# Properties

 Label 825.2.bx.h Level $825$ Weight $2$ Character orbit 825.bx Analytic conductor $6.588$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.bx (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1$$ x^16 - x^14 + 15*x^12 - 59*x^10 + 104*x^8 - 59*x^6 + 15*x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{15} + \beta_{13} - \beta_{8}) q^{2} - \beta_{14} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} + \beta_{4} + \beta_1) q^{7} + ( - 4 \beta_{14} + 3 \beta_{11} + 3 \beta_{9} - 3 \beta_{8} + \beta_{4} + 3 \beta_1) q^{8} - \beta_{12} q^{9}+O(q^{10})$$ q + (-2*b15 + b13 - b8) * q^2 - b14 * q^3 + (b7 - 2*b6 - b5 + 4*b3 - 2*b2 - 1) * q^4 + (-b10 - 2*b7 - b6 - b5) * q^6 + (-b15 + b14 + 3*b13 + b4 + b1) * q^7 + (-4*b14 + 3*b11 + 3*b9 - 3*b8 + b4 + 3*b1) * q^8 - b12 * q^9 $$q + ( - 2 \beta_{15} + \beta_{13} - \beta_{8}) q^{2} - \beta_{14} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} + \beta_{4} + \beta_1) q^{7} + ( - 4 \beta_{14} + 3 \beta_{11} + 3 \beta_{9} - 3 \beta_{8} + \beta_{4} + 3 \beta_1) q^{8} - \beta_{12} q^{9} + (\beta_{12} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_{2}) q^{11} + (4 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - 3 \beta_{11} - 4 \beta_{9} + \beta_{8} - 2 \beta_{4}) q^{12} + (\beta_{14} + \beta_{13} + \beta_{9} - \beta_{8} + 2 \beta_1) q^{13} + (\beta_{12} - 2 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} + 3 \beta_{3} - 5 \beta_{2} + 1) q^{14} + (5 \beta_{12} - 5 \beta_{10} - 10 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} - 5 \beta_{3}) q^{16} + 5 \beta_{9} q^{17} + ( - 2 \beta_{11} - \beta_{4} - \beta_1) q^{18} + ( - \beta_{12} + 2 \beta_{10} - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 5 \beta_{2} + \cdots - 1) q^{19}+ \cdots + ( - \beta_{12} + 3 \beta_{10} + \beta_{7} + \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100})$$ q + (-2*b15 + b13 - b8) * q^2 - b14 * q^3 + (b7 - 2*b6 - b5 + 4*b3 - 2*b2 - 1) * q^4 + (-b10 - 2*b7 - b6 - b5) * q^6 + (-b15 + b14 + 3*b13 + b4 + b1) * q^7 + (-4*b14 + 3*b11 + 3*b9 - 3*b8 + b4 + 3*b1) * q^8 - b12 * q^9 + (b12 + b7 - b6 - b5 + b3 - 3*b2) * q^11 + (4*b15 - 3*b14 - 2*b13 - 3*b11 - 4*b9 + b8 - 2*b4) * q^12 + (b14 + b13 + b9 - b8 + 2*b1) * q^13 + (b12 - 2*b10 + 3*b7 - 2*b5 + 3*b3 - 5*b2 + 1) * q^14 + (5*b12 - 5*b10 - 10*b7 - 5*b6 - 5*b5 - 5*b3) * q^16 + 5*b9 * q^17 + (-2*b11 - b4 - b1) * q^18 + (-b12 + 2*b10 - 2*b7 + 2*b5 - 2*b3 + 5*b2 - 1) * q^19 + (b12 - 2*b10 - b7 - 2*b6 - 3*b5 - 3*b2) * q^21 + (-4*b15 - b14 + 3*b13 + 6*b11 + 4*b9 - 3*b8 + 2*b4 + 5*b1) * q^22 + (b15 + b13 - b9 - b8 + b4) * q^23 + (-4*b12 + 3*b10 + b6 - 3*b3 + 3*b2 + 3) * q^24 + (b7 - 2*b6 - 2*b5 - 2*b2 - 1) * q^26 + b9 * q^27 + (-9*b15 + 5*b13 + 9*b11 + 5*b9 - 5*b8 + 5*b4 + 5*b1) * q^28 + (4*b6 + 3*b5 - b3 + 4*b2) * q^29 + (3*b6 - 2*b3 + 2) * q^31 + (16*b15 - 7*b14 - 4*b13 - 7*b11 - 16*b9 + 8*b8 - 4*b4) * q^32 + (b15 - b14 - b13 - 2*b9 - b4 + 2*b1) * q^33 + (-5*b10 - 5*b6 - 10) * q^34 + (-3*b12 + 2*b10 + 4*b7 + 2*b5 + 4*b3 + b2 - 3) * q^36 + (-5*b15 + 5*b14 - b4 - b1) * q^37 + (9*b15 - 10*b13 - 9*b11 - 5*b9 + 7*b8 - 7*b4 - 10*b1) * q^38 + (b12 + b10 - b6 - b5 - b3) * q^39 + (-6*b12 - b10 + b7 - b5 + b3 + 3*b2 - 6) * q^41 + (3*b15 - 4*b14 + 2*b13 - 4*b9 - 2*b8 + 5*b1) * q^42 + (-3*b15 - 3*b14 + b13 - 3*b11 + 3*b9 + b4) * q^43 + (10*b12 - 7*b10 - 14*b7 - 6*b6 - 7*b5 - 3*b3 - 5*b2 + 4) * q^44 + (b12 + b10 + 3*b6 - b3 + b2 + 1) * q^46 + (-6*b14 - 3*b11 - 3*b9 - b8 - 4*b4 + b1) * q^47 + (-5*b15 + 5*b14 - 5*b11 - 5*b4 - 5*b1) * q^48 + (4*b12 + 2*b10 - b6 - b5 - 4*b3) * q^49 - 5*b3 * q^51 + (-2*b14 + b4) * q^52 + (-4*b15 + 5*b14 + 3*b13 + 5*b9 - 3*b8 - 3*b1) * q^53 + (-b10 - b6 - 2) * q^54 + (15*b12 - 8*b10 - 15*b7 - 8*b6 - 4*b5 - 4*b2 + 6) * q^56 + (-2*b15 + 3*b14 - 2*b13 + 3*b9 + 2*b8 - 5*b1) * q^57 + (6*b14 - b11 - b9 + 4*b8 + 3*b4 - 4*b1) * q^58 + (6*b7 + b6 + b3 + b2 - 6) * q^59 + (-2*b12 - 3*b10 - b7 - 6*b6 - 6*b5 + 2*b3) * q^61 + (-7*b15 + 7*b14 + 5*b13 + b4 + b1) * q^62 + (-b11 - b9 - b8 - 2*b4 + b1) * q^63 + (-8*b12 + 2*b10 + 11*b6 - 16*b3 + 2*b2 + 16) * q^64 + (-b12 + 2*b10 - 4*b7 - b6 - 3*b5 - 4*b3 + 6) * q^66 + (-4*b15 + 5*b14 + 5*b13 + 5*b11 + 4*b9 + 5*b8 + 5*b4) * q^67 + (20*b15 - 5*b14 - 10*b13 - 5*b9 + 10*b8 - 5*b1) * q^68 + (-b10 + b7 - b5 + b3) * q^69 + (-4*b12 + 3*b10 + 8*b7 - 2*b6 - 2*b5 + 4*b3) * q^71 + (-3*b15 - 2*b13 + 3*b11 + 7*b9 - b8 + b4 - 2*b1) * q^72 + (-4*b15 + 4*b14 + 8*b13 - 2*b11 + 5*b4 + 5*b1) * q^73 + (-b12 + 5*b10 + 12*b7 + 5*b5 + 12*b3 - 4*b2 - 1) * q^74 + (-19*b12 + 11*b10 + 19*b7 + 11*b6 + 12*b5 + 12*b2 - 9) * q^76 + (-9*b15 + 7*b14 - 2*b13 + 11*b11 + 6*b9 + b8 + b4 - 3*b1) * q^77 + (b14 - 2*b13 + b11 - 2*b4) * q^78 + (-2*b12 - 2*b10 - 2*b6 + 7*b3 - 2*b2 - 7) * q^79 - b3 * q^81 + (14*b15 - 7*b13 - 14*b11 - 3*b9 + 3*b8 - 3*b4 - 7*b1) * q^82 + (-8*b15 + 3*b13 + 8*b11 - 2*b9 - 5*b8 + 5*b4 + 3*b1) * q^83 + (-9*b7 - 5*b5 - 5*b3 + 9) * q^84 + (-5*b12 - 4*b10 - 8*b6 + 12*b3 - 4*b2 - 12) * q^86 + (-b15 + b14 + 4*b13 + b11 + b9 - b8 + 4*b4) * q^87 + (18*b15 - 18*b14 - 4*b13 - 8*b11 - 23*b9 + 8*b8 - 2*b4 + 7*b1) * q^88 + (3*b12 + 2*b10 - 3*b7 + 2*b6 + 2*b5 + 2*b2 - 2) * q^89 + (6*b12 + 5*b10 - 2*b7 + 5*b5 - 2*b3 + 7*b2 + 6) * q^91 + (-3*b15 + 3*b14 + 2*b13 + b11) * q^92 + (-2*b15 + 3*b13 + 2*b11 + 2*b9 - 3*b8 + 3*b4 + 3*b1) * q^93 + (-10*b12 + b10 - 3*b7 - 8*b6 - 8*b5 + 10*b3) * q^94 + (-7*b12 + 4*b10 + 16*b7 + 4*b5 + 16*b3 - 4*b2 - 7) * q^96 + (4*b15 - 3*b14 - 3*b13 - 3*b9 + 3*b8 + b1) * q^97 + (-b15 + 7*b14 - 3*b13 + 7*b11 + b9 + 3*b8 - 3*b4) * q^98 + (-b12 + 3*b10 + b7 + b5 + 2*b3 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{4} + 4 q^{9}+O(q^{10})$$ 16 * q + 4 * q^4 + 4 * q^9 $$16 q + 4 q^{4} + 4 q^{9} - 6 q^{11} + 20 q^{14} - 40 q^{16} - 12 q^{19} - 8 q^{21} + 40 q^{24} - 16 q^{26} + 6 q^{31} - 100 q^{34} - 4 q^{36} - 12 q^{39} - 50 q^{41} - 14 q^{44} - 12 q^{46} - 42 q^{49} - 20 q^{51} - 20 q^{54} + 40 q^{56} - 70 q^{59} + 42 q^{61} + 154 q^{64} + 50 q^{66} + 10 q^{69} + 50 q^{71} + 58 q^{74} - 28 q^{76} - 60 q^{79} - 4 q^{81} + 68 q^{84} - 68 q^{86} - 64 q^{89} + 74 q^{91} + 78 q^{94} + 20 q^{96} + 6 q^{99}+O(q^{100})$$ 16 * q + 4 * q^4 + 4 * q^9 - 6 * q^11 + 20 * q^14 - 40 * q^16 - 12 * q^19 - 8 * q^21 + 40 * q^24 - 16 * q^26 + 6 * q^31 - 100 * q^34 - 4 * q^36 - 12 * q^39 - 50 * q^41 - 14 * q^44 - 12 * q^46 - 42 * q^49 - 20 * q^51 - 20 * q^54 + 40 * q^56 - 70 * q^59 + 42 * q^61 + 154 * q^64 + 50 * q^66 + 10 * q^69 + 50 * q^71 + 58 * q^74 - 28 * q^76 - 60 * q^79 - 4 * q^81 + 68 * q^84 - 68 * q^86 - 64 * q^89 + 74 * q^91 + 78 * q^94 + 20 * q^96 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 21\nu^{14} + 2\nu^{12} + 289\nu^{10} - 908\nu^{8} + 772\nu^{6} + 1045\nu^{4} - 622\nu^{2} - 63 ) / 384$$ (21*v^14 + 2*v^12 + 289*v^10 - 908*v^8 + 772*v^6 + 1045*v^4 - 622*v^2 - 63) / 384 $$\beta_{3}$$ $$=$$ $$( 21\nu^{14} - 22\nu^{12} + 329\nu^{10} - 1260\nu^{8} + 2436\nu^{6} - 1995\nu^{4} + 1498\nu^{2} - 119 ) / 384$$ (21*v^14 - 22*v^12 + 329*v^10 - 1260*v^8 + 2436*v^6 - 1995*v^4 + 1498*v^2 - 119) / 384 $$\beta_{4}$$ $$=$$ $$( -21\nu^{15} + 22\nu^{13} - 329\nu^{11} + 1260\nu^{9} - 2436\nu^{7} + 1995\nu^{5} - 1498\nu^{3} + 119\nu ) / 384$$ (-21*v^15 + 22*v^13 - 329*v^11 + 1260*v^9 - 2436*v^7 + 1995*v^5 - 1498*v^3 + 119*v) / 384 $$\beta_{5}$$ $$=$$ $$( 5\nu^{14} - 11\nu^{12} + 76\nu^{10} - 384\nu^{8} + 804\nu^{6} - 671\nu^{4} + 137\nu^{2} - 40 ) / 96$$ (5*v^14 - 11*v^12 + 76*v^10 - 384*v^8 + 804*v^6 - 671*v^4 + 137*v^2 - 40) / 96 $$\beta_{6}$$ $$=$$ $$( -21\nu^{14} + 10\nu^{12} - 309\nu^{10} + 1084\nu^{8} - 1604\nu^{6} + 475\nu^{4} - 246\nu^{2} + 91 ) / 192$$ (-21*v^14 + 10*v^12 - 309*v^10 + 1084*v^8 - 1604*v^6 + 475*v^4 - 246*v^2 + 91) / 192 $$\beta_{7}$$ $$=$$ $$( 9\nu^{14} - 17\nu^{12} + 138\nu^{10} - 648\nu^{8} + 1332\nu^{6} - 1107\nu^{4} + 155\nu^{2} + 30 ) / 96$$ (9*v^14 - 17*v^12 + 138*v^10 - 648*v^8 + 1332*v^6 - 1107*v^4 + 155*v^2 + 30) / 96 $$\beta_{8}$$ $$=$$ $$( 9\nu^{15} - 17\nu^{13} + 138\nu^{11} - 648\nu^{9} + 1332\nu^{7} - 1107\nu^{5} + 155\nu^{3} + 30\nu ) / 96$$ (9*v^15 - 17*v^13 + 138*v^11 - 648*v^9 + 1332*v^7 - 1107*v^5 + 155*v^3 + 30*v) / 96 $$\beta_{9}$$ $$=$$ $$( -9\nu^{15} + 23\nu^{13} - 148\nu^{11} + 736\nu^{9} - 1748\nu^{7} + 1867\nu^{5} - 685\nu^{3} - 16\nu ) / 96$$ (-9*v^15 + 23*v^13 - 148*v^11 + 736*v^9 - 1748*v^7 + 1867*v^5 - 685*v^3 - 16*v) / 96 $$\beta_{10}$$ $$=$$ $$( 57\nu^{14} - 30\nu^{12} + 845\nu^{10} - 2972\nu^{8} + 4564\nu^{6} - 1511\nu^{4} + 466\nu^{2} + 77 ) / 384$$ (57*v^14 - 30*v^12 + 845*v^10 - 2972*v^8 + 4564*v^6 - 1511*v^4 + 466*v^2 + 77) / 384 $$\beta_{11}$$ $$=$$ $$( -63\nu^{15} + 42\nu^{13} - 947\nu^{11} + 3428\nu^{9} - 5644\nu^{7} + 2945\nu^{5} - 1990\nu^{3} + 685\nu ) / 384$$ (-63*v^15 + 42*v^13 - 947*v^11 + 3428*v^9 - 5644*v^7 + 2945*v^5 - 1990*v^3 + 685*v) / 384 $$\beta_{12}$$ $$=$$ $$( -17\nu^{14} + 5\nu^{12} - 246\nu^{10} + 824\nu^{8} - 1108\nu^{6} - 101\nu^{4} + 201\nu^{2} - 58 ) / 96$$ (-17*v^14 + 5*v^12 - 246*v^10 + 824*v^8 - 1108*v^6 - 101*v^4 + 201*v^2 - 58) / 96 $$\beta_{13}$$ $$=$$ $$( -17\nu^{15} + 5\nu^{13} - 246\nu^{11} + 824\nu^{9} - 1108\nu^{7} - 101\nu^{5} + 201\nu^{3} - 58\nu ) / 96$$ (-17*v^15 + 5*v^13 - 246*v^11 + 824*v^9 - 1108*v^7 - 101*v^5 + 201*v^3 - 58*v) / 96 $$\beta_{14}$$ $$=$$ $$( 77\nu^{15} - 134\nu^{13} + 1185\nu^{11} - 5388\nu^{9} + 10980\nu^{7} - 9107\nu^{5} + 2666\nu^{3} - 543\nu ) / 384$$ (77*v^15 - 134*v^13 + 1185*v^11 - 5388*v^9 + 10980*v^7 - 9107*v^5 + 2666*v^3 - 543*v) / 384 $$\beta_{15}$$ $$=$$ $$( 40\nu^{15} - 35\nu^{13} + 589\nu^{11} - 2284\nu^{9} + 3776\nu^{7} - 1556\nu^{5} - 71\nu^{3} + 97\nu ) / 96$$ (40*v^15 - 35*v^13 + 589*v^11 - 2284*v^9 + 3776*v^7 - 1556*v^5 - 71*v^3 + 97*v) / 96
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{3} + \beta_{2}$$ b6 + b3 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{8} - 3\beta_{4} - \beta_1$$ b11 + b9 + b8 - 3*b4 - b1 $$\nu^{4}$$ $$=$$ $$\beta_{12} + 5\beta_{10} - 3\beta_{7} + 4\beta_{6} + 4\beta_{5} - \beta_{3}$$ b12 + 5*b10 - 3*b7 + 4*b6 + 4*b5 - b3 $$\nu^{5}$$ $$=$$ $$5\beta_{15} - \beta_{14} + 6\beta_{13} - \beta_{11} - 5\beta_{9} - 12\beta_{8} + 6\beta_{4}$$ 5*b15 - b14 + 6*b13 - b11 - 5*b9 - 12*b8 + 6*b4 $$\nu^{6}$$ $$=$$ $$-6\beta_{12} - 16\beta_{10} - 23\beta_{6} - 6\beta_{3} - 16\beta_{2} + 6$$ -6*b12 - 16*b10 - 23*b6 - 6*b3 - 16*b2 + 6 $$\nu^{7}$$ $$=$$ $$-16\beta_{15} + 16\beta_{14} - 22\beta_{13} - 7\beta_{11} + 29\beta_{4} + 29\beta_1$$ -16*b15 + 16*b14 - 22*b13 - 7*b11 + 29*b4 + 29*b1 $$\nu^{8}$$ $$=$$ $$-22\beta_{12} - 67\beta_{10} + 51\beta_{7} - 67\beta_{5} + 51\beta_{3} + 36\beta_{2} - 22$$ -22*b12 - 67*b10 + 51*b7 - 67*b5 + 51*b3 + 36*b2 - 22 $$\nu^{9}$$ $$=$$ $$-67\beta_{15} - 89\beta_{13} + 67\beta_{11} + 103\beta_{9} + 221\beta_{8} - 221\beta_{4} - 89\beta_1$$ -67*b15 - 89*b13 + 67*b11 + 103*b9 + 221*b8 - 221*b4 - 89*b1 $$\nu^{10}$$ $$=$$ $$132\beta_{12} + 456\beta_{10} - 132\beta_{7} + 456\beta_{6} + 168\beta_{5} + 168\beta_{2} - 89$$ 132*b12 + 456*b10 - 132*b7 + 456*b6 + 168*b5 + 168*b2 - 89 $$\nu^{11}$$ $$=$$ $$456\beta_{15} - 288\beta_{14} + 588\beta_{13} - 288\beta_{9} - 588\beta_{8} - 377\beta_1$$ 456*b15 - 288*b14 + 588*b13 - 288*b9 - 588*b8 - 377*b1 $$\nu^{12}$$ $$=$$ $$-588\beta_{7} - 1253\beta_{6} + 756\beta_{5} - 965\beta_{3} - 1253\beta_{2} + 588$$ -588*b7 - 1253*b6 + 756*b5 - 965*b3 - 1253*b2 + 588 $$\nu^{13}$$ $$=$$ $$756\beta_{14} - 1253\beta_{11} - 1253\beta_{9} - 2597\beta_{8} + 4227\beta_{4} + 2597\beta_1$$ 756*b14 - 1253*b11 - 1253*b9 - 2597*b8 + 4227*b4 + 2597*b1 $$\nu^{14}$$ $$=$$ $$-2597\beta_{12} - 8833\beta_{10} + 4227\beta_{7} - 5480\beta_{6} - 5480\beta_{5} + 2597\beta_{3}$$ -2597*b12 - 8833*b10 + 4227*b7 - 5480*b6 - 5480*b5 + 2597*b3 $$\nu^{15}$$ $$=$$ $$- 8833 \beta_{15} + 3353 \beta_{14} - 11430 \beta_{13} + 3353 \beta_{11} + 8833 \beta_{9} + 18540 \beta_{8} - 11430 \beta_{4}$$ -8833*b15 + 3353*b14 - 11430*b13 + 3353*b11 + 8833*b9 + 18540*b8 - 11430*b4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.701538 + 0.227943i 1.28932 − 0.418926i −1.28932 + 0.418926i 0.701538 − 0.227943i 0.280526 + 0.386111i 1.23158 + 1.69513i −1.23158 − 1.69513i −0.280526 − 0.386111i 0.280526 − 0.386111i 1.23158 − 1.69513i −1.23158 + 1.69513i −0.280526 + 0.386111i −0.701538 − 0.227943i 1.28932 + 0.418926i −1.28932 − 0.418926i 0.701538 + 0.227943i
−2.33569 + 0.758911i −0.587785 + 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i −1.93196 2.65911i −2.93237 + 4.03606i −0.309017 0.951057i 0
49.2 −1.10527 + 0.359123i −0.587785 + 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i 2.51974 + 3.46813i 1.80980 2.49097i −0.309017 0.951057i 0
49.3 1.10527 0.359123i 0.587785 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i −2.51974 3.46813i −1.80980 + 2.49097i −0.309017 0.951057i 0
49.4 2.33569 0.758911i 0.587785 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i 1.93196 + 2.65911i 2.93237 4.03606i −0.309017 0.951057i 0
124.1 −1.62947 2.24278i 0.951057 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i −2.16612 0.703814i 9.71623 3.15700i 0.809017 0.587785i 0
124.2 −0.817172 1.12474i −0.951057 + 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i −1.21506 0.394797i −2.73326 + 0.888090i 0.809017 0.587785i 0
124.3 0.817172 + 1.12474i 0.951057 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i 1.21506 + 0.394797i 2.73326 0.888090i 0.809017 0.587785i 0
124.4 1.62947 + 2.24278i −0.951057 + 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i 2.16612 + 0.703814i −9.71623 + 3.15700i 0.809017 0.587785i 0
499.1 −1.62947 + 2.24278i 0.951057 + 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i −2.16612 + 0.703814i 9.71623 + 3.15700i 0.809017 + 0.587785i 0
499.2 −0.817172 + 1.12474i −0.951057 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i −1.21506 + 0.394797i −2.73326 0.888090i 0.809017 + 0.587785i 0
499.3 0.817172 1.12474i 0.951057 + 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i 1.21506 0.394797i 2.73326 + 0.888090i 0.809017 + 0.587785i 0
499.4 1.62947 2.24278i −0.951057 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i 2.16612 0.703814i −9.71623 3.15700i 0.809017 + 0.587785i 0
724.1 −2.33569 0.758911i −0.587785 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i −1.93196 + 2.65911i −2.93237 4.03606i −0.309017 + 0.951057i 0
724.2 −1.10527 0.359123i −0.587785 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i 2.51974 3.46813i 1.80980 + 2.49097i −0.309017 + 0.951057i 0
724.3 1.10527 + 0.359123i 0.587785 + 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i −2.51974 + 3.46813i −1.80980 2.49097i −0.309017 + 0.951057i 0
724.4 2.33569 + 0.758911i 0.587785 + 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i 1.93196 2.65911i 2.93237 + 4.03606i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 724.4 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
11.c even 5 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bx.h 16
5.b even 2 1 inner 825.2.bx.h 16
5.c odd 4 1 165.2.m.a 8
5.c odd 4 1 825.2.n.k 8
11.c even 5 1 inner 825.2.bx.h 16
15.e even 4 1 495.2.n.d 8
55.j even 10 1 inner 825.2.bx.h 16
55.k odd 20 1 165.2.m.a 8
55.k odd 20 1 825.2.n.k 8
55.k odd 20 1 1815.2.a.x 4
55.k odd 20 1 9075.2.a.cl 4
55.l even 20 1 1815.2.a.o 4
55.l even 20 1 9075.2.a.dj 4
165.u odd 20 1 5445.2.a.bv 4
165.v even 20 1 495.2.n.d 8
165.v even 20 1 5445.2.a.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 5.c odd 4 1
165.2.m.a 8 55.k odd 20 1
495.2.n.d 8 15.e even 4 1
495.2.n.d 8 165.v even 20 1
825.2.n.k 8 5.c odd 4 1
825.2.n.k 8 55.k odd 20 1
825.2.bx.h 16 1.a even 1 1 trivial
825.2.bx.h 16 5.b even 2 1 inner
825.2.bx.h 16 11.c even 5 1 inner
825.2.bx.h 16 55.j even 10 1 inner
1815.2.a.o 4 55.l even 20 1
1815.2.a.x 4 55.k odd 20 1
5445.2.a.be 4 165.v even 20 1
5445.2.a.bv 4 165.u odd 20 1
9075.2.a.cl 4 55.k odd 20 1
9075.2.a.dj 4 55.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{16} - 6T_{2}^{14} + 57T_{2}^{12} - 473T_{2}^{10} + 2730T_{2}^{8} - 3647T_{2}^{6} + 9087T_{2}^{4} - 15609T_{2}^{2} + 14641$$ T2^16 - 6*T2^14 + 57*T2^12 - 473*T2^10 + 2730*T2^8 - 3647*T2^6 + 9087*T2^4 - 15609*T2^2 + 14641 $$T_{13}^{16} - 42T_{13}^{14} + 665T_{13}^{12} + 683T_{13}^{10} + 2874T_{13}^{8} - 103T_{13}^{6} + 75T_{13}^{4} - 13T_{13}^{2} + 1$$ T13^16 - 42*T13^14 + 665*T13^12 + 683*T13^10 + 2874*T13^8 - 103*T13^6 + 75*T13^4 - 13*T13^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 6 T^{14} + 57 T^{12} + \cdots + 14641$$
$3$ $$(T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 7 T^{14} + 383 T^{12} + \cdots + 2825761$$
$11$ $$(T^{8} + 3 T^{7} + 8 T^{6} + T^{5} + \cdots + 14641)^{2}$$
$13$ $$T^{16} - 42 T^{14} + 665 T^{12} + 683 T^{10} + \cdots + 1$$
$17$ $$(T^{8} - 25 T^{6} + 625 T^{4} + \cdots + 390625)^{2}$$
$19$ $$(T^{8} + 6 T^{7} + 9 T^{6} - 123 T^{5} + \cdots + 961)^{2}$$
$23$ $$(T^{8} + 27 T^{6} + 49 T^{4} + 23 T^{2} + \cdots + 1)^{2}$$
$29$ $$(T^{8} + 17 T^{6} - 95 T^{5} + \cdots + 290521)^{2}$$
$31$ $$(T^{8} - 3 T^{7} + 11 T^{6} + T^{5} + \cdots + 19321)^{2}$$
$37$ $$T^{16} - 149 T^{14} + \cdots + 34507149121$$
$41$ $$(T^{8} + 25 T^{7} + 327 T^{6} + \cdots + 4289041)^{2}$$
$43$ $$(T^{8} + 188 T^{6} + 12438 T^{4} + \cdots + 3463321)^{2}$$
$47$ $$T^{16} - 21 T^{14} + 15567 T^{12} + \cdots + 14641$$
$53$ $$T^{16} + 77 T^{14} + \cdots + 2609649624481$$
$59$ $$(T^{8} + 35 T^{7} + 633 T^{6} + \cdots + 5285401)^{2}$$
$61$ $$(T^{8} - 21 T^{7} + 242 T^{6} + \cdots + 3575881)^{2}$$
$67$ $$(T^{8} + 441 T^{6} + 57706 T^{4} + \cdots + 11417641)^{2}$$
$71$ $$(T^{8} - 25 T^{7} + 347 T^{6} + \cdots + 5527201)^{2}$$
$73$ $$T^{16} - 59 T^{14} + \cdots + 71\!\cdots\!01$$
$79$ $$(T^{8} + 30 T^{7} + 487 T^{6} + \cdots + 1437601)^{2}$$
$83$ $$T^{16} - 597 T^{14} + \cdots + 22\!\cdots\!81$$
$89$ $$(T^{4} + 16 T^{3} + 45 T^{2} - 132 T - 271)^{4}$$
$97$ $$T^{16} + 65 T^{14} + 8000 T^{12} + \cdots + 390625$$
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