# Properties

 Label 95.2.i.b Level $95$ Weight $2$ Character orbit 95.i Analytic conductor $0.759$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 95.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1$$ x^12 - 6*x^10 + 29*x^8 - 40*x^6 + 43*x^4 - 7*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -36\nu^{10} + 174\nu^{8} - 841\nu^{6} + 258\nu^{4} - 42\nu^{2} - 2207 ) / 1205$$ (-36*v^10 + 174*v^8 - 841*v^6 + 258*v^4 - 42*v^2 - 2207) / 1205 $$\beta_{3}$$ $$=$$ $$( 138\nu^{10} - 667\nu^{8} + 3023\nu^{6} - 989\nu^{4} + 161\nu^{2} + 2636 ) / 1205$$ (138*v^10 - 667*v^8 + 3023*v^6 - 989*v^4 + 161*v^2 + 2636) / 1205 $$\beta_{4}$$ $$=$$ $$( 138\nu^{11} - 667\nu^{9} + 3023\nu^{7} - 989\nu^{5} + 161\nu^{3} + 2636\nu ) / 1205$$ (138*v^11 - 667*v^9 + 3023*v^7 - 989*v^5 + 161*v^3 + 2636*v) / 1205 $$\beta_{5}$$ $$=$$ $$( 174\nu^{11} - 841\nu^{9} + 3864\nu^{7} - 1247\nu^{5} + 203\nu^{3} + 7253\nu ) / 1205$$ (174*v^11 - 841*v^9 + 3864*v^7 - 1247*v^5 + 203*v^3 + 7253*v) / 1205 $$\beta_{6}$$ $$=$$ $$( 203\nu^{10} - 1182\nu^{8} + 5713\nu^{6} - 7279\nu^{4} + 8471\nu^{2} - 174 ) / 1205$$ (203*v^10 - 1182*v^8 + 5713*v^6 - 7279*v^4 + 8471*v^2 - 174) / 1205 $$\beta_{7}$$ $$=$$ $$( -203\nu^{11} + 1182\nu^{9} - 5713\nu^{7} + 7279\nu^{5} - 8471\nu^{3} + 174\nu ) / 1205$$ (-203*v^11 + 1182*v^9 - 5713*v^7 + 7279*v^5 - 8471*v^3 + 174*v) / 1205 $$\beta_{8}$$ $$=$$ $$( -74\nu^{10} + 438\nu^{8} - 2117\nu^{6} + 2860\nu^{4} - 3139\nu^{2} + 511 ) / 241$$ (-74*v^10 + 438*v^8 - 2117*v^6 + 2860*v^4 - 3139*v^2 + 511) / 241 $$\beta_{9}$$ $$=$$ $$( -429\nu^{10} + 2676\nu^{8} - 12934\nu^{6} + 19342\nu^{4} - 19178\nu^{2} + 3122 ) / 1205$$ (-429*v^10 + 2676*v^8 - 12934*v^6 + 19342*v^4 - 19178*v^2 + 3122) / 1205 $$\beta_{10}$$ $$=$$ $$( -429\nu^{11} + 2676\nu^{9} - 12934\nu^{7} + 19342\nu^{5} - 19178\nu^{3} + 3122\nu ) / 1205$$ (-429*v^11 + 2676*v^9 - 12934*v^7 + 19342*v^5 - 19178*v^3 + 3122*v) / 1205 $$\beta_{11}$$ $$=$$ $$( -1002\nu^{11} + 6048\nu^{9} - 29232\nu^{7} + 40921\nu^{5} - 43344\nu^{3} + 7056\nu ) / 1205$$ (-1002*v^11 + 6048*v^9 - 29232*v^7 + 40921*v^5 - 43344*v^3 + 7056*v) / 1205
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + 2\beta_{6} + \beta_{2}$$ b8 + 2*b6 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{10} - 3\beta_{7} - \beta_{5} + \beta_{4} + \beta_1$$ b11 - b10 - 3*b7 - b5 + b4 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 5\beta_{8} + 7\beta_{6} - 7$$ -b9 + 5*b8 + 7*b6 - 7 $$\nu^{5}$$ $$=$$ $$5\beta_{11} - 6\beta_{10} - 12\beta_{7} - 12\beta_1$$ 5*b11 - 6*b10 - 12*b7 - 12*b1 $$\nu^{6}$$ $$=$$ $$-6\beta_{3} - 23\beta_{2} - 29$$ -6*b3 - 23*b2 - 29 $$\nu^{7}$$ $$=$$ $$23\beta_{5} - 29\beta_{4} - 75\beta_1$$ 23*b5 - 29*b4 - 75*b1 $$\nu^{8}$$ $$=$$ $$29\beta_{9} - 104\beta_{8} - 127\beta_{6} - 29\beta_{3} - 104\beta_{2}$$ 29*b9 - 104*b8 - 127*b6 - 29*b3 - 104*b2 $$\nu^{9}$$ $$=$$ $$-104\beta_{11} + 133\beta_{10} + 231\beta_{7} + 104\beta_{5} - 133\beta_{4} - 104\beta_1$$ -104*b11 + 133*b10 + 231*b7 + 104*b5 - 133*b4 - 104*b1 $$\nu^{10}$$ $$=$$ $$133\beta_{9} - 468\beta_{8} - 566\beta_{6} + 566$$ 133*b9 - 468*b8 - 566*b6 + 566 $$\nu^{11}$$ $$=$$ $$-468\beta_{11} + 601\beta_{10} + 1034\beta_{7} + 1034\beta_1$$ -468*b11 + 601*b10 + 1034*b7 + 1034*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.352587 − 0.203566i −1.00376 − 0.579521i −1.83525 − 1.05958i 1.83525 + 1.05958i 1.00376 + 0.579521i 0.352587 + 0.203566i −0.352587 + 0.203566i −1.00376 + 0.579521i −1.83525 + 1.05958i 1.83525 − 1.05958i 1.00376 − 0.579521i 0.352587 − 0.203566i
−2.12713 1.22810i 1.35190 + 0.780522i 2.01647 + 3.49262i −1.45193 + 1.70056i −1.91712 3.32055i 4.50527i 4.99330i −0.281570 0.487693i 5.17691 1.83419i
49.2 −0.747190 0.431391i 2.66661 + 1.53957i −0.627804 1.08739i −0.0476457 2.23556i −1.32831 2.30070i 0.566520i 2.80888i 3.24054 + 5.61278i −0.928799 + 1.69094i
49.3 −0.408663 0.235942i −0.900858 0.520111i −0.888663 1.53921i −2.19202 0.441641i 0.245432 + 0.425100i 1.17540i 1.78246i −0.958970 1.66098i 0.791597 + 0.697672i
49.4 0.408663 + 0.235942i 0.900858 + 0.520111i −0.888663 1.53921i 1.47848 + 1.67752i 0.245432 + 0.425100i 1.17540i 1.78246i −0.958970 1.66098i 0.208403 + 1.03438i
49.5 0.747190 + 0.431391i −2.66661 1.53957i −0.627804 1.08739i 1.95987 1.07652i −1.32831 2.30070i 0.566520i 2.80888i 3.24054 + 5.61278i 1.92880 + 0.0411078i
49.6 2.12713 + 1.22810i −1.35190 0.780522i 2.01647 + 3.49262i −0.746759 + 2.10769i −1.91712 3.32055i 4.50527i 4.99330i −0.281570 0.487693i −4.17691 + 3.56624i
64.1 −2.12713 + 1.22810i 1.35190 0.780522i 2.01647 3.49262i −1.45193 1.70056i −1.91712 + 3.32055i 4.50527i 4.99330i −0.281570 + 0.487693i 5.17691 + 1.83419i
64.2 −0.747190 + 0.431391i 2.66661 1.53957i −0.627804 + 1.08739i −0.0476457 + 2.23556i −1.32831 + 2.30070i 0.566520i 2.80888i 3.24054 5.61278i −0.928799 1.69094i
64.3 −0.408663 + 0.235942i −0.900858 + 0.520111i −0.888663 + 1.53921i −2.19202 + 0.441641i 0.245432 0.425100i 1.17540i 1.78246i −0.958970 + 1.66098i 0.791597 0.697672i
64.4 0.408663 0.235942i 0.900858 0.520111i −0.888663 + 1.53921i 1.47848 1.67752i 0.245432 0.425100i 1.17540i 1.78246i −0.958970 + 1.66098i 0.208403 1.03438i
64.5 0.747190 0.431391i −2.66661 + 1.53957i −0.627804 + 1.08739i 1.95987 + 1.07652i −1.32831 + 2.30070i 0.566520i 2.80888i 3.24054 5.61278i 1.92880 0.0411078i
64.6 2.12713 1.22810i −1.35190 + 0.780522i 2.01647 3.49262i −0.746759 2.10769i −1.91712 + 3.32055i 4.50527i 4.99330i −0.281570 + 0.487693i −4.17691 3.56624i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.6 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
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.i.b 12
3.b odd 2 1 855.2.be.d 12
5.b even 2 1 inner 95.2.i.b 12
5.c odd 4 2 475.2.e.g 12
15.d odd 2 1 855.2.be.d 12
19.c even 3 1 inner 95.2.i.b 12
19.c even 3 1 1805.2.b.f 6
19.d odd 6 1 1805.2.b.g 6
57.h odd 6 1 855.2.be.d 12
95.h odd 6 1 1805.2.b.g 6
95.i even 6 1 inner 95.2.i.b 12
95.i even 6 1 1805.2.b.f 6
95.l even 12 2 9025.2.a.bt 6
95.m odd 12 2 475.2.e.g 12
95.m odd 12 2 9025.2.a.bu 6
285.n odd 6 1 855.2.be.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 1.a even 1 1 trivial
95.2.i.b 12 5.b even 2 1 inner
95.2.i.b 12 19.c even 3 1 inner
95.2.i.b 12 95.i even 6 1 inner
475.2.e.g 12 5.c odd 4 2
475.2.e.g 12 95.m odd 12 2
855.2.be.d 12 3.b odd 2 1
855.2.be.d 12 15.d odd 2 1
855.2.be.d 12 57.h odd 6 1
855.2.be.d 12 285.n odd 6 1
1805.2.b.f 6 19.c even 3 1
1805.2.b.f 6 95.i even 6 1
1805.2.b.g 6 19.d odd 6 1
1805.2.b.g 6 95.h odd 6 1
9025.2.a.bt 6 95.l even 12 2
9025.2.a.bu 6 95.m odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 7T_{2}^{10} + 43T_{2}^{8} - 40T_{2}^{6} + 29T_{2}^{4} - 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 7 T^{10} + 43 T^{8} - 40 T^{6} + \cdots + 1$$
$3$ $$T^{12} - 13 T^{10} + 133 T^{8} + \cdots + 625$$
$5$ $$T^{12} + 2 T^{11} + 5 T^{10} + \cdots + 15625$$
$7$ $$(T^{6} + 22 T^{4} + 35 T^{2} + 9)^{2}$$
$11$ $$(T^{3} - T^{2} - 4 T + 3)^{4}$$
$13$ $$T^{12} - 31 T^{10} + 722 T^{8} + \cdots + 81$$
$17$ $$T^{12} - 35 T^{10} + 1027 T^{8} + \cdots + 6561$$
$19$ $$(T^{6} + 6 T^{5} + 30 T^{4} + 115 T^{3} + \cdots + 6859)^{2}$$
$23$ $$T^{12} - 12 T^{10} + 137 T^{8} - 82 T^{6} + \cdots + 1$$
$29$ $$(T^{6} + 6 T^{5} + 63 T^{4} - 108 T^{3} + \cdots + 729)^{2}$$
$31$ $$(T^{3} - 15 T^{2} + 70 T - 97)^{4}$$
$37$ $$(T^{6} + 98 T^{4} + 887 T^{2} + 729)^{2}$$
$41$ $$(T^{6} + 6 T^{5} + 77 T^{4} - 240 T^{3} + \cdots + 9)^{2}$$
$43$ $$T^{12} - 127 T^{10} + 13613 T^{8} + \cdots + 194481$$
$47$ $$T^{12} - 214 T^{10} + \cdots + 47562811921$$
$53$ $$T^{12} - 99 T^{10} + \cdots + 131079601$$
$59$ $$(T^{6} - 10 T^{5} + 155 T^{4} + \cdots + 84681)^{2}$$
$61$ $$(T^{6} - T^{5} + 42 T^{4} - 185 T^{3} + \cdots + 12769)^{2}$$
$67$ $$T^{12} - 76 T^{10} + \cdots + 207360000$$
$71$ $$(T^{6} - T^{5} + 125 T^{4} + 1078 T^{3} + \cdots + 227529)^{2}$$
$73$ $$T^{12} - 236 T^{10} + \cdots + 9845600625$$
$79$ $$(T^{6} - 12 T^{5} + 176 T^{4} + \cdots + 200704)^{2}$$
$83$ $$(T^{6} + 459 T^{4} + 56302 T^{2} + \cdots + 966289)^{2}$$
$89$ $$(T^{6} - 18 T^{5} + 296 T^{4} - 648 T^{3} + \cdots + 5184)^{2}$$
$97$ $$T^{12} - 529 T^{10} + \cdots + 1534548635361$$