + 1.05464i −4.23677 + 3.07819i −0.304192 0.526876i 1.90889 3.30629i 0.180508 + 1.71742i 7.04481 + 5.11835i 0.103062 0.980572i −1.09513 + 1.21627i
20.2 0.640321 + 1.97070i −1.43153 1.58988i −1.85563 + 1.34820i −1.17396 2.03335i 2.21654 3.83916i 0.384094 + 3.65441i −0.492333 0.357701i −0.164841 + 1.56836i 3.25543 3.61552i
28.1 −1.02470 + 0.744490i −0.155153 + 1.47618i −0.122284 + 0.376353i 1.90016 3.29117i −0.940018 1.62816i −2.14115 + 0.455117i −0.937688 2.88591i 0.779397 + 0.165666i 0.503147 + 4.78712i
28.2 −0.284315 + 0.206567i 0.302431 2.87744i −0.579869 + 1.78465i −1.48661 + 2.57489i 0.508398 + 0.880572i 1.05848 0.224987i −0.420982 1.29565i −5.25377 1.11672i −0.109221 1.03917i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.2.g.a 16
3.b odd 2 1 279.2.y.c 16
4.b odd 2 1 496.2.bg.c 16
5.b even 2 1 775.2.bl.a 16
5.c odd 4 2 775.2.ck.a 32
31.b odd 2 1 961.2.g.l 16
31.c even 3 1 961.2.d.o 16
31.c even 3 1 961.2.g.k 16
31.d even 5 1 961.2.c.j 16
31.d even 5 1 961.2.g.k 16
31.d even 5 1 961.2.g.s 16
31.d even 5 1 961.2.g.t 16
31.e odd 6 1 961.2.d.n 16
31.e odd 6 1 961.2.g.j 16
31.f odd 10 1 961.2.c.i 16
31.f odd 10 1 961.2.g.j 16
31.f odd 10 1 961.2.g.m 16
31.f odd 10 1 961.2.g.n 16
31.g even 15 1 inner 31.2.g.a 16
31.g even 15 1 961.2.a.i 8
31.g even 15 1 961.2.c.j 16
31.g even 15 1 961.2.d.o 16
31.g even 15 2 961.2.d.p 16
31.g even 15 1 961.2.g.s 16
31.g even 15 1 961.2.g.t 16
31.h odd 30 1 961.2.a.j 8
31.h odd 30 1 961.2.c.i 16
31.h odd 30 1 961.2.d.n 16
31.h odd 30 2 961.2.d.q 16
31.h odd 30 1 961.2.g.l 16
31.h odd 30 1 961.2.g.m 16
31.h odd 30 1 961.2.g.n 16
93.o odd 30 1 279.2.y.c 16
93.o odd 30 1 8649.2.a.bf 8
93.p even 30 1 8649.2.a.be 8
124.n odd 30 1 496.2.bg.c 16
155.u even 30 1 775.2.bl.a 16
155.w odd 60 2 775.2.ck.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a 16 1.a even 1 1 trivial
31.2.g.a 16 31.g even 15 1 inner
279.2.y.c 16 3.b odd 2 1
279.2.y.c 16 93.o odd 30 1
496.2.bg.c 16 4.b odd 2 1
496.2.bg.c 16 124.n odd 30 1
775.2.bl.a 16 5.b even 2 1
775.2.bl.a 16 155.u even 30 1
775.2.ck.a 32 5.c odd 4 2
775.2.ck.a 32 155.w odd 60 2
961.2.a.i 8 31.g even 15 1
961.2.a.j 8 31.h odd 30 1
961.2.c.i 16 31.f odd 10 1
961.2.c.i 16 31.h odd 30 1
961.2.c.j 16 31.d even 5 1
961.2.c.j 16 31.g even 15 1
961.2.d.n 16 31.e odd 6 1
961.2.d.n 16 31.h odd 30 1
961.2.d.o 16 31.c even 3 1
961.2.d.o 16 31.g even 15 1
961.2.d.p 16 31.g even 15 2
961.2.d.q 16 31.h odd 30 2
961.2.g.j 16 31.e odd 6 1
961.2.g.j 16 31.f odd 10 1
961.2.g.k 16 31.c even 3 1
961.2.g.k 16 31.d even 5 1
961.2.g.l 16 31.b odd 2 1
961.2.g.l 16 31.h odd 30 1
961.2.g.m 16 31.f odd 10 1
961.2.g.m 16 31.h odd 30 1
961.2.g.n 16 31.f odd 10 1
961.2.g.n 16 31.h odd 30 1
961.2.g.s 16 31.d even 5 1
961.2.g.s 16 31.g even 15 1
961.2.g.t 16 31.d even 5 1
961.2.g.t 16 31.g even 15 1
8649.2.a.be 8 93.p even 30 1
8649.2.a.bf 8 93.o odd 30 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(31, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 6 T^{15} + 29 T^{14} + 91 T^{13} + \cdots + 81$$
$3$ $$T^{16} + 12 T^{15} + 74 T^{14} + \cdots + 961$$
$5$ $$T^{16} + 3 T^{15} + 31 T^{14} + \cdots + 77841$$
$7$ $$T^{16} - 2 T^{15} - 6 T^{14} + \cdots + 68121$$
$11$ $$T^{16} + 7 T^{15} + 13 T^{14} + \cdots + 77841$$
$13$ $$T^{16} + 7 T^{15} + 18 T^{14} + \cdots + 77841$$
$17$ $$T^{16} + 6 T^{15} + 41 T^{14} + \cdots + 74805201$$
$19$ $$T^{16} - 16 T^{15} + 157 T^{14} + \cdots + 361201$$
$23$ $$T^{16} - T^{15} + 60 T^{14} - 310 T^{13} + \cdots + 77841$$
$29$ $$T^{16} + 14 T^{15} + 181 T^{14} + \cdots + 77841$$
$31$ $$T^{16} - 15 T^{15} + \cdots + 852891037441$$
$37$ $$T^{16} + 8 T^{15} + 176 T^{14} + \cdots + 344807761$$
$41$ $$T^{16} + 8 T^{15} + 24 T^{14} + 74 T^{13} + \cdots + 81$$
$43$ $$T^{16} - 23 T^{15} + 203 T^{14} + \cdots + 7612081$$
$47$ $$T^{16} - 14 T^{15} + \cdots + 3306365001$$
$53$ $$T^{16} - 6 T^{15} + \cdots + 366207732801$$
$59$ $$T^{16} - 4 T^{15} + 39 T^{14} + \cdots + 167728401$$
$61$ $$(T^{8} + 30 T^{7} + 288 T^{6} + \cdots + 38161)^{2}$$
$67$ $$T^{16} - 13 T^{15} + \cdots + 7485883441$$
$71$ $$T^{16} + 14 T^{15} + \cdots + 214944921$$
$73$ $$T^{16} - 2 T^{15} + \cdots + 17441907675201$$
$79$ $$T^{16} - 18 T^{15} + \cdots + 84609661119201$$
$83$ $$T^{16} + 16 T^{15} + \cdots + 1446653267361$$
$89$ $$T^{16} - T^{15} + \cdots + 117957215601$$
$97$ $$T^{16} - 3 T^{15} + \cdots + 7131992195241$$