# Properties

 Label 230.3.d.a Level $230$ Weight $3$ Character orbit 230.d Analytic conductor $6.267$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 230.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26704608029$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529$$ x^16 + 78*x^14 + 2165*x^12 + 28310*x^10 + 184804*x^8 + 569634*x^6 + 696037*x^4 + 285578*x^2 + 529 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} - \beta_{5} q^{3} + 2 q^{4} + \beta_{2} q^{5} - \beta_{10} q^{6} + (\beta_{7} + \beta_1) q^{7} - 2 \beta_{6} q^{8} + ( - \beta_{14} - \beta_{11} - \beta_{10} + \beta_{6} + 5) q^{9}+O(q^{10})$$ q - b6 * q^2 - b5 * q^3 + 2 * q^4 + b2 * q^5 - b10 * q^6 + (b7 + b1) * q^7 - 2*b6 * q^8 + (-b14 - b11 - b10 + b6 + 5) * q^9 $$q - \beta_{6} q^{2} - \beta_{5} q^{3} + 2 q^{4} + \beta_{2} q^{5} - \beta_{10} q^{6} + (\beta_{7} + \beta_1) q^{7} - 2 \beta_{6} q^{8} + ( - \beta_{14} - \beta_{11} - \beta_{10} + \beta_{6} + 5) q^{9} - \beta_{8} q^{10} + ( - \beta_{9} + \beta_{7} + \beta_{4} + \beta_{2} + \beta_1) q^{11} - 2 \beta_{5} q^{12} + (\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + ( - 6 \beta_{9} - 16 \beta_{8} + 4 \beta_{7} + 14 \beta_{4} - 8 \beta_{3} + \cdots + 4 \beta_1) q^{99}+O(q^{100})$$ q - b6 * q^2 - b5 * q^3 + 2 * q^4 + b2 * q^5 - b10 * q^6 + (b7 + b1) * q^7 - 2*b6 * q^8 + (-b14 - b11 - b10 + b6 + 5) * q^9 - b8 * q^10 + (-b9 + b7 + b4 + b2 + b1) * q^11 - 2*b5 * q^12 + (b15 - b14 + b13 - b12 + b11 + b10 - b7 - 3*b6 - b5 + b4 + b2 + 1) * q^13 + (-b12 - b9 + b8 + 2*b2) * q^14 + (-b12 + b8 + b7 + b4 - b3 + b2 + b1) * q^15 + 4 * q^16 + (b12 + b9 - 2*b8 - 3*b7 - 2*b2) * q^17 + (-b15 + b14 - b13 + b11 + b10 - 4*b6 - 3) * q^18 + (b9 + 4*b8 - b7 - b4 - 2*b3 + 3*b2 + 3*b1) * q^19 + 2*b2 * q^20 + (3*b12 + 2*b9 - b8 - b7 - 4*b4 + 5*b3 + b2 - 2*b1) * q^21 + (-b12 - b9 + b8 - b4 - b3 + 2*b2 + b1) * q^22 + (b15 - b14 - b13 - b10 - b9 + 2*b8 + b7 + 5*b6 + b5 - 3*b3 - 2*b2 + b1 + 1) * q^23 - 2*b10 * q^24 - 5 * q^25 + (2*b15 + 2*b11 - b10 + 2*b5 + 6) * q^26 + (-2*b15 - b12 + 4*b10 - b7 - 8*b6 - b5 + b4 + b2 - 8) * q^27 + (2*b7 + 2*b1) * q^28 + (b15 + b14 + 2*b13 + b12 - 2*b11 - b10 + b7 - 6*b6 - b5 - b4 - b2 - 6) * q^29 + (b9 - b7 - b4 + 2*b3 - b2) * q^30 + (3*b15 - b14 + b12 + 2*b10 + b7 - 2*b6 - 5*b5 - b4 - b2 - 8) * q^31 - 4*b6 * q^32 + (5*b12 + 4*b9 + b8 - b7 - 8*b4 + 9*b3 - b2 - 4*b1) * q^33 + (-2*b7 + 3*b4 - 3*b3 + 2*b2 + b1) * q^34 + (-b15 - b14 - b12 - b7 + 4*b6 + 2*b5 + b4 + b2 + 4) * q^35 + (-2*b14 - 2*b11 - 2*b10 + 2*b6 + 10) * q^36 + (5*b9 + 11*b8 + 2*b7 - 4*b4 + b3 + 3*b2 + 4*b1) * q^37 + (-b12 - 3*b9 - 3*b8 - 2*b7 + 5*b4 - b3 - 4*b2 + b1) * q^38 + (-b14 + 4*b13 - 5*b11 - 2*b10 + 11*b6 - 2*b5 + 18) * q^39 - 2*b8 * q^40 + (-b15 + b14 - 4*b13 + 3*b12 - 2*b11 + b10 + 3*b7 + 2*b6 - 3*b5 - 3*b4 - 3*b2 - 10) * q^41 + (-3*b12 - 3*b9 - 3*b8 + 4*b7 + 3*b4 - 5*b3 + 2*b2 - b1) * q^42 + (5*b12 + b9 - 2*b8 + 2*b7 - 2*b4 + 4*b2 - b1) * q^43 + (-2*b9 + 2*b7 + 2*b4 + 2*b2 + 2*b1) * q^44 + (2*b9 + b8 - 2*b7 - 2*b4 - b3 + 3*b2) * q^45 + (-b15 - b14 + 2*b11 + 2*b10 - b9 + b8 - 4*b7 + b5 + 2*b4 + 2*b3 - 2*b2 - b1 - 9) * q^46 + (-2*b15 + 2*b13 - 4*b12 + 4*b11 + 2*b10 - 4*b7 + 6*b6 + b5 + 4*b4 + 4*b2 - 10) * q^47 - 4*b5 * q^48 + (-3*b15 - 2*b13 + b12 - 3*b11 + b7 - 9*b6 - b5 - b4 - b2 - 1) * q^49 + 5*b6 * q^50 + (4*b9 - 6*b8 - 2*b7 + 2*b4 + 4*b3 - 12*b2 - 2*b1) * q^51 + (2*b15 - 2*b14 + 2*b13 - 2*b12 + 2*b11 + 2*b10 - 2*b7 - 6*b6 - 2*b5 + 2*b4 + 2*b2 + 2) * q^52 + (-b12 + b9 + b7 - 6*b4 + 6*b3 - 12*b2 - 6*b1) * q^53 + (-2*b13 + 2*b12 - 2*b11 - 3*b10 + 2*b7 + 8*b6 + 6*b5 - 2*b4 - 2*b2 + 16) * q^54 + (-3*b14 - b12 - b11 - 2*b10 - b7 + 3*b6 + b5 + b4 + b2 + 2) * q^55 + (-2*b12 - 2*b9 + 2*b8 + 4*b2) * q^56 + (-5*b12 - 2*b9 - 3*b8 + 11*b7 + 10*b4 - 11*b3 + 3*b2 - 2*b1) * q^57 + (4*b14 + 2*b13 + b12 + 2*b10 + b7 + 5*b6 - 2*b5 - b4 - b2 + 8) * q^58 + (b15 + 8*b14 + 3*b12 + 3*b11 + 3*b7 + b6 + 3*b5 - 3*b4 - 3*b2 + 10) * q^59 + (-2*b12 + 2*b8 + 2*b7 + 2*b4 - 2*b3 + 2*b2 + 2*b1) * q^60 + (5*b12 + 8*b9 - 5*b8 - 9*b7 - 2*b4 + 9*b3 - 3*b2 - 4*b1) * q^61 + (2*b13 - 3*b12 + 4*b11 - 2*b10 - 3*b7 + 9*b6 + 8*b5 + 3*b4 + 3*b2 + 4) * q^62 + (-5*b9 - 8*b8 - 2*b7 + 11*b4 - 19*b2 - 2*b1) * q^63 + 8 * q^64 + (-3*b12 - 2*b9 - 2*b8 + 5*b7 - 2*b3 + 3*b2 - 2*b1) * q^65 + (-5*b12 - 5*b9 - 3*b8 + 8*b7 + 5*b4 - 7*b3 - 2*b2 - 3*b1) * q^66 + (-4*b12 - 6*b9 - 12*b8 + 3*b7 - 2*b4 - 8*b3 + 10*b2 - 5*b1) * q^67 + (2*b12 + 2*b9 - 4*b8 - 6*b7 - 4*b2) * q^68 + (-3*b15 + 6*b14 - 2*b13 + 2*b12 - b11 + 6*b10 - 3*b9 + 9*b8 + 6*b7 - 7*b6 - 5*b5 - 5*b4 - 5*b3 - 2*b2 - 4*b1 - 21) * q^69 + (-2*b13 + b12 + b10 + b7 - 3*b6 - b4 - b2 - 8) * q^70 + (-b15 + b14 - 4*b13 + 3*b12 - 2*b11 - 2*b10 + 3*b7 + 16*b6 + 13*b5 - 3*b4 - 3*b2 + 16) * q^71 + (-2*b15 + 2*b14 - 2*b13 + 2*b11 + 2*b10 - 8*b6 - 6) * q^72 + (3*b15 - b14 - 3*b13 + 3*b12 - b11 - 7*b10 + 3*b7 - 9*b6 + 7*b5 - 3*b4 - 3*b2 - 3) * q^73 + (-5*b12 - 3*b9 - 3*b8 + 6*b4 + 2*b3 - 14*b2 - 2*b1) * q^74 + 5*b5 * q^75 + (2*b9 + 8*b8 - 2*b7 - 2*b4 - 4*b3 + 6*b2 + 6*b1) * q^76 + (-8*b15 + 4*b14 - 2*b13 - 2*b12 + 2*b11 + 4*b10 - 2*b7 - 2*b6 + 2*b5 + 2*b4 + 2*b2 - 62) * q^77 + (-b15 + 9*b14 - b13 + 4*b12 + b11 + 3*b10 + 4*b7 - 17*b6 - 2*b5 - 4*b4 - 4*b2 - 31) * q^78 + (6*b12 - b9 + 2*b8 - 17*b7 - b4 + 14*b3 - 9*b2 - 5*b1) * q^79 + 4*b2 * q^80 + (2*b15 + 4*b13 - 2*b12 + 2*b11 - 8*b10 - 2*b7 + 38*b6 + 6*b5 + 2*b4 + 2*b2 - 5) * q^81 + (-6*b15 - 2*b14 - 3*b12 - 2*b11 - 2*b10 - 3*b7 + 9*b6 + 6*b5 + 3*b4 + 3*b2 - 2) * q^82 + (-8*b12 - 3*b9 + 16*b8 - 8*b7 + 7*b4 - 4*b3 + 3*b2 - 2*b1) * q^83 + (6*b12 + 4*b9 - 2*b8 - 2*b7 - 8*b4 + 10*b3 + 2*b2 - 4*b1) * q^84 + (3*b15 + 2*b13 - b12 + 3*b11 + 2*b10 - b7 + 5*b6 - 3*b5 + b4 + b2 + 2) * q^85 + (b12 - 5*b9 - 11*b8 - 4*b7 - b4 - b3 - 2*b2 - 9*b1) * q^86 + (6*b15 - 4*b14 + 2*b13 - 5*b12 + 10*b11 + 6*b10 - 5*b7 - 8*b6 + 11*b5 + 5*b4 + 5*b2 - 14) * q^87 + (-2*b12 - 2*b9 + 2*b8 - 2*b4 - 2*b3 + 4*b2 + 2*b1) * q^88 + (3*b12 - 10*b9 - 7*b8 - 7*b7 + 10*b4 - 9*b3 - 13*b2 - 4*b1) * q^89 + (b12 - 6*b8 - b7 + 4*b4 + b3 - 3*b2 - b1) * q^90 + (-4*b12 - 3*b9 - 2*b8 + b7 - 9*b4 + 16*b3 + 3*b2 + 3*b1) * q^91 + (2*b15 - 2*b14 - 2*b13 - 2*b10 - 2*b9 + 4*b8 + 2*b7 + 10*b6 + 2*b5 - 6*b3 - 4*b2 + 2*b1 + 2) * q^92 + (3*b15 - b14 + 3*b13 + 3*b12 - 7*b11 - 3*b10 + 3*b7 + 41*b6 + 7*b5 - 3*b4 - 3*b2 + 57) * q^93 + (6*b15 - 2*b14 - 2*b13 + 4*b12 - 2*b11 - 5*b10 + 4*b7 + 10*b6 - 4*b5 - 4*b4 - 4*b2 - 10) * q^94 + (b14 - 3*b12 + 7*b11 + 4*b10 - 3*b7 - 11*b6 + 3*b5 + 3*b4 + 3*b2 - 14) * q^95 - 4*b10 * q^96 + (-3*b12 + 10*b9 - b8 - 7*b7 - 6*b4 + 3*b3 - 9*b2 + 10*b1) * q^97 + (-5*b15 + b14 - 3*b13 + b12 - 3*b11 - b10 + b7 + b6 + 2*b5 - b4 - b2 + 17) * q^98 + (-6*b9 - 16*b8 + 4*b7 + 14*b4 - 8*b3 - 38*b2 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10})$$ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 $$16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98}+O(q^{100})$$ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 + 24 * q^13 + 64 * q^16 - 32 * q^18 + 4 * q^23 - 16 * q^24 - 80 * q^25 + 96 * q^26 - 96 * q^27 - 108 * q^29 - 116 * q^31 + 60 * q^35 + 128 * q^36 + 248 * q^39 - 156 * q^41 - 124 * q^46 - 128 * q^47 - 28 * q^49 + 48 * q^52 + 224 * q^54 + 160 * q^58 + 204 * q^59 + 64 * q^62 + 128 * q^64 - 268 * q^69 - 120 * q^70 + 236 * q^71 - 64 * q^72 - 112 * q^73 - 936 * q^77 - 432 * q^78 - 136 * q^81 - 64 * q^82 + 60 * q^85 - 152 * q^87 + 8 * q^92 + 856 * q^93 - 216 * q^94 - 160 * q^95 - 32 * q^96 + 256 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + \cdots + 76734676377 \nu ) / 1473983980$$ (249266*v^15 + 19450521*v^13 + 540300528*v^11 + 7075107694*v^9 + 46294879486*v^7 + 143284317912*v^5 + 176966386308*v^3 + 76734676377*v) / 1473983980 $$\beta_{3}$$ $$=$$ $$( - 3655930587 \nu^{15} - 279912880683 \nu^{13} - 7519929098840 \nu^{11} - 93142815057018 \nu^{9} - 550737169153146 \nu^{7} + \cdots + 192163143584185 \nu ) / 13607820103360$$ (-3655930587*v^15 - 279912880683*v^13 - 7519929098840*v^11 - 93142815057018*v^9 - 550737169153146*v^7 - 1359775399344612*v^5 - 747373718410547*v^3 + 192163143584185*v) / 13607820103360 $$\beta_{4}$$ $$=$$ $$( - 2028139777 \nu^{15} - 159386627629 \nu^{13} - 4483504013264 \nu^{11} - 59956611391678 \nu^{9} - 407018725514790 \nu^{7} + \cdots - 890813001228721 \nu ) / 3401955025840$$ (-2028139777*v^15 - 159386627629*v^13 - 4483504013264*v^11 - 59956611391678*v^9 - 407018725514790*v^7 - 1350423473295780*v^5 - 1906976451293273*v^3 - 890813001228721*v) / 3401955025840 $$\beta_{5}$$ $$=$$ $$( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680$$ (3266665611*v^14 + 251231034675*v^12 + 6803101750504*v^10 + 85389448480554*v^8 + 517393068635482*v^6 + 1353800445494324*v^4 + 988318890938067*v^2 + 38708131230591) / 6803910051680 $$\beta_{6}$$ $$=$$ $$( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + \cdots + 885659459 ) / 505359680$$ (328071*v^14 + 25254583*v^12 + 684448696*v^10 + 8586649314*v^8 + 51832164194*v^6 + 133890163508*v^4 + 92129883455*v^2 + 885659459) / 505359680 $$\beta_{7}$$ $$=$$ $$( 651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + \cdots - 93749086185413 \nu ) / 485993575120$$ (651909559*v^15 + 49847669703*v^13 + 1334253632328*v^11 + 16362993734546*v^9 + 94189913828650*v^7 + 212449351122740*v^5 + 42951096904351*v^3 - 93749086185413*v) / 485993575120 $$\beta_{8}$$ $$=$$ $$( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8504894308903 \nu ) / 114351429440$$ (191987899*v^15 + 14808207659*v^13 + 402812007992*v^11 + 5087217548666*v^9 + 31116072823834*v^7 + 83034850253508*v^5 + 65752777188627*v^3 + 8504894308903*v) / 114351429440 $$\beta_{9}$$ $$=$$ $$( - 229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} + \cdots - 4166619657821 \nu ) / 121498393780$$ (-229620557*v^15 - 17699955559*v^13 - 480902133464*v^11 - 6059544792898*v^9 - 36877647447520*v^7 - 97036544153580*v^5 - 71790826408533*v^3 - 4166619657821*v) / 121498393780 $$\beta_{10}$$ $$=$$ $$( 14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + \cdots - 44834433467221 ) / 6803910051680$$ (14365389855*v^14 + 1105876709199*v^12 + 29972322851736*v^10 + 375976473404450*v^8 + 2268003524367106*v^6 + 5840712005282452*v^4 + 3940822109052359*v^2 - 44834433467221) / 6803910051680 $$\beta_{11}$$ $$=$$ $$( 45221309975 \nu^{15} - 21002456926 \nu^{14} + 3479923444575 \nu^{13} - 1617708410646 \nu^{12} + 94261526625960 \nu^{11} + \cdots - 82437933075262 ) / 13607820103360$$ (45221309975*v^15 - 21002456926*v^14 + 3479923444575*v^13 - 1617708410646*v^12 + 94261526625960*v^11 - 43888163873888*v^10 + 1181548675553650*v^9 - 551450943599924*v^8 + 7119963817216130*v^7 - 3335789410501956*v^6 + 18295459522336580*v^5 - 8640632102007032*v^4 + 12205236948457855*v^3 - 5987597192457438*v^2 - 277533627658085*v - 82437933075262) / 13607820103360 $$\beta_{12}$$ $$=$$ $$( 33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + \cdots - 392465157363513 \nu ) / 6803910051680$$ (33188908451*v^15 + 2553066418411*v^13 + 69108994984088*v^11 + 865212237602154*v^9 + 5200964736292826*v^7 + 13281722071508452*v^5 + 8606845528408123*v^3 - 392465157363513*v) / 6803910051680 $$\beta_{13}$$ $$=$$ $$( 45221309975 \nu^{15} - 28278344419 \nu^{14} + 3479923444575 \nu^{13} - 2174316072571 \nu^{12} + 94261526625960 \nu^{11} + \cdots + 28012979344985 ) / 13607820103360$$ (45221309975*v^15 - 28278344419*v^14 + 3479923444575*v^13 - 2174316072571*v^12 + 94261526625960*v^11 - 58816507643080*v^10 + 1181548675553650*v^9 - 735781705952346*v^8 + 7119963817216130*v^7 - 4420040893973322*v^6 + 18295459522336580*v^5 - 11297228352398164*v^4 + 12205236948457855*v^3 - 7477536169138859*v^2 - 277533627658085*v + 28012979344985) / 13607820103360 $$\beta_{14}$$ $$=$$ $$( - 45221309975 \nu^{15} + 40620036785 \nu^{14} - 3479923444575 \nu^{13} + 3124787270393 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 6504254102973 ) / 13607820103360$$ (-45221309975*v^15 + 40620036785*v^14 - 3479923444575*v^13 + 3124787270393*v^12 - 94261526625960*v^11 + 84613633553592*v^10 - 1181548675553650*v^9 + 1060681057319710*v^8 - 7119963817216130*v^7 + 6398166721254382*v^6 - 18295459522336580*v^5 + 16500140438754044*v^4 - 12205236948457855*v^3 + 11210557588727913*v^2 + 277533627658085*v + 6504254102973) / 13607820103360 $$\beta_{15}$$ $$=$$ $$( - 45221309975 \nu^{15} + 68821388859 \nu^{14} - 3479923444575 \nu^{13} + 5299632441843 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 78181554830047 ) / 13607820103360$$ (-45221309975*v^15 + 68821388859*v^14 - 3479923444575*v^13 + 5299632441843*v^12 - 94261526625960*v^11 + 143717482361288*v^10 - 1181548675553650*v^9 + 1804651216508106*v^8 - 7119963817216130*v^7 + 10906087984591930*v^6 - 18295459522336580*v^5 + 28188544007262900*v^4 - 12205236948457855*v^3 + 19272909387076131*v^2 + 277533627658085*v + 78181554830047) / 13607820103360
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{14} + \beta_{12} - \beta_{11} - 3\beta_{10} + \beta_{7} + 5\beta_{6} - 3\beta_{5} - \beta_{4} - \beta_{2} - 18 ) / 2$$ (b14 + b12 - b11 - 3*b10 + b7 + 5*b6 - 3*b5 - b4 - b2 - 18) / 2 $$\nu^{3}$$ $$=$$ $$( 7\beta_{12} + 5\beta_{9} - 19\beta_{8} + 2\beta_{7} - 7\beta_{4} + 5\beta_{3} + 10\beta_{2} - 25\beta_1 ) / 2$$ (7*b12 + 5*b9 - 19*b8 + 2*b7 - 7*b4 + 5*b3 + 10*b2 - 25*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 31 \beta_{12} + 43 \beta_{11} + 79 \beta_{10} - 31 \beta_{7} - 179 \beta_{6} + 117 \beta_{5} + 31 \beta_{4} + 31 \beta_{2} + 400 ) / 2$$ (14*b15 - 47*b14 - 14*b13 - 31*b12 + 43*b11 + 79*b10 - 31*b7 - 179*b6 + 117*b5 + 31*b4 + 31*b2 + 400) / 2 $$\nu^{5}$$ $$=$$ $$( - 349 \beta_{12} - 193 \beta_{9} + 929 \beta_{8} - 28 \beta_{7} + 313 \beta_{4} - 223 \beta_{3} - 342 \beta_{2} + 834 \beta_1 ) / 2$$ (-349*b12 - 193*b9 + 929*b8 - 28*b7 + 313*b4 - 223*b3 - 342*b2 + 834*b1) / 2 $$\nu^{6}$$ $$=$$ $$- 359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 543 \beta_{12} - 872 \beta_{11} - 1235 \beta_{10} + 543 \beta_{7} + 3226 \beta_{6} - 2129 \beta_{5} - 543 \beta_{4} - 543 \beta_{2} - 5998$$ -359*b15 + 898*b14 + 325*b13 + 543*b12 - 872*b11 - 1235*b10 + 543*b7 + 3226*b6 - 2129*b5 - 543*b4 - 543*b2 - 5998 $$\nu^{7}$$ $$=$$ $$( 13910 \beta_{12} + 6966 \beta_{9} - 36654 \beta_{8} + 52 \beta_{7} - 11798 \beta_{4} + 8514 \beta_{3} + 12044 \beta_{2} - 29577 \beta_1 ) / 2$$ (13910*b12 + 6966*b9 - 36654*b8 + 52*b7 - 11798*b4 + 8514*b3 + 12044*b2 - 29577*b1) / 2 $$\nu^{8}$$ $$=$$ $$( 28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 39353 \beta_{12} + 66439 \beta_{11} + 84671 \beta_{10} - 39353 \beta_{7} - 232271 \beta_{6} + 154427 \beta_{5} + 39353 \beta_{4} + \cdots + 406476 ) / 2$$ (28822*b15 - 66115*b14 - 25026*b13 - 39353*b12 + 66439*b11 + 84671*b10 - 39353*b7 - 232271*b6 + 154427*b5 + 39353*b4 + 39353*b2 + 406476) / 2 $$\nu^{9}$$ $$=$$ $$( - 520505 \beta_{12} - 250339 \beta_{9} + 1366801 \beta_{8} + 12974 \beta_{7} + 430885 \beta_{4} - 313151 \beta_{3} - 434946 \beta_{2} + 1065837 \beta_1 ) / 2$$ (-520505*b12 - 250339*b9 + 1366801*b8 + 12974*b7 + 430885*b4 - 313151*b3 - 434946*b2 + 1065837*b1) / 2 $$\nu^{10}$$ $$=$$ $$( - 1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 1431797 \beta_{12} - 2458153 \beta_{11} - 3010165 \beta_{10} + 1431797 \beta_{7} + 8384705 \beta_{6} + \cdots - 14401944 ) / 2$$ (-1081278*b15 + 2410481*b14 + 923762*b13 + 1431797*b12 - 2458153*b11 - 3010165*b10 + 1431797*b7 + 8384705*b6 - 5600419*b5 - 1431797*b4 - 1431797*b2 - 14401944) / 2 $$\nu^{11}$$ $$=$$ $$( 19083917 \beta_{12} + 9029821 \beta_{9} - 50060453 \beta_{8} - 679532 \beta_{7} - 15643385 \beta_{4} + 11400387 \beta_{3} + 15800198 \beta_{2} - 38587470 \beta_1 ) / 2$$ (19083917*b12 + 9029821*b9 - 50060453*b8 - 679532*b7 - 15643385*b4 + 11400387*b3 + 15800198*b2 - 38587470*b1) / 2 $$\nu^{12}$$ $$=$$ $$19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 26024800 \beta_{12} + 44933550 \beta_{11} + 54218408 \beta_{10} - 26024800 \beta_{7} - 151719058 \beta_{6} + \cdots + 259136963$$ 19844966*b15 - 43809734*b14 - 16848718*b13 - 26024800*b12 + 44933550*b11 + 54218408*b10 - 26024800*b7 - 151719058*b6 + 101571480*b5 + 26024800*b4 + 26024800*b2 + 259136963 $$\nu^{13}$$ $$=$$ $$( - 694927048 \beta_{12} - 326698624 \beta_{9} + 1822403440 \beta_{8} + 27487912 \beta_{7} + 567460000 \beta_{4} - 413915040 \beta_{3} - 574139472 \beta_{2} + 1399069645 \beta_1 ) / 2$$ (-694927048*b12 - 326698624*b9 + 1822403440*b8 + 27487912*b7 + 567460000*b4 - 413915040*b3 - 574139472*b2 + 1399069645*b1) / 2 $$\nu^{14}$$ $$=$$ $$( - 1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 1890378909 \beta_{12} - 3270129841 \beta_{11} - 3924573439 \beta_{10} + \cdots - 18751391662 ) / 2$$ (-1446079476*b15 + 3181503425*b14 + 1224795972*b13 + 1890378909*b12 - 3270129841*b11 - 3924573439*b10 + 1890378909*b7 + 10997064301*b6 - 7369713143*b5 - 1890378909*b4 - 1890378909*b2 - 18751391662) / 2 $$\nu^{15}$$ $$=$$ $$( 25246432839 \beta_{12} + 11839142477 \beta_{9} - 66202882547 \beta_{8} - 1035217254 \beta_{7} - 20585412727 \beta_{4} + 15018968005 \beta_{3} + \cdots - 50751416961 \beta_1 ) / 2$$ (25246432839*b12 + 11839142477*b9 - 66202882547*b8 - 1035217254*b7 - 20585412727*b4 + 15018968005*b3 + 20850847362*b2 - 50751416961*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 − 0.0431371i 0.0431371i 2.98291i − 2.98291i − 1.01877i 1.01877i 3.47734i − 3.47734i 6.02373i − 6.02373i − 3.68124i 3.68124i − 2.26343i 2.26343i − 1.00527i 1.00527i
−1.41421 −5.41949 2.00000 2.23607i 7.66432 8.24199i −2.82843 20.3709 3.16228i
91.2 −1.41421 −5.41949 2.00000 2.23607i 7.66432 8.24199i −2.82843 20.3709 3.16228i
91.3 −1.41421 −0.278523 2.00000 2.23607i 0.393890 8.51262i −2.82843 −8.92243 3.16228i
91.4 −1.41421 −0.278523 2.00000 2.23607i 0.393890 8.51262i −2.82843 −8.92243 3.16228i
91.5 −1.41421 2.34854 2.00000 2.23607i −3.32134 7.61815i −2.82843 −3.48436 3.16228i
91.6 −1.41421 2.34854 2.00000 2.23607i −3.32134 7.61815i −2.82843 −3.48436 3.16228i
91.7 −1.41421 4.76369 2.00000 2.23607i −6.73687 7.05858i −2.82843 13.6927 3.16228i
91.8 −1.41421 4.76369 2.00000 2.23607i −6.73687 7.05858i −2.82843 13.6927 3.16228i
91.9 1.41421 −3.79379 2.00000 2.23607i −5.36524 7.10180i 2.82843 5.39287 3.16228i
91.10 1.41421 −3.79379 2.00000 2.23607i −5.36524 7.10180i 2.82843 5.39287 3.16228i
91.11 1.41421 −3.36596 2.00000 2.23607i −4.76019 1.16919i 2.82843 2.32968 3.16228i
91.12 1.41421 −3.36596 2.00000 2.23607i −4.76019 1.16919i 2.82843 2.32968 3.16228i
91.13 1.41421 1.43837 2.00000 2.23607i 2.03417 10.1866i 2.82843 −6.93108 3.16228i
91.14 1.41421 1.43837 2.00000 2.23607i 2.03417 10.1866i 2.82843 −6.93108 3.16228i
91.15 1.41421 4.30716 2.00000 2.23607i 6.09125 1.47532i 2.82843 9.55167 3.16228i
91.16 1.41421 4.30716 2.00000 2.23607i 6.09125 1.47532i 2.82843 9.55167 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.d.a 16
3.b odd 2 1 2070.3.c.a 16
4.b odd 2 1 1840.3.k.d 16
5.b even 2 1 1150.3.d.b 16
5.c odd 4 2 1150.3.c.c 32
23.b odd 2 1 inner 230.3.d.a 16
69.c even 2 1 2070.3.c.a 16
92.b even 2 1 1840.3.k.d 16
115.c odd 2 1 1150.3.d.b 16
115.e even 4 2 1150.3.c.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a 16 1.a even 1 1 trivial
230.3.d.a 16 23.b odd 2 1 inner
1150.3.c.c 32 5.c odd 4 2
1150.3.c.c 32 115.e even 4 2
1150.3.d.b 16 5.b even 2 1
1150.3.d.b 16 115.c odd 2 1
1840.3.k.d 16 4.b odd 2 1
1840.3.k.d 16 92.b even 2 1
2070.3.c.a 16 3.b odd 2 1
2070.3.c.a 16 69.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(230, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{8}$$
$3$ $$(T^{8} - 52 T^{6} + 16 T^{5} + 829 T^{4} + \cdots + 1336)^{2}$$
$5$ $$(T^{2} + 5)^{8}$$
$7$ $$T^{16} + 406 T^{14} + \cdots + 221645107264$$
$11$ $$T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44$$
$13$ $$(T^{8} - 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2}$$
$17$ $$T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00$$
$19$ $$T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24$$
$23$ $$T^{16} - 4 T^{15} + \cdots + 61\!\cdots\!61$$
$29$ $$(T^{8} + 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2}$$
$31$ $$(T^{8} + 58 T^{7} + \cdots + 229759835104)^{2}$$
$37$ $$T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04$$
$41$ $$(T^{8} + 78 T^{7} + \cdots - 212194449184)^{2}$$
$43$ $$T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24$$
$47$ $$(T^{8} + 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2}$$
$53$ $$T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64$$
$59$ $$(T^{8} - 102 T^{7} + \cdots - 42922529206784)^{2}$$
$61$ $$T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84$$
$67$ $$T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04$$
$71$ $$(T^{8} - 118 T^{7} + \cdots - 24390990617024)^{2}$$
$73$ $$(T^{8} + 56 T^{7} + \cdots + 1317400530416)^{2}$$
$79$ $$T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24$$
$83$ $$T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64$$
$89$ $$T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00$$
$97$ $$T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04$$