# Properties

 Label 930.2.h.d Level $930$ Weight $2$ Character orbit 930.h Analytic conductor $7.426$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4x^{14} + 6x^{12} + 36x^{10} - 142x^{8} + 324x^{6} + 486x^{4} - 2916x^{2} + 6561$$ x^16 - 4*x^14 + 6*x^12 + 36*x^10 - 142*x^8 + 324*x^6 + 486*x^4 - 2916*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{31}]$$ Coefficient ring index: $$2^{5}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4x^{14} + 6x^{12} + 36x^{10} - 142x^{8} + 324x^{6} + 486x^{4} - 2916x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$( 5\nu^{15} - 101\nu^{13} + 111\nu^{11} - 63\nu^{9} - 4355\nu^{7} + 2187\nu^{5} - 22113\nu^{3} - 57591\nu ) / 69984$$ (5*v^15 - 101*v^13 + 111*v^11 - 63*v^9 - 4355*v^7 + 2187*v^5 - 22113*v^3 - 57591*v) / 69984 $$\beta_{4}$$ $$=$$ $$( 5\nu^{14} - 101\nu^{12} + 111\nu^{10} - 63\nu^{8} - 4355\nu^{6} + 2187\nu^{4} + 1215\nu^{2} - 80919 ) / 23328$$ (5*v^14 - 101*v^12 + 111*v^10 - 63*v^8 - 4355*v^6 + 2187*v^4 + 1215*v^2 - 80919) / 23328 $$\beta_{5}$$ $$=$$ $$( \nu^{15} - 4\nu^{13} + 6\nu^{11} + 36\nu^{9} - 142\nu^{7} + 324\nu^{5} + 486\nu^{3} - 2916\nu ) / 2187$$ (v^15 - 4*v^13 + 6*v^11 + 36*v^9 - 142*v^7 + 324*v^5 + 486*v^3 - 2916*v) / 2187 $$\beta_{6}$$ $$=$$ $$( 14\nu^{14} - 65\nu^{12} - 42\nu^{10} + 369\nu^{8} - 1826\nu^{6} + 711\nu^{4} + 5022\nu^{2} - 45927 ) / 11664$$ (14*v^14 - 65*v^12 - 42*v^10 + 369*v^8 - 1826*v^6 + 711*v^4 + 5022*v^2 - 45927) / 11664 $$\beta_{7}$$ $$=$$ $$( -\nu^{14} + 4\nu^{12} - 6\nu^{10} - 36\nu^{8} + 142\nu^{6} - 324\nu^{4} - 486\nu^{2} + 2916 ) / 729$$ (-v^14 + 4*v^12 - 6*v^10 - 36*v^8 + 142*v^6 - 324*v^4 - 486*v^2 + 2916) / 729 $$\beta_{8}$$ $$=$$ $$( -23\nu^{14} - 61\nu^{12} - 93\nu^{10} - 207\nu^{8} - 1999\nu^{6} - 2493\nu^{4} - 10125\nu^{2} - 57591 ) / 11664$$ (-23*v^14 - 61*v^12 - 93*v^10 - 207*v^8 - 1999*v^6 - 2493*v^4 - 10125*v^2 - 57591) / 11664 $$\beta_{9}$$ $$=$$ $$( -19\nu^{14} + 22\nu^{12} + 21\nu^{10} - 684\nu^{8} + 997\nu^{6} + 54\nu^{4} - 14499\nu^{2} + 20412 ) / 5832$$ (-19*v^14 + 22*v^12 + 21*v^10 - 684*v^8 + 997*v^6 + 54*v^4 - 14499*v^2 + 20412) / 5832 $$\beta_{10}$$ $$=$$ $$( -14\nu^{15} + 65\nu^{13} + 42\nu^{11} - 369\nu^{9} + 1826\nu^{7} - 711\nu^{5} - 5022\nu^{3} + 45927\nu ) / 11664$$ (-14*v^15 + 65*v^13 + 42*v^11 - 369*v^9 + 1826*v^7 - 711*v^5 - 5022*v^3 + 45927*v) / 11664 $$\beta_{11}$$ $$=$$ $$( -61\nu^{15} - 17\nu^{13} - 51\nu^{11} - 1575\nu^{9} - 5\nu^{7} - 2385\nu^{5} - 39123\nu^{3} - 5103\nu ) / 34992$$ (-61*v^15 - 17*v^13 - 51*v^11 - 1575*v^9 - 5*v^7 - 2385*v^5 - 39123*v^3 - 5103*v) / 34992 $$\beta_{12}$$ $$=$$ $$( 7\nu^{15} - 14\nu^{13} - 23\nu^{11} + 210\nu^{9} - 625\nu^{7} + 442\nu^{5} + 4113\nu^{3} - 15390\nu ) / 3888$$ (7*v^15 - 14*v^13 - 23*v^11 + 210*v^9 - 625*v^7 + 442*v^5 + 4113*v^3 - 15390*v) / 3888 $$\beta_{13}$$ $$=$$ $$( 85 \nu^{15} - 168 \nu^{14} - 43 \nu^{13} - 192 \nu^{12} - 273 \nu^{11} + 504 \nu^{10} + 3951 \nu^{9} - 3456 \nu^{8} - 4051 \nu^{7} - 1416 \nu^{6} + 2133 \nu^{5} - 1728 \nu^{4} + \cdots - 117369 \nu ) / 69984$$ (85*v^15 - 168*v^14 - 43*v^13 - 192*v^12 - 273*v^11 + 504*v^10 + 3951*v^9 - 3456*v^8 - 4051*v^7 - 1416*v^6 + 2133*v^5 - 1728*v^4 + 64719*v^3 - 138024*v^2 - 117369*v) / 69984 $$\beta_{14}$$ $$=$$ $$( - 179 \nu^{15} - 168 \nu^{14} + 635 \nu^{13} - 192 \nu^{12} + 303 \nu^{11} + 504 \nu^{10} - 6039 \nu^{9} - 3456 \nu^{8} + 23717 \nu^{7} - 1416 \nu^{6} - 10773 \nu^{5} + \cdots + 508113 \nu ) / 69984$$ (-179*v^15 - 168*v^14 + 635*v^13 - 192*v^12 + 303*v^11 + 504*v^10 - 6039*v^9 - 3456*v^8 + 23717*v^7 - 1416*v^6 - 10773*v^5 - 1728*v^4 - 120609*v^3 - 138024*v^2 + 508113*v) / 69984 $$\beta_{15}$$ $$=$$ $$( - 179 \nu^{15} + 168 \nu^{14} + 635 \nu^{13} + 192 \nu^{12} + 303 \nu^{11} - 504 \nu^{10} - 6039 \nu^{9} + 3456 \nu^{8} + 23717 \nu^{7} + 1416 \nu^{6} - 10773 \nu^{5} + \cdots + 508113 \nu ) / 69984$$ (-179*v^15 + 168*v^14 + 635*v^13 + 192*v^12 + 303*v^11 - 504*v^10 - 6039*v^9 + 3456*v^8 + 23717*v^7 + 1416*v^6 - 10773*v^5 + 1728*v^4 - 120609*v^3 + 138024*v^2 + 508113*v) / 69984
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$-\beta_{15} - \beta_{13} + \beta_{10} - 2\beta_{3}$$ -b15 - b13 + b10 - 2*b3 $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{14} + 2\beta_{9} + \beta_{8} - 3\beta_{7} - 2\beta_{4} + 3$$ b15 - b14 + 2*b9 + b8 - 3*b7 - 2*b4 + 3 $$\nu^{5}$$ $$=$$ $$\beta_{15} + 2\beta_{14} - \beta_{13} + 5\beta_{12} + 2\beta_{11} + 8\beta_{5} - 4\beta_{3} + 4\beta_1$$ b15 + 2*b14 - b13 + 5*b12 + 2*b11 + 8*b5 - 4*b3 + 4*b1 $$\nu^{6}$$ $$=$$ $$-3\beta_{15} + 3\beta_{14} - 4\beta_{8} + \beta_{7} + 8\beta_{6} - 8\beta_{4} + 6\beta_{2} - 20$$ -3*b15 + 3*b14 - 4*b8 + b7 + 8*b6 - 8*b4 + 6*b2 - 20 $$\nu^{7}$$ $$=$$ $$5\beta_{15} + \beta_{14} + 4\beta_{13} - 8\beta_{12} - 2\beta_{11} - 16\beta_{10} + 7\beta_{5} - 14\beta_{3} - 8\beta_1$$ 5*b15 + b14 + 4*b13 - 8*b12 - 2*b11 - 16*b10 + 7*b5 - 14*b3 - 8*b1 $$\nu^{8}$$ $$=$$ $$-3\beta_{15} + 3\beta_{14} - 12\beta_{9} + 8\beta_{8} - 10\beta_{7} - 16\beta_{6} - 16\beta_{4} - 10\beta_{2} + 3$$ -3*b15 + 3*b14 - 12*b9 + 8*b8 - 10*b7 - 16*b6 - 16*b4 - 10*b2 + 3 $$\nu^{9}$$ $$=$$ $$29 \beta_{15} - 23 \beta_{14} + 52 \beta_{13} - 32 \beta_{12} + 10 \beta_{11} - 16 \beta_{10} + 16 \beta_{5} + 10 \beta_{3} + 11 \beta_1$$ 29*b15 - 23*b14 + 52*b13 - 32*b12 + 10*b11 - 16*b10 + 16*b5 + 10*b3 + 11*b1 $$\nu^{10}$$ $$=$$ $$- 39 \beta_{15} + 39 \beta_{14} - 48 \beta_{9} - 52 \beta_{8} + 26 \beta_{7} - 40 \beta_{6} + 56 \beta_{4} + 21 \beta_{2} - 156$$ -39*b15 + 39*b14 - 48*b9 - 52*b8 + 26*b7 - 40*b6 + 56*b4 + 21*b2 - 156 $$\nu^{11}$$ $$=$$ $$- 38 \beta_{15} - 35 \beta_{14} - 3 \beta_{13} - 152 \beta_{12} - 74 \beta_{11} + 39 \beta_{10} + 4 \beta_{5} + 92 \beta_{3} - 160 \beta_1$$ -38*b15 - 35*b14 - 3*b13 - 152*b12 - 74*b11 + 39*b10 + 4*b5 + 92*b3 - 160*b1 $$\nu^{12}$$ $$=$$ $$112 \beta_{15} - 112 \beta_{14} + 2 \beta_{9} + 111 \beta_{8} - 55 \beta_{7} - 352 \beta_{6} + 162 \beta_{4} - 234 \beta_{2} - 63$$ 112*b15 - 112*b14 + 2*b9 + 111*b8 - 55*b7 - 352*b6 + 162*b4 - 234*b2 - 63 $$\nu^{13}$$ $$=$$ $$- 40 \beta_{15} + 3 \beta_{14} - 43 \beta_{13} + 339 \beta_{12} + 504 \beta_{10} - 168 \beta_{5} + 406 \beta_{3} - 336 \beta_1$$ -40*b15 + 3*b14 - 43*b13 + 339*b12 + 504*b10 - 168*b5 + 406*b3 - 336*b1 $$\nu^{14}$$ $$=$$ $$40 \beta_{15} - 40 \beta_{14} + 80 \beta_{9} - 424 \beta_{8} + 369 \beta_{7} + 544 \beta_{6} + 400 \beta_{4} - 336 \beta_{2} - 320$$ 40*b15 - 40*b14 + 80*b9 - 424*b8 + 369*b7 + 544*b6 + 400*b4 - 336*b2 - 320 $$\nu^{15}$$ $$=$$ $$- 104 \beta_{15} + 544 \beta_{14} - 648 \beta_{13} + 664 \beta_{12} - 848 \beta_{11} - 400 \beta_{10} - 683 \beta_{5} + 992 \beta_{3} - 296 \beta_1$$ -104*b15 + 544*b14 - 648*b13 + 664*b12 - 848*b11 - 400*b10 - 683*b5 + 992*b3 - 296*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
371.1
 0.206738 + 1.71967i −1.29725 + 1.14767i 1.60627 + 0.647994i −1.64143 + 0.552909i 1.64143 − 0.552909i −1.60627 − 0.647994i 1.29725 − 1.14767i −0.206738 − 1.71967i 0.206738 − 1.71967i −1.29725 − 1.14767i 1.60627 − 0.647994i −1.64143 − 0.552909i 1.64143 + 0.552909i −1.60627 + 0.647994i 1.29725 + 1.14767i −0.206738 + 1.71967i
1.00000i −1.71967 0.206738i −1.00000 1.00000i −0.206738 + 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.2 1.00000i −1.14767 + 1.29725i −1.00000 1.00000i 1.29725 + 1.14767i 3.69457 1.00000i −0.365730 2.97762i 1.00000
371.3 1.00000i −0.647994 1.60627i −1.00000 1.00000i −1.60627 + 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.4 1.00000i −0.552909 + 1.64143i −1.00000 1.00000i 1.64143 + 0.552909i −2.30718 1.00000i −2.38858 1.81512i 1.00000
371.5 1.00000i 0.552909 1.64143i −1.00000 1.00000i −1.64143 0.552909i −2.30718 1.00000i −2.38858 1.81512i 1.00000
371.6 1.00000i 0.647994 + 1.60627i −1.00000 1.00000i 1.60627 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.7 1.00000i 1.14767 1.29725i −1.00000 1.00000i −1.29725 1.14767i 3.69457 1.00000i −0.365730 2.97762i 1.00000
371.8 1.00000i 1.71967 + 0.206738i −1.00000 1.00000i 0.206738 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.9 1.00000i −1.71967 + 0.206738i −1.00000 1.00000i −0.206738 1.71967i −1.18164 1.00000i 2.91452 0.711040i 1.00000
371.10 1.00000i −1.14767 1.29725i −1.00000 1.00000i 1.29725 1.14767i 3.69457 1.00000i −0.365730 + 2.97762i 1.00000
371.11 1.00000i −0.647994 + 1.60627i −1.00000 1.00000i −1.60627 0.647994i 0.794255 1.00000i −2.16021 2.08171i 1.00000
371.12 1.00000i −0.552909 1.64143i −1.00000 1.00000i 1.64143 0.552909i −2.30718 1.00000i −2.38858 + 1.81512i 1.00000
371.13 1.00000i 0.552909 + 1.64143i −1.00000 1.00000i −1.64143 + 0.552909i −2.30718 1.00000i −2.38858 + 1.81512i 1.00000
371.14 1.00000i 0.647994 1.60627i −1.00000 1.00000i 1.60627 + 0.647994i 0.794255 1.00000i −2.16021 2.08171i 1.00000
371.15 1.00000i 1.14767 + 1.29725i −1.00000 1.00000i −1.29725 + 1.14767i 3.69457 1.00000i −0.365730 + 2.97762i 1.00000
371.16 1.00000i 1.71967 0.206738i −1.00000 1.00000i 0.206738 + 1.71967i −1.18164 1.00000i 2.91452 0.711040i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 371.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.d 16
3.b odd 2 1 inner 930.2.h.d 16
31.b odd 2 1 inner 930.2.h.d 16
93.c even 2 1 inner 930.2.h.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.d 16 1.a even 1 1 trivial
930.2.h.d 16 3.b odd 2 1 inner
930.2.h.d 16 31.b odd 2 1 inner
930.2.h.d 16 93.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - T_{7}^{3} - 10T_{7}^{2} - 2T_{7} + 8$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$T^{16} + 4 T^{14} + 6 T^{12} - 36 T^{10} + \cdots + 6561$$
$5$ $$(T^{2} + 1)^{8}$$
$7$ $$(T^{4} - T^{3} - 10 T^{2} - 2 T + 8)^{4}$$
$11$ $$(T^{8} - 43 T^{6} + 134 T^{4} - 86 T^{2} + \cdots + 8)^{2}$$
$13$ $$(T^{8} + 20 T^{6} + 114 T^{4} + 216 T^{2} + \cdots + 128)^{2}$$
$17$ $$(T^{8} - 60 T^{6} + 1184 T^{4} + \cdots + 8192)^{2}$$
$19$ $$(T^{4} - 5 T^{3} - 36 T^{2} + 128 T + 256)^{4}$$
$23$ $$(T^{8} - 27 T^{6} + 228 T^{4} - 640 T^{2} + \cdots + 512)^{2}$$
$29$ $$(T^{8} - 152 T^{6} + 7154 T^{4} + \cdots + 61952)^{2}$$
$31$ $$(T^{8} + 4 T^{7} - 56 T^{6} - 12 T^{5} + \cdots + 923521)^{2}$$
$37$ $$(T^{8} + 264 T^{6} + 22258 T^{4} + \cdots + 3463712)^{2}$$
$41$ $$(T^{8} + 100 T^{6} + 2340 T^{4} + \cdots + 1024)^{2}$$
$43$ $$(T^{8} + 135 T^{6} + 5994 T^{4} + \cdots + 209952)^{2}$$
$47$ $$(T^{8} + 176 T^{6} + 6580 T^{4} + \cdots + 135424)^{2}$$
$53$ $$(T^{8} - 279 T^{6} + 21810 T^{4} + \cdots + 4892192)^{2}$$
$59$ $$(T^{8} + 272 T^{6} + 22336 T^{4} + \cdots + 4734976)^{2}$$
$61$ $$(T^{8} + 192 T^{6} + 7842 T^{4} + \cdots + 247808)^{2}$$
$67$ $$(T^{4} + 6 T^{3} - 128 T^{2} - 336 T - 64)^{4}$$
$71$ $$(T^{8} + 357 T^{6} + 33388 T^{4} + \cdots + 7744)^{2}$$
$73$ $$(T^{8} + 287 T^{6} + 6352 T^{4} + \cdots + 2048)^{2}$$
$79$ $$(T^{8} + 419 T^{6} + 59332 T^{4} + \cdots + 59710592)^{2}$$
$83$ $$(T^{8} - 420 T^{6} + 57714 T^{4} + \cdots + 31490048)^{2}$$
$89$ $$(T^{8} - 179 T^{6} + 9348 T^{4} + \cdots + 1131008)^{2}$$
$97$ $$(T^{4} + 2 T^{3} - 74 T^{2} + 20 T + 592)^{4}$$