# Properties

 Label 125.2.d.b Level $125$ Weight $2$ Character orbit 125.d Analytic conductor $0.998$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.d (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.998130025266$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625$$ x^16 + x^14 - 4*x^12 - 49*x^10 + 11*x^8 + 395*x^6 + 900*x^4 + 1125*x^2 + 625 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} - 420964990 \nu^{5} - 210825325 \nu^{3} - 35093875 \nu ) / 858900625$$ (22849*v^15 - 2422021*v^13 + 6573709*v^11 - 1538146*v^9 + 97097069*v^7 - 420964990*v^5 - 210825325*v^3 - 35093875*v) / 858900625 $$\beta_{2}$$ $$=$$ $$( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + 77053830 \nu^{4} + 27131400 \nu^{2} + \cdots + 181387875 ) / 171780125$$ (948392*v^14 - 2081693*v^12 - 1547103*v^10 - 35207443*v^8 + 136185777*v^6 + 77053830*v^4 + 27131400*v^2 + 181387875) / 171780125 $$\beta_{3}$$ $$=$$ $$( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + 54634025 \nu^{4} + 77779875 \nu^{2} + 52412250 ) / 15616375$$ (122987*v^14 + 97172*v^12 - 701513*v^10 - 5799603*v^8 + 3932417*v^6 + 54634025*v^4 + 77779875*v^2 + 52412250) / 15616375 $$\beta_{4}$$ $$=$$ $$( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + \cdots + 563878625 ) / 171780125$$ (1379032*v^14 - 312743*v^12 - 4990978*v^10 - 61900293*v^8 + 94590252*v^6 + 424939915*v^4 + 771306350*v^2 + 563878625) / 171780125 $$\beta_{5}$$ $$=$$ $$( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475 ) / 34356025$$ (-281280*v^14 + 69219*v^12 + 939009*v^10 + 12381889*v^8 - 18275316*v^6 - 83450271*v^4 - 152170830*v^2 - 167096475) / 34356025 $$\beta_{6}$$ $$=$$ $$( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} - 137248467 \nu^{7} - 474206065 \nu^{5} + 222668250 \nu^{3} + \cdots + 846395125 \nu ) / 858900625$$ (-1157122*v^15 + 1428203*v^13 + 8678488*v^11 + 43919978*v^9 - 137248467*v^7 - 474206065*v^5 + 222668250*v^3 + 846395125*v) / 858900625 $$\beta_{7}$$ $$=$$ $$( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} + \cdots - 215410900 ) / 34356025$$ (-441862*v^14 + 108648*v^12 + 1544473*v^10 + 20140738*v^8 - 29602957*v^6 - 135351515*v^4 - 246968035*v^2 - 215410900) / 34356025 $$\beta_{8}$$ $$=$$ $$( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} - 716617430 \nu^{4} - 884417625 \nu^{2} + \cdots - 860602375 ) / 171780125$$ (-2817591*v^14 + 2379174*v^12 + 8884104*v^10 + 118381374*v^8 - 262226761*v^6 - 716617430*v^4 - 884417625*v^2 - 860602375) / 171780125 $$\beta_{9}$$ $$=$$ $$( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} - 210462216 \nu^{4} - 281904075 \nu^{2} + \cdots - 222107450 ) / 34356025$$ (-626638*v^14 + 144621*v^12 + 2783831*v^10 + 27217946*v^8 - 41997039*v^6 - 210462216*v^4 - 281904075*v^2 - 222107450) / 34356025 $$\beta_{10}$$ $$=$$ $$( - 257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} - 113117275 \nu^{5} - 254856700 \nu^{3} + \cdots - 189940875 \nu ) / 78081875$$ (-257689*v^15 - 260409*v^13 + 1184711*v^11 + 13501191*v^9 - 4440499*v^7 - 113117275*v^5 - 254856700*v^3 - 189940875*v) / 78081875 $$\beta_{11}$$ $$=$$ $$( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + 137417033 \nu^{7} + 1546922530 \nu^{5} + 2678832450 \nu^{3} + \cdots + 1573606875 \nu ) / 858900625$$ (3711283*v^15 + 2442853*v^13 - 18411087*v^11 - 175485672*v^9 + 137417033*v^7 + 1546922530*v^5 + 2678832450*v^3 + 1573606875*v) / 858900625 $$\beta_{12}$$ $$=$$ $$( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + 752782897 \nu^{7} + 3788808280 \nu^{5} + 5075395150 \nu^{3} + \cdots + 4625990250 \nu ) / 858900625$$ (11296137*v^15 - 2500148*v^13 - 48798533*v^11 - 491452273*v^9 + 752782897*v^7 + 3788808280*v^5 + 5075395150*v^3 + 4625990250*v) / 858900625 $$\beta_{13}$$ $$=$$ $$( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + 285294535 \nu^{5} + 432663800 \nu^{3} + \cdots + 416073000 \nu ) / 78081875$$ (1033909*v^15 - 651386*v^13 - 3517956*v^11 - 44316636*v^9 + 84515054*v^7 + 285294535*v^5 + 432663800*v^3 + 416073000*v) / 78081875 $$\beta_{14}$$ $$=$$ $$( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + 466969935 \nu^{5} + 843562400 \nu^{3} + \cdots + 722601875 \nu ) / 78081875$$ (1468861*v^15 - 201274*v^13 - 5426879*v^11 - 66369199*v^9 + 90901286*v^7 + 466969935*v^5 + 843562400*v^3 + 722601875*v) / 78081875 $$\beta_{15}$$ $$=$$ $$( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + 1385305413 \nu^{7} + 5454301045 \nu^{5} + 7862490225 \nu^{3} + \cdots + 7417196625 \nu ) / 858900625$$ (18299323*v^15 - 8550017*v^13 - 66698807*v^11 - 789938892*v^9 + 1385305413*v^7 + 5454301045*v^5 + 7862490225*v^3 + 7417196625*v) / 858900625
 $$\nu$$ $$=$$ $$( -\beta_{15} - 5\beta_{14} + 4\beta_{13} + 4\beta_{12} - 5\beta_{10} + 5\beta_{6} + \beta_1 ) / 5$$ (-b15 - 5*b14 + 4*b13 + 4*b12 - 5*b10 + 5*b6 + b1) / 5 $$\nu^{2}$$ $$=$$ $$( -2\beta_{9} + 2\beta_{8} + 3\beta_{7} + 2\beta_{5} + 11\beta_{4} - 4\beta_{3} - \beta_{2} + 4 ) / 5$$ (-2*b9 + 2*b8 + 3*b7 + 2*b5 + 11*b4 - 4*b3 - b2 + 4) / 5 $$\nu^{3}$$ $$=$$ $$( -6\beta_{15} + 14\beta_{13} - 6\beta_{12} + 5\beta_{11} - 9\beta_1 ) / 5$$ (-6*b15 + 14*b13 - 6*b12 + 5*b11 - 9*b1) / 5 $$\nu^{4}$$ $$=$$ $$( -7\beta_{9} - 13\beta_{8} + 3\beta_{7} + 7\beta_{5} + 6\beta_{4} - 19\beta_{3} - 26\beta_{2} + 14 ) / 5$$ (-7*b9 - 13*b8 + 3*b7 + 7*b5 + 6*b4 - 19*b3 - 26*b2 + 14) / 5 $$\nu^{5}$$ $$=$$ $$( 39\beta_{15} - 20\beta_{14} - 6\beta_{13} - 36\beta_{12} - 30\beta_{10} - 29\beta_1 ) / 5$$ (39*b15 - 20*b14 - 6*b13 - 36*b12 - 30*b10 - 29*b1) / 5 $$\nu^{6}$$ $$=$$ $$( 23\beta_{9} - 23\beta_{8} + 63\beta_{7} + 52\beta_{5} + 156\beta_{4} + 11\beta_{3} - 11\beta_{2} + 144 ) / 5$$ (23*b9 - 23*b8 + 63*b7 + 52*b5 + 156*b4 + 11*b3 - 11*b2 + 144) / 5 $$\nu^{7}$$ $$=$$ $$( - 26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + 115 \beta_{6} + 66 \beta_1 ) / 5$$ (-26*b15 - 190*b14 + 184*b13 + 49*b12 + 190*b11 - 115*b10 + 115*b6 + 66*b1) / 5 $$\nu^{8}$$ $$=$$ $$( -57\beta_{9} - 18\beta_{8} + 363\beta_{7} - 208\beta_{5} + 381\beta_{4} - 114\beta_{3} - 96\beta_{2} + 39 ) / 5$$ (-57*b9 - 18*b8 + 363*b7 - 208*b5 + 381*b4 - 114*b3 - 96*b2 + 39) / 5 $$\nu^{9}$$ $$=$$ $$( -\beta_{15} + 284\beta_{13} - 286\beta_{12} + 285\beta_{11} + 285\beta_{10} + 190\beta_{6} - 479\beta_1 ) / 5$$ (-b15 + 284*b13 - 286*b12 + 285*b11 + 285*b10 + 190*b6 - 479*b1) / 5 $$\nu^{10}$$ $$=$$ $$( 208\beta_{9} - 688\beta_{8} + 208\beta_{7} - 493\beta_{5} - 149\beta_{4} - 344\beta_{3} - 896\beta_{2} - 606 ) / 5$$ (208*b9 - 688*b8 + 208*b7 - 493*b5 - 149*b4 - 344*b3 - 896*b2 - 606) / 5 $$\nu^{11}$$ $$=$$ $$( 2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + 95 \beta_{6} - 624 \beta_1 ) / 5$$ (2344*b15 - 290*b14 - 1651*b13 - 1846*b12 + 195*b11 - 290*b10 + 95*b6 - 624*b1) / 5 $$\nu^{12}$$ $$=$$ $$( 2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + 1104 \beta_{2} + 1784 ) / 5$$ (2208*b9 - 683*b8 + 2888*b7 - 683*b5 + 3956*b4 + 2891*b3 + 1104*b2 + 1784) / 5 $$\nu^{13}$$ $$=$$ $$( - 906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + 4675 \beta_{6} + 5166 \beta_1 ) / 5$$ (-906*b15 - 4675*b14 + 2364*b13 + 3769*b12 + 7530*b11 + 4675*b6 + 5166*b1) / 5 $$\nu^{14}$$ $$=$$ $$( 393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + 2024 \beta_{2} - 18511 ) / 5$$ (393*b9 + 1012*b8 + 10778*b7 - 18118*b5 - 619*b4 + 1631*b3 + 2024*b2 - 18511) / 5 $$\nu^{15}$$ $$=$$ $$( 1639\beta_{15} + 19130\beta_{14} - 12431\beta_{13} - 9336\beta_{12} + 30920\beta_{10} - 13429\beta_1 ) / 5$$ (1639*b15 + 19130*b14 - 12431*b13 - 9336*b12 + 30920*b10 - 13429*b1) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/125\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.644389 − 0.983224i 0.0566033 − 1.17421i −0.0566033 + 1.17421i −0.644389 + 0.983224i 1.86824 + 0.357358i 0.917186 + 1.66637i −0.917186 − 1.66637i −1.86824 − 0.357358i 1.86824 − 0.357358i 0.917186 − 1.66637i −0.917186 + 1.66637i −1.86824 + 0.357358i 0.644389 + 0.983224i 0.0566033 + 1.17421i −0.0566033 − 1.17421i −0.644389 − 0.983224i
−0.644389 1.98322i 1.77862 1.29224i −1.89991 + 1.38036i 0 −3.70892 2.69469i 0.992398 0.587785 + 0.427051i 0.566541 1.74363i 0
26.2 −0.0566033 0.174207i 1.19083 0.865190i 1.59089 1.15585i 0 −0.218127 0.158479i −3.26086 −0.587785 0.427051i −0.257524 + 0.792578i 0
26.3 0.0566033 + 0.174207i −1.19083 + 0.865190i 1.59089 1.15585i 0 −0.218127 0.158479i 3.26086 0.587785 + 0.427051i −0.257524 + 0.792578i 0
26.4 0.644389 + 1.98322i −1.77862 + 1.29224i −1.89991 + 1.38036i 0 −3.70892 2.69469i −0.992398 −0.587785 0.427051i 0.566541 1.74363i 0
51.1 −1.86824 + 1.35736i 0.146753 0.451659i 1.02988 3.16963i 0 0.338893 + 1.04301i 3.03582 0.951057 + 2.92705i 2.24459 + 1.63079i 0
51.2 −0.917186 + 0.666375i −0.804303 + 2.47539i −0.220859 + 0.679734i 0 −0.911842 2.80636i 0.407162 −0.951057 2.92705i −3.05361 2.21858i 0
51.3 0.917186 0.666375i 0.804303 2.47539i −0.220859 + 0.679734i 0 −0.911842 2.80636i −0.407162 0.951057 + 2.92705i −3.05361 2.21858i 0
51.4 1.86824 1.35736i −0.146753 + 0.451659i 1.02988 3.16963i 0 0.338893 + 1.04301i −3.03582 −0.951057 2.92705i 2.24459 + 1.63079i 0
76.1 −1.86824 1.35736i 0.146753 + 0.451659i 1.02988 + 3.16963i 0 0.338893 1.04301i 3.03582 0.951057 2.92705i 2.24459 1.63079i 0
76.2 −0.917186 0.666375i −0.804303 2.47539i −0.220859 0.679734i 0 −0.911842 + 2.80636i 0.407162 −0.951057 + 2.92705i −3.05361 + 2.21858i 0
76.3 0.917186 + 0.666375i 0.804303 + 2.47539i −0.220859 0.679734i 0 −0.911842 + 2.80636i −0.407162 0.951057 2.92705i −3.05361 + 2.21858i 0
76.4 1.86824 + 1.35736i −0.146753 0.451659i 1.02988 + 3.16963i 0 0.338893 1.04301i −3.03582 −0.951057 + 2.92705i 2.24459 1.63079i 0
101.1 −0.644389 + 1.98322i 1.77862 + 1.29224i −1.89991 1.38036i 0 −3.70892 + 2.69469i 0.992398 0.587785 0.427051i 0.566541 + 1.74363i 0
101.2 −0.0566033 + 0.174207i 1.19083 + 0.865190i 1.59089 + 1.15585i 0 −0.218127 + 0.158479i −3.26086 −0.587785 + 0.427051i −0.257524 0.792578i 0
101.3 0.0566033 0.174207i −1.19083 0.865190i 1.59089 + 1.15585i 0 −0.218127 + 0.158479i 3.26086 0.587785 0.427051i −0.257524 0.792578i 0
101.4 0.644389 1.98322i −1.77862 1.29224i −1.89991 1.38036i 0 −3.70892 + 2.69469i −0.992398 −0.587785 + 0.427051i 0.566541 + 1.74363i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.4 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.d.b 16
5.b even 2 1 inner 125.2.d.b 16
5.c odd 4 1 25.2.e.a 8
5.c odd 4 1 125.2.e.b 8
15.e even 4 1 225.2.m.a 8
20.e even 4 1 400.2.y.c 8
25.d even 5 1 inner 125.2.d.b 16
25.d even 5 1 625.2.a.f 8
25.d even 5 2 625.2.d.o 16
25.e even 10 1 inner 125.2.d.b 16
25.e even 10 1 625.2.a.f 8
25.e even 10 2 625.2.d.o 16
25.f odd 20 1 25.2.e.a 8
25.f odd 20 1 125.2.e.b 8
25.f odd 20 2 625.2.b.c 8
25.f odd 20 2 625.2.e.a 8
25.f odd 20 2 625.2.e.i 8
75.h odd 10 1 5625.2.a.x 8
75.j odd 10 1 5625.2.a.x 8
75.l even 20 1 225.2.m.a 8
100.h odd 10 1 10000.2.a.bj 8
100.j odd 10 1 10000.2.a.bj 8
100.l even 20 1 400.2.y.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 5.c odd 4 1
25.2.e.a 8 25.f odd 20 1
125.2.d.b 16 1.a even 1 1 trivial
125.2.d.b 16 5.b even 2 1 inner
125.2.d.b 16 25.d even 5 1 inner
125.2.d.b 16 25.e even 10 1 inner
125.2.e.b 8 5.c odd 4 1
125.2.e.b 8 25.f odd 20 1
225.2.m.a 8 15.e even 4 1
225.2.m.a 8 75.l even 20 1
400.2.y.c 8 20.e even 4 1
400.2.y.c 8 100.l even 20 1
625.2.a.f 8 25.d even 5 1
625.2.a.f 8 25.e even 10 1
625.2.b.c 8 25.f odd 20 2
625.2.d.o 16 25.d even 5 2
625.2.d.o 16 25.e even 10 2
625.2.e.a 8 25.f odd 20 2
625.2.e.i 8 25.f odd 20 2
5625.2.a.x 8 75.h odd 10 1
5625.2.a.x 8 75.j odd 10 1
10000.2.a.bj 8 100.h odd 10 1
10000.2.a.bj 8 100.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 3T_{2}^{14} + 23T_{2}^{12} + 126T_{2}^{10} + 475T_{2}^{8} - 174T_{2}^{6} + 878T_{2}^{4} + 48T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(125, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 3 T^{14} + 23 T^{12} + 126 T^{10} + \cdots + 1$$
$3$ $$T^{16} + 7 T^{14} + 33 T^{12} + 119 T^{10} + \cdots + 256$$
$5$ $$T^{16}$$
$7$ $$(T^{8} - 21 T^{6} + 121 T^{4} - 116 T^{2} + \cdots + 16)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{4}$$
$13$ $$T^{16} + 17 T^{14} + 108 T^{12} - 101 T^{10} + \cdots + 1$$
$17$ $$T^{16} - 12 T^{14} + 698 T^{12} + \cdots + 3748096$$
$19$ $$(T^{8} - 5 T^{7} + 30 T^{6} - 40 T^{5} + \cdots + 400)^{2}$$
$23$ $$T^{16} + 27 T^{14} + 333 T^{12} + \cdots + 65536$$
$29$ $$(T^{8} - 5 T^{7} + 30 T^{6} + 5 T^{5} + \cdots + 483025)^{2}$$
$31$ $$(T^{8} + 9 T^{7} + 117 T^{6} + 917 T^{5} + \cdots + 1936)^{2}$$
$37$ $$T^{16} + 88 T^{14} + \cdots + 13521270961$$
$41$ $$(T^{8} + 4 T^{7} + 52 T^{6} + 457 T^{5} + \cdots + 13456)^{2}$$
$43$ $$(T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016)^{2}$$
$47$ $$T^{16} - 32 T^{14} + \cdots + 4294967296$$
$53$ $$T^{16} + 112 T^{14} + \cdots + 76661949773761$$
$59$ $$(T^{8} - 15 T^{5} + 5635 T^{4} + \cdots + 4080400)^{2}$$
$61$ $$(T^{8} + 9 T^{7} - 43 T^{6} - 1068 T^{5} + \cdots + 116281)^{2}$$
$67$ $$T^{16} + 168 T^{14} + \cdots + 60523872256$$
$71$ $$(T^{8} - 6 T^{7} + 142 T^{6} + \cdots + 24245776)^{2}$$
$73$ $$T^{16} + 127 T^{14} + 6183 T^{12} + \cdots + 1$$
$79$ $$(T^{8} + 15 T^{7} + 100 T^{6} + \cdots + 33408400)^{2}$$
$83$ $$T^{16} + 127 T^{14} + \cdots + 9971220736$$
$89$ $$(T^{8} - 25 T^{7} + 520 T^{6} + \cdots + 1392400)^{2}$$
$97$ $$T^{16} + 328 T^{14} + \cdots + 90\!\cdots\!61$$