# Properties

 Label 936.2.m.h Level $936$ Weight $2$ Character orbit 936.m Analytic conductor $7.474$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$936 = 2^{3} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 936.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.47399762919$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625$$ x^16 + 2*x^14 - 16*x^12 - 72*x^10 + 26*x^8 + 360*x^6 + 725*x^4 + 1000*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} - \beta_{5} q^{7} + (\beta_{14} + \beta_{12} - \beta_{10} - \beta_{9}) q^{8}+O(q^{10})$$ q - b10 * q^2 + b3 * q^4 + b14 * q^5 - b5 * q^7 + (b14 + b12 - b10 - b9) * q^8 $$q - \beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} - \beta_{5} q^{7} + (\beta_{14} + \beta_{12} - \beta_{10} - \beta_{9}) q^{8} + (\beta_{4} - \beta_{2} + 1) q^{10} + ( - \beta_{14} - \beta_{12} - \beta_{10}) q^{11} + (\beta_{15} + \beta_{4} + \beta_{2}) q^{13} + (\beta_{11} - \beta_{6} - \beta_1) q^{14} - 2 \beta_{2} q^{16} + ( - \beta_{11} + \beta_{7} - 2 \beta_1) q^{17} + ( - \beta_{15} + \beta_{13}) q^{19} + ( - 2 \beta_{12} - 2 \beta_{9}) q^{20} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{22} + (\beta_{11} - 2 \beta_{6}) q^{23} + ( - \beta_{4} - \beta_{3}) q^{25} + ( - 2 \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_1) q^{26} + ( - \beta_{15} + 2 \beta_{13} - \beta_{8}) q^{28} - \beta_{7} q^{29} + (2 \beta_{8} - \beta_{5}) q^{31} - 4 \beta_{9} q^{32} + ( - 2 \beta_{15} - \beta_{13} - \beta_{8} + \beta_{5}) q^{34} + ( - \beta_{11} - 2 \beta_{7}) q^{35} + ( - 2 \beta_{15} - 2 \beta_{13}) q^{37} + ( - \beta_{11} - \beta_{6} - \beta_1) q^{38} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 6) q^{40} + ( - 3 \beta_{12} + 3 \beta_{10} - 3 \beta_{9}) q^{41} + (\beta_{4} - \beta_{3} + 1) q^{43} + (\beta_{14} + 3 \beta_{12} - 3 \beta_{10} + \beta_{9}) q^{44} + (\beta_{15} + 3 \beta_{13} + \beta_{5}) q^{46} + (4 \beta_{12} - 4 \beta_{10} - 3 \beta_{9}) q^{47} + (\beta_{4} + \beta_{3} - 8) q^{49} + ( - \beta_{14} + \beta_{12} + \beta_{9}) q^{50} + (\beta_{13} + \beta_{8} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{52} + ( - \beta_{11} + 2 \beta_{7}) q^{53} + (3 \beta_{4} + 3 \beta_{3} - 5) q^{55} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{56} + (\beta_{15} - \beta_{8}) q^{58} + (\beta_{14} - 3 \beta_{12} - 3 \beta_{10}) q^{59} + (3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 3) q^{61} + ( - \beta_{11} + 4 \beta_{7} - \beta_{6} - 3 \beta_1) q^{62} + ( - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4) q^{64} + ( - \beta_{12} + \beta_{10} + 3 \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{65} + (\beta_{15} - 3 \beta_{13}) q^{67} + ( - 2 \beta_{11} - 2 \beta_{7} + 2 \beta_{6}) q^{68} + (3 \beta_{15} - \beta_{13} - 2 \beta_{8} + \beta_{5}) q^{70} + (4 \beta_{12} - 4 \beta_{10} - 5 \beta_{9}) q^{71} + (2 \beta_{8} - 2 \beta_{5}) q^{73} + ( - 2 \beta_{11} + 2 \beta_{6} - 2 \beta_1) q^{74} + (\beta_{15} - \beta_{8} + 2 \beta_{5}) q^{76} + (3 \beta_{11} + \beta_{7}) q^{77} + ( - 4 \beta_{4} - 4 \beta_{3} + 2) q^{79} + ( - 2 \beta_{14} + 2 \beta_{12} - 6 \beta_{10} - 2 \beta_{9}) q^{80} + ( - 3 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 9) q^{82} + (3 \beta_{14} + \beta_{12} + \beta_{10}) q^{83} + (2 \beta_{15} - 4 \beta_{13}) q^{85} + ( - \beta_{14} - 3 \beta_{12} + \beta_{10} + \beta_{9}) q^{86} + (2 \beta_{4} + 4 \beta_{3} - 6) q^{88} + (\beta_{12} - \beta_{10} + 7 \beta_{9}) q^{89} + ( - \beta_{15} - 3 \beta_{13} + \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - 5) q^{91} + ( - 2 \beta_{6} + 2 \beta_1) q^{92} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5) q^{94} + (\beta_{11} - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_1) q^{95} + 2 \beta_{8} q^{97} + (\beta_{14} - \beta_{12} + 8 \beta_{10} - \beta_{9}) q^{98}+O(q^{100})$$ q - b10 * q^2 + b3 * q^4 + b14 * q^5 - b5 * q^7 + (b14 + b12 - b10 - b9) * q^8 + (b4 - b2 + 1) * q^10 + (-b14 - b12 - b10) * q^11 + (b15 + b4 + b2) * q^13 + (b11 - b6 - b1) * q^14 - 2*b2 * q^16 + (-b11 + b7 - 2*b1) * q^17 + (-b15 + b13) * q^19 + (-2*b12 - 2*b9) * q^20 + (-b4 + b3 + b2 + 1) * q^22 + (b11 - 2*b6) * q^23 + (-b4 - b3) * q^25 + (-2*b12 + b11 + b10 + 2*b9 + b1) * q^26 + (-b15 + 2*b13 - b8) * q^28 - b7 * q^29 + (2*b8 - b5) * q^31 - 4*b9 * q^32 + (-2*b15 - b13 - b8 + b5) * q^34 + (-b11 - 2*b7) * q^35 + (-2*b15 - 2*b13) * q^37 + (-b11 - b6 - b1) * q^38 + (-2*b4 - 2*b3 - 2*b2 + 6) * q^40 + (-3*b12 + 3*b10 - 3*b9) * q^41 + (b4 - b3 + 1) * q^43 + (b14 + 3*b12 - 3*b10 + b9) * q^44 + (b15 + 3*b13 + b5) * q^46 + (4*b12 - 4*b10 - 3*b9) * q^47 + (b4 + b3 - 8) * q^49 + (-b14 + b12 + b9) * q^50 + (b13 + b8 - b5 + 2*b4 + b3 + 2*b2 + 2) * q^52 + (-b11 + 2*b7) * q^53 + (3*b4 + 3*b3 - 5) * q^55 + (-2*b7 - 2*b6) * q^56 + (b15 - b8) * q^58 + (b14 - 3*b12 - 3*b10) * q^59 + (3*b4 + 3*b3 + 6*b2 - 3) * q^61 + (-b11 + 4*b7 - b6 - 3*b1) * q^62 + (-4*b4 - 4*b3 - 4*b2 + 4) * q^64 + (-b12 + b10 + 3*b9 + b7 - 2*b6 - 2*b1) * q^65 + (b15 - 3*b13) * q^67 + (-2*b11 - 2*b7 + 2*b6) * q^68 + (3*b15 - b13 - 2*b8 + b5) * q^70 + (4*b12 - 4*b10 - 5*b9) * q^71 + (2*b8 - 2*b5) * q^73 + (-2*b11 + 2*b6 - 2*b1) * q^74 + (b15 - b8 + 2*b5) * q^76 + (3*b11 + b7) * q^77 + (-4*b4 - 4*b3 + 2) * q^79 + (-2*b14 + 2*b12 - 6*b10 - 2*b9) * q^80 + (-3*b4 - 6*b3 - 3*b2 + 9) * q^82 + (3*b14 + b12 + b10) * q^83 + (2*b15 - 4*b13) * q^85 + (-b14 - 3*b12 + b10 + b9) * q^86 + (2*b4 + 4*b3 - 6) * q^88 + (b12 - b10 + 7*b9) * q^89 + (-b15 - 3*b13 + b4 + 5*b3 + 6*b2 - 5) * q^91 + (-2*b6 + 2*b1) * q^92 + (-3*b4 + b3 - 3*b2 - 5) * q^94 + (b11 - 2*b7 + 2*b6 + 4*b1) * q^95 + 2*b8 * q^97 + (b14 - b12 + 8*b10 - b9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4}+O(q^{10})$$ 16 * q + 8 * q^4 $$16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94}+O(q^{100})$$ 16 * q + 8 * q^4 - 16 * q^16 + 40 * q^22 + 80 * q^40 - 128 * q^49 + 40 * q^52 - 80 * q^55 + 32 * q^64 + 32 * q^79 + 96 * q^82 - 80 * q^88 - 72 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( 310 \nu^{14} - 2816 \nu^{12} - 9222 \nu^{10} + 18376 \nu^{8} + 130292 \nu^{6} - 53906 \nu^{4} + 290050 \nu^{2} + 217000 ) / 176275$$ (310*v^14 - 2816*v^12 - 9222*v^10 + 18376*v^8 + 130292*v^6 - 53906*v^4 + 290050*v^2 + 217000) / 176275 $$\beta_{2}$$ $$=$$ $$( 11992 \nu^{14} + 9614 \nu^{12} - 190612 \nu^{10} - 554504 \nu^{8} + 1001932 \nu^{6} + 2217000 \nu^{4} + 3496000 \nu^{2} + 5574000 ) / 881375$$ (11992*v^14 + 9614*v^12 - 190612*v^10 - 554504*v^8 + 1001932*v^6 + 2217000*v^4 + 3496000*v^2 + 5574000) / 881375 $$\beta_{3}$$ $$=$$ $$( 16948 \nu^{14} + 5486 \nu^{12} - 299188 \nu^{10} - 779596 \nu^{8} + 1911868 \nu^{6} + 4019270 \nu^{4} + 6074000 \nu^{2} + 6351000 ) / 881375$$ (16948*v^14 + 5486*v^12 - 299188*v^10 - 779596*v^8 + 1911868*v^6 + 4019270*v^4 + 6074000*v^2 + 6351000) / 881375 $$\beta_{4}$$ $$=$$ $$( - 17344 \nu^{14} - 7298 \nu^{12} + 306684 \nu^{10} + 805078 \nu^{8} - 1876924 \nu^{6} - 3981500 \nu^{4} - 6068600 \nu^{2} - 8342875 ) / 881375$$ (-17344*v^14 - 7298*v^12 + 306684*v^10 + 805078*v^8 - 1876924*v^6 - 3981500*v^4 - 6068600*v^2 - 8342875) / 881375 $$\beta_{5}$$ $$=$$ $$( - 10661 \nu^{15} - 14207 \nu^{13} + 225956 \nu^{11} + 888177 \nu^{9} - 521116 \nu^{7} - 5727895 \nu^{5} - 13225300 \nu^{3} - 26244000 \nu ) / 4406875$$ (-10661*v^15 - 14207*v^13 + 225956*v^11 + 888177*v^9 - 521116*v^7 - 5727895*v^5 - 13225300*v^3 - 26244000*v) / 4406875 $$\beta_{6}$$ $$=$$ $$( - 25832 \nu^{14} - 21274 \nu^{12} + 450392 \nu^{10} + 1381764 \nu^{8} - 2270262 \nu^{6} - 7392230 \nu^{4} - 12703750 \nu^{2} - 13499250 ) / 881375$$ (-25832*v^14 - 21274*v^12 + 450392*v^10 + 1381764*v^8 - 2270262*v^6 - 7392230*v^4 - 12703750*v^2 - 13499250) / 881375 $$\beta_{7}$$ $$=$$ $$( - 28806 \nu^{14} - 29832 \nu^{12} + 448156 \nu^{10} + 1496702 \nu^{8} - 2047066 \nu^{6} - 6175530 \nu^{4} - 12354350 \nu^{2} - 13465750 ) / 881375$$ (-28806*v^14 - 29832*v^12 + 448156*v^10 + 1496702*v^8 - 2047066*v^6 - 6175530*v^4 - 12354350*v^2 - 13465750) / 881375 $$\beta_{8}$$ $$=$$ $$( 15392 \nu^{15} + 52754 \nu^{13} - 145257 \nu^{11} - 987344 \nu^{9} - 868348 \nu^{7} + 1700315 \nu^{5} + 6288975 \nu^{3} + 15325500 \nu ) / 4406875$$ (15392*v^15 + 52754*v^13 - 145257*v^11 - 987344*v^9 - 868348*v^7 + 1700315*v^5 + 6288975*v^3 + 15325500*v) / 4406875 $$\beta_{9}$$ $$=$$ $$( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu ) / 400625$$ (-1496*v^15 - 2572*v^13 + 24401*v^11 + 91992*v^9 - 100886*v^7 - 478015*v^5 - 412275*v^3 - 598500*v) / 400625 $$\beta_{10}$$ $$=$$ $$( - 22194 \nu^{15} - 11523 \nu^{13} + 339409 \nu^{11} + 1024903 \nu^{9} - 1669899 \nu^{7} - 3769000 \nu^{5} - 11127550 \nu^{3} - 9814875 \nu ) / 4406875$$ (-22194*v^15 - 11523*v^13 + 339409*v^11 + 1024903*v^9 - 1669899*v^7 - 3769000*v^5 - 11127550*v^3 - 9814875*v) / 4406875 $$\beta_{11}$$ $$=$$ $$( - 44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000 ) / 881375$$ (-44196*v^14 - 27412*v^12 + 754046*v^10 + 2232082*v^8 - 3966856*v^6 - 11081230*v^4 - 21249600*v^2 - 22476000) / 881375 $$\beta_{12}$$ $$=$$ $$( - 46447 \nu^{15} - 41694 \nu^{13} + 802127 \nu^{11} + 2508384 \nu^{9} - 4035597 \nu^{7} - 13232745 \nu^{5} - 19604750 \nu^{3} - 21119125 \nu ) / 4406875$$ (-46447*v^15 - 41694*v^13 + 802127*v^11 + 2508384*v^9 - 4035597*v^7 - 13232745*v^5 - 19604750*v^3 - 21119125*v) / 4406875 $$\beta_{13}$$ $$=$$ $$( 78923 \nu^{15} + 34611 \nu^{13} - 1339113 \nu^{11} - 3553821 \nu^{9} + 8128668 \nu^{7} + 15701920 \nu^{5} + 26400825 \nu^{3} + 32586750 \nu ) / 4406875$$ (78923*v^15 + 34611*v^13 - 1339113*v^11 - 3553821*v^9 + 8128668*v^7 + 15701920*v^5 + 26400825*v^3 + 32586750*v) / 4406875 $$\beta_{14}$$ $$=$$ $$( - 108813 \nu^{15} - 85331 \nu^{13} + 1829673 \nu^{11} + 5646091 \nu^{9} - 9386828 \nu^{7} - 27713410 \nu^{5} - 49911275 \nu^{3} - 50144500 \nu ) / 4406875$$ (-108813*v^15 - 85331*v^13 + 1829673*v^11 + 5646091*v^9 - 9386828*v^7 - 27713410*v^5 - 49911275*v^3 - 50144500*v) / 4406875 $$\beta_{15}$$ $$=$$ $$( - 128244 \nu^{15} - 47208 \nu^{13} + 2180589 \nu^{11} + 5658588 \nu^{9} - 13459254 \nu^{7} - 25923185 \nu^{5} - 43314975 \nu^{3} - 53041500 \nu ) / 4406875$$ (-128244*v^15 - 47208*v^13 + 2180589*v^11 + 5658588*v^9 - 13459254*v^7 - 25923185*v^5 - 43314975*v^3 - 53041500*v) / 4406875
 $$\nu$$ $$=$$ $$( -\beta_{15} - \beta_{13} + 2\beta_{12} - 2\beta_{9} - \beta_{5} ) / 4$$ (-b15 - b13 + 2*b12 - 2*b9 - b5) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 2$$ (b11 - b7 - b6 - b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{14} - \beta_{13} + 5\beta_{12} + 7\beta_{10} + 5\beta_{9} + \beta_{8} + \beta_{5} ) / 4$$ (-5*b14 - b13 + 5*b12 + 7*b10 + 5*b9 + b8 + b5) / 4 $$\nu^{4}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 8\beta_{4} + 5\beta_{3} + 5\beta_{2} + 8 ) / 2$$ (2*b7 - 2*b6 + 8*b4 + 5*b3 + 5*b2 + 8) / 2 $$\nu^{5}$$ $$=$$ $$( -7\beta_{15} - 12\beta_{13} - 3\beta_{9} ) / 2$$ (-7*b15 - 12*b13 - 3*b9) / 2 $$\nu^{6}$$ $$=$$ $$( 11\beta_{11} - 11\beta_{7} - 6\beta_{6} + 11\beta_{3} - 15\beta_{2} + 5\beta _1 + 30 ) / 2$$ (11*b11 - 11*b7 - 6*b6 + 11*b3 - 15*b2 + 5*b1 + 30) / 2 $$\nu^{7}$$ $$=$$ $$( -5\beta_{15} - 55\beta_{14} - 4\beta_{13} + 127\beta_{12} + 55\beta_{10} - 55\beta_{9} - 9\beta_{8} - 4\beta_{5} ) / 4$$ (-5*b15 - 55*b14 - 4*b13 + 127*b12 + 55*b10 - 55*b9 - 9*b8 - 4*b5) / 4 $$\nu^{8}$$ $$=$$ $$( 32\beta_{11} - 52\beta_{6} + 53\beta_{4} + 18\beta_{3} + 53\beta_{2} + 32\beta _1 + 17 ) / 2$$ (32*b11 - 52*b6 + 53*b4 + 18*b3 + 53*b2 + 32*b1 + 17) / 2 $$\nu^{9}$$ $$=$$ $$( - 137 \beta_{15} - 57 \beta_{14} - 224 \beta_{13} - 57 \beta_{12} + 247 \beta_{10} + 247 \beta_{9} + 87 \beta_{8} + 50 \beta_{5} ) / 4$$ (-137*b15 - 57*b14 - 224*b13 - 57*b12 + 247*b10 + 247*b9 + 87*b8 + 50*b5) / 4 $$\nu^{10}$$ $$=$$ $$( 21\beta_{11} - 36\beta_{7} + 275\beta_{4} + 275\beta_{3} + 607 ) / 2$$ (21*b11 - 36*b7 + 275*b4 + 275*b3 + 607) / 2 $$\nu^{11}$$ $$=$$ $$( - 384 \beta_{15} - 341 \beta_{14} - 623 \beta_{13} + 1421 \beta_{12} - 341 \beta_{10} - 1421 \beta_{9} - 239 \beta_{8} - 145 \beta_{5} ) / 4$$ (-384*b15 - 341*b14 - 623*b13 + 1421*b12 - 341*b10 - 1421*b9 - 239*b8 - 145*b5) / 4 $$\nu^{12}$$ $$=$$ $$( 752\beta_{11} - 462\beta_{7} - 752\beta_{6} - 110\beta_{4} - 177\beta_{2} + 462\beta _1 + 110 ) / 2$$ (752*b11 - 462*b7 - 752*b6 - 110*b4 - 177*b2 + 462*b1 + 110) / 2 $$\nu^{13}$$ $$=$$ $$( - 395 \beta_{15} - 2315 \beta_{14} - 642 \beta_{13} + 2315 \beta_{12} + 5199 \beta_{10} + 2315 \beta_{9} + 1037 \beta_{8} + 642 \beta_{5} ) / 4$$ (-395*b15 - 2315*b14 - 642*b13 + 2315*b12 + 5199*b10 + 2315*b9 + 1037*b8 + 642*b5) / 4 $$\nu^{14}$$ $$=$$ $$( 639\beta_{7} - 639\beta_{6} + 5431\beta_{4} + 3360\beta_{3} + 3360\beta_{2} + 400\beta _1 + 5431 ) / 2$$ (639*b7 - 639*b6 + 5431*b4 + 3360*b3 + 3360*b2 + 400*b1 + 5431) / 2 $$\nu^{15}$$ $$=$$ $$( -4399\beta_{15} - 7109\beta_{13} + 2475\beta_{12} - 2475\beta_{10} - 3996\beta_{9} ) / 2$$ (-4399*b15 - 7109*b13 + 2475*b12 - 2475*b10 - 3996*b9) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$469$$ $$703$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −0.556839 − 1.81878i 1.90184 − 0.0324487i 1.90184 + 0.0324487i −0.556839 + 1.81878i 0.752864 − 0.902863i 0.0783900 + 1.17295i 0.0783900 − 1.17295i 0.752864 + 0.902863i −0.752864 − 0.902863i −0.0783900 + 1.17295i −0.0783900 − 1.17295i −0.752864 + 0.902863i 0.556839 − 1.81878i −1.90184 − 0.0324487i −1.90184 + 0.0324487i 0.556839 + 1.81878i
−1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.2 −1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.3 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.4 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.5 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.6 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.7 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.8 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.9 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.10 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.11 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.12 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.13 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.14 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.15 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
181.16 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.16 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
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.m.h 16
3.b odd 2 1 inner 936.2.m.h 16
4.b odd 2 1 3744.2.m.h 16
8.b even 2 1 inner 936.2.m.h 16
8.d odd 2 1 3744.2.m.h 16
12.b even 2 1 3744.2.m.h 16
13.b even 2 1 inner 936.2.m.h 16
24.f even 2 1 3744.2.m.h 16
24.h odd 2 1 inner 936.2.m.h 16
39.d odd 2 1 inner 936.2.m.h 16
52.b odd 2 1 3744.2.m.h 16
104.e even 2 1 inner 936.2.m.h 16
104.h odd 2 1 3744.2.m.h 16
156.h even 2 1 3744.2.m.h 16
312.b odd 2 1 inner 936.2.m.h 16
312.h even 2 1 3744.2.m.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 1.a even 1 1 trivial
936.2.m.h 16 3.b odd 2 1 inner
936.2.m.h 16 8.b even 2 1 inner
936.2.m.h 16 13.b even 2 1 inner
936.2.m.h 16 24.h odd 2 1 inner
936.2.m.h 16 39.d odd 2 1 inner
936.2.m.h 16 104.e even 2 1 inner
936.2.m.h 16 312.b odd 2 1 inner
3744.2.m.h 16 4.b odd 2 1
3744.2.m.h 16 8.d odd 2 1
3744.2.m.h 16 12.b even 2 1
3744.2.m.h 16 24.f even 2 1
3744.2.m.h 16 52.b odd 2 1
3744.2.m.h 16 104.h odd 2 1
3744.2.m.h 16 156.h even 2 1
3744.2.m.h 16 312.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 10T_{5}^{2} + 20$$ acting on $$S_{2}^{\mathrm{new}}(936, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2}$$
$3$ $$T^{16}$$
$5$ $$(T^{4} - 10 T^{2} + 20)^{4}$$
$7$ $$(T^{4} + 30 T^{2} + 220)^{4}$$
$11$ $$(T^{4} - 20 T^{2} + 20)^{4}$$
$13$ $$(T^{8} - 12 T^{6} + 54 T^{4} - 2028 T^{2} + \cdots + 28561)^{2}$$
$17$ $$(T^{4} - 60 T^{2} + 880)^{4}$$
$19$ $$(T^{4} - 34 T^{2} + 44)^{4}$$
$23$ $$(T^{4} - 80 T^{2} + 880)^{4}$$
$29$ $$(T^{4} + 28 T^{2} + 176)^{4}$$
$31$ $$(T^{4} + 150 T^{2} + 5500)^{4}$$
$37$ $$(T^{4} - 104 T^{2} + 704)^{4}$$
$41$ $$(T^{4} + 126 T^{2} + 324)^{4}$$
$43$ $$(T^{4} + 20 T^{2} + 80)^{4}$$
$47$ $$(T^{4} + 84 T^{2} + 1444)^{4}$$
$53$ $$(T^{4} + 128 T^{2} + 176)^{4}$$
$59$ $$(T^{4} - 100 T^{2} + 500)^{4}$$
$61$ $$(T^{4} + 180 T^{2} + 6480)^{4}$$
$67$ $$(T^{4} - 154 T^{2} + 5324)^{4}$$
$71$ $$(T^{4} + 116 T^{2} + 484)^{4}$$
$73$ $$(T^{4} + 200 T^{2} + 3520)^{4}$$
$79$ $$(T^{2} - 4 T - 76)^{8}$$
$83$ $$(T^{4} - 100 T^{2} + 500)^{4}$$
$89$ $$(T^{4} + 230 T^{2} + 12100)^{4}$$
$97$ $$(T^{4} + 160 T^{2} + 3520)^{4}$$