# Properties

 Label 475.2.e.g Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ 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: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2}) q^{4} + (\beta_{8} - 2 \beta_{6} + \beta_{2}) q^{6} + ( - 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{7} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{8} + (2 \beta_{9} + \beta_{8} - 2 \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b5 + b4) * q^3 + (-b9 + b8 - b3 + b2) * q^4 + (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_{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} + \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_{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_{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_{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_{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} + (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} + ( - 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_{3} + 1) q^{46} + (4 \beta_{11} + 4 \beta_{10} - \beta_{7} + \beta_1) q^{47} + (3 \beta_{11} - \beta_{7} + \beta_1) q^{48} + (4 \beta_{3} - 3 \beta_{2} - 1) q^{49} + (4 \beta_{9} + \beta_{8} - 8 \beta_{6} + 4 \beta_{3} + \beta_{2}) q^{51} + (7 \beta_{5} - 5 \beta_1) q^{52} + (\beta_{11} - 2 \beta_{10} + 3 \beta_{7} - 3 \beta_1) q^{53} + ( - 5 \beta_{9} + 2 \beta_{8} + \beta_{6} - 1) q^{54} + (5 \beta_{3} - 3 \beta_{2} + 6) q^{56} + ( - 2 \beta_{11} - 3 \beta_{10} + 6 \beta_{7} - \beta_{5} - 4 \beta_{4} - \beta_1) q^{57} + ( - 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1) q^{58} + (\beta_{9} - 4 \beta_{8} + 3 \beta_{6} - 3) q^{59} + ( - 2 \beta_{9} + 2 \beta_{8} + \beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{61} + ( - 7 \beta_{5} + \beta_1) q^{62} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \beta_1) q^{63} + ( - 3 \beta_{3} - 1) q^{64} + ( - 3 \beta_{9} + \beta_{8} - 3 \beta_{3} + \beta_{2}) q^{66} + (2 \beta_{11} - 4 \beta_{7} + 4 \beta_1) q^{67} + (3 \beta_{10} + 3 \beta_{4}) q^{68} + ( - 3 \beta_{3} - \beta_{2} + 4) q^{69} + ( - \beta_{9} - 5 \beta_{8}) q^{71} + ( - 3 \beta_{11} - 6 \beta_{10} + 5 \beta_{7} - 5 \beta_1) q^{72} + (2 \beta_{5} + \beta_{4} - 7 \beta_1) q^{73} + ( - 2 \beta_{9} - 3 \beta_{6} + 3) q^{74} + (4 \beta_{8} - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{76} + ( - 4 \beta_{11} + \beta_{10} + 2 \beta_{7} + 4 \beta_{5} + \beta_{4} - 4 \beta_1) q^{77} + (6 \beta_{5} + \beta_{4} - 3 \beta_1) q^{78} + ( - 4 \beta_{8} + 4 \beta_{6} - 4) q^{79} + ( - 5 \beta_{9} + 4 \beta_{6} - 4) q^{81} + (\beta_{11} + 3 \beta_{10} - 2 \beta_{7} + 2 \beta_1) q^{82} + (5 \beta_{11} - 2 \beta_{10} - 9 \beta_{7} - 5 \beta_{5} - 2 \beta_{4} + 5 \beta_1) q^{83} + (6 \beta_{3} - 5 \beta_{2} + 6) q^{84} + ( - \beta_{9} + \beta_{8} + 3 \beta_{6} - \beta_{3} + \beta_{2}) q^{86} + (9 \beta_{10} - 6 \beta_{7} + 9 \beta_{4}) q^{87} + ( - 5 \beta_{11} - \beta_{10} + 4 \beta_{7} + 5 \beta_{5} - \beta_{4} - 5 \beta_1) q^{88} + ( - 4 \beta_{8} - 6 \beta_{6} - 4 \beta_{2}) q^{89} + (3 \beta_{9} - 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{3} - 4 \beta_{2}) q^{91} + (\beta_{5} - \beta_{4} + \beta_1) q^{92} + ( - 6 \beta_{5} + 5 \beta_{4} - \beta_1) q^{93} + ( - 4 \beta_{2} + 5) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2} + 6) q^{96} + ( - 5 \beta_{5} + 7 \beta_{4} + 4 \beta_1) q^{97} + ( - 13 \beta_{5} - 4 \beta_{4} + 7 \beta_1) q^{98} + ( - 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 + (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 + 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 + (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 + (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*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 + (-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 + (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 + (-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 + (-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 + (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 + (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 + 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*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 + (-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 + (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 + (-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 + (-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 + (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 + (-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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{4} - 12 q^{6} - 8 q^{9}+O(q^{10})$$ 12 * q - 2 * q^4 - 12 * q^6 - 8 * q^9 $$12 q - 2 q^{4} - 12 q^{6} - 8 q^{9} + 4 q^{11} - 22 q^{14} - 14 q^{16} + 12 q^{19} - 20 q^{21} - 2 q^{24} - 44 q^{26} + 12 q^{29} + 60 q^{31} - 10 q^{34} + 14 q^{36} - 4 q^{39} - 12 q^{41} - 20 q^{44} + 8 q^{46} + 4 q^{49} - 40 q^{51} + 4 q^{54} + 92 q^{56} - 20 q^{59} + 2 q^{61} - 24 q^{64} - 6 q^{66} + 36 q^{69} + 2 q^{71} + 22 q^{74} - 70 q^{76} - 24 q^{79} - 14 q^{81} + 96 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} + 60 q^{94} + 52 q^{96} + 30 q^{99}+O(q^{100})$$ 12 * q - 2 * q^4 - 12 * q^6 - 8 * q^9 + 4 * q^11 - 22 * q^14 - 14 * q^16 + 12 * q^19 - 20 * q^21 - 2 * q^24 - 44 * q^26 + 12 * q^29 + 60 * q^31 - 10 * q^34 + 14 * q^36 - 4 * q^39 - 12 * q^41 - 20 * q^44 + 8 * q^46 + 4 * q^49 - 40 * q^51 + 4 * q^54 + 92 * q^56 - 20 * q^59 + 2 * q^61 - 24 * q^64 - 6 * q^66 + 36 * q^69 + 2 * q^71 + 22 * q^74 - 70 * q^76 - 24 * q^79 - 14 * q^81 + 96 * q^84 + 16 * q^86 - 36 * q^89 + 24 * q^91 + 60 * q^94 + 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 $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\beta_{6}$$

## 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
 0.203566 + 0.352587i 0.579521 + 1.00376i 1.05958 + 1.83525i −1.05958 − 1.83525i −0.579521 − 1.00376i −0.203566 − 0.352587i 0.203566 − 0.352587i 0.579521 − 1.00376i 1.05958 − 1.83525i −1.05958 + 1.83525i −0.579521 + 1.00376i −0.203566 + 0.352587i
−1.22810 2.12713i −0.780522 1.35190i −2.01647 + 3.49262i 0 −1.91712 + 3.32055i 4.50527 4.99330 0.281570 0.487693i 0
26.2 −0.431391 0.747190i −1.53957 2.66661i 0.627804 1.08739i 0 −1.32831 + 2.30070i 0.566520 −2.80888 −3.24054 + 5.61278i 0
26.3 −0.235942 0.408663i 0.520111 + 0.900858i 0.888663 1.53921i 0 0.245432 0.425100i −1.17540 −1.78246 0.958970 1.66098i 0
26.4 0.235942 + 0.408663i −0.520111 0.900858i 0.888663 1.53921i 0 0.245432 0.425100i 1.17540 1.78246 0.958970 1.66098i 0
26.5 0.431391 + 0.747190i 1.53957 + 2.66661i 0.627804 1.08739i 0 −1.32831 + 2.30070i −0.566520 2.80888 −3.24054 + 5.61278i 0
26.6 1.22810 + 2.12713i 0.780522 + 1.35190i −2.01647 + 3.49262i 0 −1.91712 + 3.32055i −4.50527 −4.99330 0.281570 0.487693i 0
201.1 −1.22810 + 2.12713i −0.780522 + 1.35190i −2.01647 3.49262i 0 −1.91712 3.32055i 4.50527 4.99330 0.281570 + 0.487693i 0
201.2 −0.431391 + 0.747190i −1.53957 + 2.66661i 0.627804 + 1.08739i 0 −1.32831 2.30070i 0.566520 −2.80888 −3.24054 5.61278i 0
201.3 −0.235942 + 0.408663i 0.520111 0.900858i 0.888663 + 1.53921i 0 0.245432 + 0.425100i −1.17540 −1.78246 0.958970 + 1.66098i 0
201.4 0.235942 0.408663i −0.520111 + 0.900858i 0.888663 + 1.53921i 0 0.245432 + 0.425100i 1.17540 1.78246 0.958970 + 1.66098i 0
201.5 0.431391 0.747190i 1.53957 2.66661i 0.627804 + 1.08739i 0 −1.32831 2.30070i −0.566520 2.80888 −3.24054 5.61278i 0
201.6 1.22810 2.12713i 0.780522 1.35190i −2.01647 3.49262i 0 −1.91712 3.32055i −4.50527 −4.99330 0.281570 + 0.487693i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.6 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 475.2.e.g 12
5.b even 2 1 inner 475.2.e.g 12
5.c odd 4 2 95.2.i.b 12
15.e even 4 2 855.2.be.d 12
19.c even 3 1 inner 475.2.e.g 12
19.c even 3 1 9025.2.a.bu 6
19.d odd 6 1 9025.2.a.bt 6
95.h odd 6 1 9025.2.a.bt 6
95.i even 6 1 inner 475.2.e.g 12
95.i even 6 1 9025.2.a.bu 6
95.l even 12 2 1805.2.b.g 6
95.m odd 12 2 95.2.i.b 12
95.m odd 12 2 1805.2.b.f 6
285.v even 12 2 855.2.be.d 12

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

## 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 $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

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