# Properties

 Label 384.3.i.d Level $384$ Weight $3$ Character orbit 384.i Analytic conductor $10.463$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.i (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576$$ x^20 - 2*x^18 + 6*x^16 - 24*x^14 - 24*x^12 + 1216*x^10 - 384*x^8 - 6144*x^6 + 24576*x^4 - 131072*x^2 + 1048576 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{23}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{5} + \beta_{6} q^{7} - \beta_{16} q^{9}+O(q^{10})$$ q + b5 * q^3 + (-b10 - b9 + b8) * q^5 + b6 * q^7 - b16 * q^9 $$q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{5} + \beta_{6} q^{7} - \beta_{16} q^{9} + (\beta_{18} - \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10}) q^{11} + (\beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_1 - 5) q^{13} + ( - \beta_{15} + \beta_{14} - \beta_{12} - 6) q^{15} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + \beta_{2}) q^{17} + (\beta_{19} - \beta_{16} + \beta_{14} + \beta_{12} + \beta_{7} + \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 4 \beta_1 + 4) q^{19} + ( - \beta_{18} - \beta_{16} + \beta_{15} - \beta_{14} + 4 \beta_{13} - 3 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots - 4) q^{21}+ \cdots + ( - 4 \beta_{19} - \beta_{18} - 18 \beta_{17} - \beta_{15} - 4 \beta_{12} + 9 \beta_{5} + \cdots + 24) q^{99}+O(q^{100})$$ q + b5 * q^3 + (-b10 - b9 + b8) * q^5 + b6 * q^7 - b16 * q^9 + (b18 - b15 - b13 + b11 - b10) * q^11 + (b5 + b4 - b3 + 5*b1 - 5) * q^13 + (-b15 + b14 - b12 - 6) * q^15 + (-b14 + b13 + b12 - b9 + b8 + b5 - b4 + b2) * q^17 + (b19 - b16 + b14 + b12 + b7 + b6 - 3*b5 - 3*b4 - 4*b1 + 4) * q^19 + (-b18 - b16 + b15 - b14 + 4*b13 - 3*b11 - b10 - 2*b9 - 2*b8 - b7 + b6 - 4*b1 - 4) * q^21 + (3*b17 + 3*b13 - 3*b10 - b9 + b8 - b5 + b4 - 3*b2) * q^23 + (b19 - b16 - b11 - b9 - b8 - b5 - b4 - b3 + 3*b1) * q^25 + (3*b13 + 3*b11 + 3*b10 + b9) * q^27 + (-b19 - 2*b18 + b17 - b16 - 2*b15 + b14 - b12 - b5 + b4 - 2*b3 + 2*b2) * q^29 + (2*b14 + 2*b12 + 3*b11 + b7 - 3*b3 - 4) * q^31 + (-4*b17 - 2*b15 + b13 - b12 + 3*b11 - 4*b10 - 2*b9 - 2*b8 - 2*b7 + 2*b5 + 2*b4 - 3*b3 + b2 + 1) * q^33 + (-2*b19 - b18 + 5*b17 - 2*b16 - b15 + 2*b14 - 2*b12 - 3*b5 + 3*b4 - b3 - 7*b2) * q^35 + (b19 - b16 - b14 - b12 + 3*b11 + b9 + b8 - 2*b7 + 2*b6 + 7*b1 + 7) * q^37 + (-b19 + b18 - b17 - b16 - b13 + b10 - 5*b9 - b8 + b6 - 5*b5 - b4 + b2 - 10*b1) * q^39 + (-2*b19 - 4*b18 + 2*b17 - 2*b16 - b13 - 2*b11 - 2*b10 + b9 - b8 + b5 - b4 - 2*b3 + b2) * q^41 + (-b19 + b16 + b14 + b12 - 2*b11 - 3*b9 - 3*b8 - b7 + b6 - 12*b1 - 12) * q^43 + (b19 - b18 - 7*b17 - b15 + b12 - b7 - b6 - 3*b5 - 7*b4 - 2*b2 + b1 - 1) * q^45 + (b17 - 2*b15 + 2*b14 - 5*b13 - 2*b12 + b11 + b10 + b9 - b8 - b5 + b4 - b3 - 5*b2) * q^47 + (-b14 - b12 - 3*b11 + b9 + b8 - 4*b7 - b5 - b4 + 3*b3 - 19) * q^49 + (b19 + b17 - b16 + b14 + b12 + b7 + b6 - b5 + 7*b4 - 3*b3 + 5*b2 + 8*b1 - 8) * q^51 + (-2*b19 - 2*b16 - 2*b14 - 6*b13 + 2*b12 + b10 + 3*b9 - 3*b8) * q^53 + (2*b19 - 2*b16 + 3*b11 + 2*b9 + 2*b8 + 2*b5 + 2*b4 + 3*b3 + 4*b1) * q^55 + (-2*b18 - 2*b17 + b16 + 7*b13 + 2*b10 + 3*b9 + 7*b8 + 2*b6 + 3*b5 + 7*b4 - 7*b2 + 19*b1) * q^57 + (2*b19 + 2*b16 + 2*b14 - 2*b12 - 12*b10 - 3*b9 + 3*b8) * q^59 + (b19 - b16 + b14 + b12 - 2*b7 - 2*b6 + 7*b5 + 7*b4 - 3*b3 - 7*b1 + 7) * q^61 + (5*b17 + 7*b13 + 4*b12 - 3*b11 + 5*b10 + 5*b9 + b8 + b7 - 5*b5 - b4 + 3*b3 + 7*b2 + 16) * q^63 + (6*b17 + 2*b14 - b13 - 2*b12 + 6*b10 + 3*b9 - 3*b8 - 3*b5 + 3*b4 - b2) * q^65 + (-2*b19 + 2*b16 - 2*b14 - 2*b12 - b5 - b4 + 2*b3 + 16*b1 - 16) * q^67 + (3*b18 + b16 - 3*b15 + b14 + 6*b13 - 6*b10 + 6*b8 + 3*b7 - 3*b6 + 3*b1 + 3) * q^69 + (-2*b19 + 2*b18 + 4*b17 - 2*b16 + b11 - 4*b10 + 4*b9 - 4*b8 + 4*b5 - 4*b4 + b3) * q^71 + (-3*b11 - 7*b9 - 7*b8 + 4*b6 - 7*b5 - 7*b4 - 3*b3 - 20*b1) * q^73 + (-b19 + b18 + b16 - b15 + b14 + b13 + b12 + 3*b11 + 5*b10 - 2*b9 + 7*b8 - b7 + b6 + 8*b1 + 8) * q^75 + (2*b19 + 4*b18 + 14*b17 + 2*b16 + 4*b15 - 2*b14 + 2*b12 + 8*b5 - 8*b4 + 4*b3 + 6*b2) * q^77 + (-4*b14 - 4*b12 + b11 - 2*b9 - 2*b8 - b7 + 2*b5 + 2*b4 - b3 + 20) * q^79 + (-6*b17 + 6*b15 - b14 + 6*b13 + 3*b12 - 6*b10 + 6*b8 + 6*b7 - 6*b4 + 6*b2 - 3) * q^81 + (4*b19 + b18 - b17 + 4*b16 + b15 - 4*b14 + 4*b12 + 8*b5 - 8*b4 + b3 - 9*b2) * q^83 + (2*b11 - 6*b9 - 6*b8 + 4*b7 - 4*b6 - 4*b1 - 4) * q^85 + (5*b19 - b18 - 15*b17 + b16 - 3*b13 - 3*b11 + 15*b10 + 5*b9 - 9*b8 + 5*b5 - 9*b4 - 3*b3 + 3*b2 - 6*b1) * q^87 + (3*b19 + 8*b18 + 6*b17 + 3*b16 - 14*b13 + 4*b11 - 6*b10 - 8*b9 + 8*b8 - 8*b5 + 8*b4 + 4*b3 + 14*b2) * q^89 + (3*b19 - 3*b16 - 3*b14 - 3*b12 - 6*b11 + 2*b9 + 2*b8 + b7 - b6 - 4*b1 - 4) * q^91 + (-3*b19 + 2*b18 - 11*b17 - b16 + 2*b15 + b14 - 3*b12 + 4*b7 + 4*b6 - 4*b5 + 10*b4 + 3*b3 - 4*b2 - 7*b1 + 7) * q^93 + (-13*b17 - 4*b14 - 11*b13 + 4*b12 - 13*b10 - 5*b9 + 5*b8 + 5*b5 - 5*b4 - 11*b2) * q^95 + (b14 + b12 - 2*b11 - 6*b9 - 6*b8 + 8*b7 + 6*b5 + 6*b4 + 2*b3 + 18) * q^97 + (-4*b19 - b18 - 18*b17 - b15 - 4*b12 + 9*b5 - 9*b4 + 18*b2 - 24*b1 + 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 6 q^{3}+O(q^{10})$$ 20 * q + 6 * q^3 $$20 q + 6 q^{3} - 92 q^{13} - 116 q^{15} + 52 q^{19} - 48 q^{21} - 18 q^{27} - 80 q^{31} + 60 q^{33} + 116 q^{37} - 172 q^{43} - 60 q^{45} - 364 q^{49} - 128 q^{51} + 244 q^{61} + 296 q^{63} - 356 q^{67} + 20 q^{69} + 146 q^{75} + 384 q^{79} - 188 q^{81} - 48 q^{85} - 136 q^{91} + 132 q^{93} + 472 q^{97} + 452 q^{99}+O(q^{100})$$ 20 * q + 6 * q^3 - 92 * q^13 - 116 * q^15 + 52 * q^19 - 48 * q^21 - 18 * q^27 - 80 * q^31 + 60 * q^33 + 116 * q^37 - 172 * q^43 - 60 * q^45 - 364 * q^49 - 128 * q^51 + 244 * q^61 + 296 * q^63 - 356 * q^67 + 20 * q^69 + 146 * q^75 + 384 * q^79 - 188 * q^81 - 48 * q^85 - 136 * q^91 + 132 * q^93 + 472 * q^97 + 452 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576$$ :

 $$\beta_{1}$$ $$=$$ $$( - 33 \nu^{18} - 42 \nu^{16} + 1090 \nu^{14} - 528 \nu^{12} - 11816 \nu^{10} + 6496 \nu^{8} - 64512 \nu^{6} + 1111040 \nu^{4} + 1482752 \nu^{2} - 16842752 ) / 10158080$$ (-33*v^18 - 42*v^16 + 1090*v^14 - 528*v^12 - 11816*v^10 + 6496*v^8 - 64512*v^6 + 1111040*v^4 + 1482752*v^2 - 16842752) / 10158080 $$\beta_{2}$$ $$=$$ $$( - 3 \nu^{19} - 90 \nu^{17} + 430 \nu^{15} + 1288 \nu^{13} - 5816 \nu^{11} - 3904 \nu^{9} - 19328 \nu^{7} + 278528 \nu^{5} + 2621440 \nu^{3} - 1638400 \nu ) / 4063232$$ (-3*v^19 - 90*v^17 + 430*v^15 + 1288*v^13 - 5816*v^11 - 3904*v^9 - 19328*v^7 + 278528*v^5 + 2621440*v^3 - 1638400*v) / 4063232 $$\beta_{3}$$ $$=$$ $$( 4 \nu^{18} + 51 \nu^{16} - 74 \nu^{14} - 790 \nu^{12} + 2736 \nu^{10} + 5240 \nu^{8} + 45664 \nu^{6} + 11520 \nu^{4} - 942080 \nu^{2} + 2064384 ) / 507904$$ (4*v^18 + 51*v^16 - 74*v^14 - 790*v^12 + 2736*v^10 + 5240*v^8 + 45664*v^6 + 11520*v^4 - 942080*v^2 + 2064384) / 507904 $$\beta_{4}$$ $$=$$ $$( - 4 \nu^{19} - 112 \nu^{18} - 101 \nu^{17} + 542 \nu^{16} - 410 \nu^{15} - 980 \nu^{14} + 1946 \nu^{13} - 5372 \nu^{12} - 368 \nu^{11} + 13216 \nu^{10} + 24248 \nu^{9} + \cdots + 18481152 ) / 10158080$$ (-4*v^19 - 112*v^18 - 101*v^17 + 542*v^16 - 410*v^15 - 980*v^14 + 1946*v^13 - 5372*v^12 - 368*v^11 + 13216*v^10 + 24248*v^9 - 64976*v^8 - 17696*v^7 + 29632*v^6 - 398080*v^5 + 750080*v^4 + 1181696*v^3 - 7708672*v^2 + 753664*v + 18481152) / 10158080 $$\beta_{5}$$ $$=$$ $$( 4 \nu^{19} - 112 \nu^{18} + 101 \nu^{17} + 542 \nu^{16} + 410 \nu^{15} - 980 \nu^{14} - 1946 \nu^{13} - 5372 \nu^{12} + 368 \nu^{11} + 13216 \nu^{10} - 24248 \nu^{9} - 64976 \nu^{8} + \cdots + 18481152 ) / 10158080$$ (4*v^19 - 112*v^18 + 101*v^17 + 542*v^16 + 410*v^15 - 980*v^14 - 1946*v^13 - 5372*v^12 + 368*v^11 + 13216*v^10 - 24248*v^9 - 64976*v^8 + 17696*v^7 + 29632*v^6 + 398080*v^5 + 750080*v^4 - 1181696*v^3 - 7708672*v^2 - 753664*v + 18481152) / 10158080 $$\beta_{6}$$ $$=$$ $$( - 187 \nu^{18} - 858 \nu^{16} + 6590 \nu^{14} - 21592 \nu^{12} - 90104 \nu^{10} - 159936 \nu^{8} - 782208 \nu^{6} + 3809280 \nu^{4} + 4169728 \nu^{2} + \cdots - 139460608 ) / 10158080$$ (-187*v^18 - 858*v^16 + 6590*v^14 - 21592*v^12 - 90104*v^10 - 159936*v^8 - 782208*v^6 + 3809280*v^4 + 4169728*v^2 - 139460608) / 10158080 $$\beta_{7}$$ $$=$$ $$( - 43 \nu^{18} - 106 \nu^{16} - 258 \nu^{14} - 1400 \nu^{12} + 7432 \nu^{10} - 17984 \nu^{8} - 125824 \nu^{6} + 706560 \nu^{4} - 1171456 \nu^{2} + 1966080 ) / 2031616$$ (-43*v^18 - 106*v^16 - 258*v^14 - 1400*v^12 + 7432*v^10 - 17984*v^8 - 125824*v^6 + 706560*v^4 - 1171456*v^2 + 1966080) / 2031616 $$\beta_{8}$$ $$=$$ $$( 79 \nu^{19} + 592 \nu^{18} - 34 \nu^{17} - 3112 \nu^{16} - 2030 \nu^{15} + 2320 \nu^{14} - 4896 \nu^{13} + 18832 \nu^{12} - 12392 \nu^{11} - 77056 \nu^{10} + 68512 \nu^{9} + \cdots - 96468992 ) / 40632320$$ (79*v^19 + 592*v^18 - 34*v^17 - 3112*v^16 - 2030*v^15 + 2320*v^14 - 4896*v^13 + 18832*v^12 - 12392*v^11 - 77056*v^10 + 68512*v^9 + 518336*v^8 + 78336*v^7 - 1806592*v^6 + 10240*v^5 - 8714240*v^4 - 6332416*v^3 + 46989312*v^2 + 10878976*v - 96468992) / 40632320 $$\beta_{9}$$ $$=$$ $$( - 79 \nu^{19} + 592 \nu^{18} + 34 \nu^{17} - 3112 \nu^{16} + 2030 \nu^{15} + 2320 \nu^{14} + 4896 \nu^{13} + 18832 \nu^{12} + 12392 \nu^{11} - 77056 \nu^{10} - 68512 \nu^{9} + \cdots - 96468992 ) / 40632320$$ (-79*v^19 + 592*v^18 + 34*v^17 - 3112*v^16 + 2030*v^15 + 2320*v^14 + 4896*v^13 + 18832*v^12 + 12392*v^11 - 77056*v^10 - 68512*v^9 + 518336*v^8 - 78336*v^7 - 1806592*v^6 - 10240*v^5 - 8714240*v^4 + 6332416*v^3 + 46989312*v^2 - 10878976*v - 96468992) / 40632320 $$\beta_{10}$$ $$=$$ $$( - 117 \nu^{19} + 282 \nu^{17} + 2210 \nu^{15} - 8552 \nu^{13} - 18184 \nu^{11} + 45504 \nu^{9} - 161408 \nu^{7} + 3051520 \nu^{5} - 7176192 \nu^{3} + \cdots - 18546688 \nu ) / 20316160$$ (-117*v^19 + 282*v^17 + 2210*v^15 - 8552*v^13 - 18184*v^11 + 45504*v^9 - 161408*v^7 + 3051520*v^5 - 7176192*v^3 - 18546688*v) / 20316160 $$\beta_{11}$$ $$=$$ $$( 29 \nu^{18} - 100 \nu^{16} - 22 \nu^{14} + 1180 \nu^{12} - 280 \nu^{10} + 11696 \nu^{8} + 6976 \nu^{6} - 372224 \nu^{4} + 3276800 \nu^{2} - 327680 ) / 1015808$$ (29*v^18 - 100*v^16 - 22*v^14 + 1180*v^12 - 280*v^10 + 11696*v^8 + 6976*v^6 - 372224*v^4 + 3276800*v^2 - 327680) / 1015808 $$\beta_{12}$$ $$=$$ $$( - 53 \nu^{19} - 48 \nu^{18} + 394 \nu^{17} + 320 \nu^{16} - 670 \nu^{15} - 2272 \nu^{14} - 392 \nu^{13} - 2624 \nu^{12} + 32824 \nu^{11} + 2944 \nu^{10} - 31936 \nu^{9} + \cdots + 15990784 ) / 8126464$$ (-53*v^19 - 48*v^18 + 394*v^17 + 320*v^16 - 670*v^15 - 2272*v^14 - 392*v^13 - 2624*v^12 + 32824*v^11 + 2944*v^10 - 31936*v^9 + 55040*v^8 + 184960*v^7 + 655360*v^6 + 1164288*v^5 - 2379776*v^4 - 7356416*v^3 - 7733248*v^2 + 31326208*v + 15990784) / 8126464 $$\beta_{13}$$ $$=$$ $$( 7 \nu^{19} - 38 \nu^{17} + 154 \nu^{15} + 136 \nu^{13} - 1640 \nu^{11} + 8448 \nu^{9} - 7808 \nu^{7} + 22016 \nu^{5} + 845824 \nu^{3} - 3457024 \nu ) / 1015808$$ (7*v^19 - 38*v^17 + 154*v^15 + 136*v^13 - 1640*v^11 + 8448*v^9 - 7808*v^7 + 22016*v^5 + 845824*v^3 - 3457024*v) / 1015808 $$\beta_{14}$$ $$=$$ $$( 53 \nu^{19} - 48 \nu^{18} - 394 \nu^{17} + 320 \nu^{16} + 670 \nu^{15} - 2272 \nu^{14} + 392 \nu^{13} - 2624 \nu^{12} - 32824 \nu^{11} + 2944 \nu^{10} + 31936 \nu^{9} + 55040 \nu^{8} + \cdots + 15990784 ) / 8126464$$ (53*v^19 - 48*v^18 - 394*v^17 + 320*v^16 + 670*v^15 - 2272*v^14 + 392*v^13 - 2624*v^12 - 32824*v^11 + 2944*v^10 + 31936*v^9 + 55040*v^8 - 184960*v^7 + 655360*v^6 - 1164288*v^5 - 2379776*v^4 + 7356416*v^3 - 7733248*v^2 - 31326208*v + 15990784) / 8126464 $$\beta_{15}$$ $$=$$ $$( - 25 \nu^{19} + 84 \nu^{18} + 1234 \nu^{17} - 808 \nu^{16} - 4022 \nu^{15} + 504 \nu^{14} - 5800 \nu^{13} + 11040 \nu^{12} + 22296 \nu^{11} - 23008 \nu^{10} - 69568 \nu^{9} + \cdots - 17825792 ) / 8126464$$ (-25*v^19 + 84*v^18 + 1234*v^17 - 808*v^16 - 4022*v^15 + 504*v^14 - 5800*v^13 + 11040*v^12 + 22296*v^11 - 23008*v^10 - 69568*v^9 + 4864*v^8 + 772736*v^7 - 337408*v^6 - 1160192*v^5 - 1581056*v^4 - 14385152*v^3 + 20643840*v^2 + 47972352*v - 17825792) / 8126464 $$\beta_{16}$$ $$=$$ $$( - 539 \nu^{19} - 1700 \nu^{18} - 386 \nu^{17} - 2840 \nu^{16} + 2190 \nu^{15} + 35560 \nu^{14} - 27464 \nu^{13} - 22240 \nu^{12} + 28232 \nu^{11} - 101280 \nu^{10} + \cdots - 399769600 ) / 40632320$$ (-539*v^19 - 1700*v^18 - 386*v^17 - 2840*v^16 + 2190*v^15 + 35560*v^14 - 27464*v^13 - 22240*v^12 + 28232*v^11 - 101280*v^10 - 311552*v^9 - 995840*v^8 - 523136*v^7 - 3965440*v^6 + 5811200*v^5 + 23490560*v^4 - 4603904*v^3 + 18513920*v^2 - 48889856*v - 399769600) / 40632320 $$\beta_{17}$$ $$=$$ $$( 17 \nu^{19} - 37 \nu^{17} - 80 \nu^{15} + 302 \nu^{13} + 904 \nu^{11} + 8936 \nu^{9} + 6368 \nu^{7} - 250240 \nu^{5} + 686592 \nu^{3} + 2428928 \nu ) / 1269760$$ (17*v^19 - 37*v^17 - 80*v^15 + 302*v^13 + 904*v^11 + 8936*v^9 + 6368*v^7 - 250240*v^5 + 686592*v^3 + 2428928*v) / 1269760 $$\beta_{18}$$ $$=$$ $$( 597 \nu^{19} - 740 \nu^{18} - 962 \nu^{17} - 40 \nu^{16} - 530 \nu^{15} + 3400 \nu^{14} + 33272 \nu^{13} + 8000 \nu^{12} - 39736 \nu^{11} - 103840 \nu^{10} + 562176 \nu^{9} + \cdots - 76021760 ) / 40632320$$ (597*v^19 - 740*v^18 - 962*v^17 - 40*v^16 - 530*v^15 + 3400*v^14 + 33272*v^13 + 8000*v^12 - 39736*v^11 - 103840*v^10 + 562176*v^9 - 443520*v^8 - 2165632*v^7 - 1966080*v^6 - 4244480*v^5 + 6983680*v^4 + 41500672*v^3 - 27852800*v^2 - 27000832*v - 76021760) / 40632320 $$\beta_{19}$$ $$=$$ $$( - 539 \nu^{19} + 1700 \nu^{18} - 386 \nu^{17} + 2840 \nu^{16} + 2190 \nu^{15} - 35560 \nu^{14} - 27464 \nu^{13} + 22240 \nu^{12} + 28232 \nu^{11} + 101280 \nu^{10} + \cdots + 399769600 ) / 40632320$$ (-539*v^19 + 1700*v^18 - 386*v^17 + 2840*v^16 + 2190*v^15 - 35560*v^14 - 27464*v^13 + 22240*v^12 + 28232*v^11 + 101280*v^10 - 311552*v^9 + 995840*v^8 - 523136*v^7 + 3965440*v^6 + 5811200*v^5 - 23490560*v^4 - 4603904*v^3 - 18513920*v^2 - 48889856*v + 399769600) / 40632320
 $$\nu$$ $$=$$ $$( \beta_{17} - \beta_{13} + \beta_{10} + \beta_{2} ) / 4$$ (b17 - b13 + b10 + b2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + \beta_{5} + \beta_{4} + 2\beta_1 ) / 2$$ (b11 + b5 + b4 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{15} + \beta_{13} + \beta_{9} - \beta_{8} + \beta_{3} + 2\beta_{2} ) / 2$$ (b19 + b18 + b17 + b16 + b15 + b13 + b9 - b8 + b3 + 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{19} - \beta_{16} - \beta_{14} - \beta_{12} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 12 \beta_1 ) / 2$$ (b19 - b16 - b14 - b12 - 2*b9 - 2*b8 + 2*b7 - 2*b5 - 2*b4 + 12*b1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{19} + 2 \beta_{18} - 5 \beta_{17} + \beta_{16} - 3 \beta_{14} + 5 \beta_{13} + 3 \beta_{12} + \beta_{11} - 5 \beta_{10} - 11 \beta_{9} + 11 \beta_{8} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + 7 \beta_{2} ) / 2$$ (b19 + 2*b18 - 5*b17 + b16 - 3*b14 + 5*b13 + 3*b12 + b11 - 5*b10 - 11*b9 + 11*b8 + 3*b5 - 3*b4 + b3 + 7*b2) / 2 $$\nu^{6}$$ $$=$$ $$3 \beta_{14} + 3 \beta_{12} + 3 \beta_{11} - 7 \beta_{9} - 7 \beta_{8} + \beta_{7} + \beta_{6} - 9 \beta_{5} - 9 \beta_{4} + \beta_{3} + 4 \beta _1 + 4$$ 3*b14 + 3*b12 + 3*b11 - 7*b9 - 7*b8 + b7 + b6 - 9*b5 - 9*b4 + b3 + 4*b1 + 4 $$\nu^{7}$$ $$=$$ $$- 3 \beta_{19} - 13 \beta_{18} + 2 \beta_{17} - 3 \beta_{16} + 3 \beta_{15} - 2 \beta_{14} + 16 \beta_{13} + 2 \beta_{12} - 8 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} - 8 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} + 7 \beta_{2}$$ -3*b19 - 13*b18 + 2*b17 - 3*b16 + 3*b15 - 2*b14 + 16*b13 + 2*b12 - 8*b11 - 3*b10 - b9 + b8 - 8*b5 + 8*b4 - 5*b3 + 7*b2 $$\nu^{8}$$ $$=$$ $$- \beta_{19} + \beta_{16} + 9 \beta_{14} + 9 \beta_{12} - 4 \beta_{11} + 28 \beta_{9} + 28 \beta_{8} + 6 \beta_{7} - 16 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} + 148 \beta _1 + 28$$ -b19 + b16 + 9*b14 + 9*b12 - 4*b11 + 28*b9 + 28*b8 + 6*b7 - 16*b6 + 14*b5 + 14*b4 + 14*b3 + 148*b1 + 28 $$\nu^{9}$$ $$=$$ $$- \beta_{19} + 4 \beta_{18} + 37 \beta_{17} - \beta_{16} + 10 \beta_{15} - 7 \beta_{14} + 31 \beta_{13} + 7 \beta_{12} - 3 \beta_{11} + 95 \beta_{10} + 19 \beta_{9} - 19 \beta_{8} - 81 \beta_{5} + 81 \beta_{4} + 7 \beta_{3} - 17 \beta_{2}$$ -b19 + 4*b18 + 37*b17 - b16 + 10*b15 - 7*b14 + 31*b13 + 7*b12 - 3*b11 + 95*b10 + 19*b9 - 19*b8 - 81*b5 + 81*b4 + 7*b3 - 17*b2 $$\nu^{10}$$ $$=$$ $$- 38 \beta_{19} + 38 \beta_{16} - 12 \beta_{14} - 12 \beta_{12} + 42 \beta_{11} + 18 \beta_{9} + 18 \beta_{8} + 30 \beta_{7} - 54 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 78 \beta_{3} - 200 \beta _1 - 576$$ -38*b19 + 38*b16 - 12*b14 - 12*b12 + 42*b11 + 18*b9 + 18*b8 + 30*b7 - 54*b6 + 14*b5 + 14*b4 + 78*b3 - 200*b1 - 576 $$\nu^{11}$$ $$=$$ $$56 \beta_{19} + 26 \beta_{18} + 30 \beta_{17} + 56 \beta_{16} - 18 \beta_{15} - 86 \beta_{14} + 182 \beta_{13} + 86 \beta_{12} + 22 \beta_{11} - 156 \beta_{10} + 148 \beta_{9} - 148 \beta_{8} + 18 \beta_{5} - 18 \beta_{4} + 4 \beta_{3} - 240 \beta_{2}$$ 56*b19 + 26*b18 + 30*b17 + 56*b16 - 18*b15 - 86*b14 + 182*b13 + 86*b12 + 22*b11 - 156*b10 + 148*b9 - 148*b8 + 18*b5 - 18*b4 + 4*b3 - 240*b2 $$\nu^{12}$$ $$=$$ $$- 26 \beta_{19} + 26 \beta_{16} - 18 \beta_{14} - 18 \beta_{12} - 236 \beta_{11} - 100 \beta_{9} - 100 \beta_{8} - 88 \beta_{7} - 276 \beta_{6} - 328 \beta_{5} - 328 \beta_{4} - 256 \beta_{3} + 168 \beta _1 - 1160$$ -26*b19 + 26*b16 - 18*b14 - 18*b12 - 236*b11 - 100*b9 - 100*b8 - 88*b7 - 276*b6 - 328*b5 - 328*b4 - 256*b3 + 168*b1 - 1160 $$\nu^{13}$$ $$=$$ $$- 486 \beta_{19} - 100 \beta_{18} - 942 \beta_{17} - 486 \beta_{16} - 248 \beta_{15} - 150 \beta_{14} - 82 \beta_{13} + 150 \beta_{12} + 74 \beta_{11} - 662 \beta_{10} + 122 \beta_{9} - 122 \beta_{8} - 466 \beta_{5} + 466 \beta_{4} + \cdots + 34 \beta_{2}$$ -486*b19 - 100*b18 - 942*b17 - 486*b16 - 248*b15 - 150*b14 - 82*b13 + 150*b12 + 74*b11 - 662*b10 + 122*b9 - 122*b8 - 466*b5 + 466*b4 - 174*b3 + 34*b2 $$\nu^{14}$$ $$=$$ $$- 712 \beta_{19} + 712 \beta_{16} - 92 \beta_{14} - 92 \beta_{12} - 300 \beta_{11} + 388 \beta_{9} + 388 \beta_{8} - 1148 \beta_{7} - 332 \beta_{6} - 492 \beta_{5} - 492 \beta_{4} + 916 \beta_{3} - 2240 \beta _1 + 7024$$ -712*b19 + 712*b16 - 92*b14 - 92*b12 - 300*b11 + 388*b9 + 388*b8 - 1148*b7 - 332*b6 - 492*b5 - 492*b4 + 916*b3 - 2240*b1 + 7024 $$\nu^{15}$$ $$=$$ $$- 1308 \beta_{19} - 1596 \beta_{18} + 2616 \beta_{17} - 1308 \beta_{16} - 908 \beta_{15} + 288 \beta_{14} - 1440 \beta_{13} - 288 \beta_{12} - 344 \beta_{11} + 4348 \beta_{10} + 5132 \beta_{9} - 5132 \beta_{8} + \cdots - 1068 \beta_{2}$$ -1308*b19 - 1596*b18 + 2616*b17 - 1308*b16 - 908*b15 + 288*b14 - 1440*b13 - 288*b12 - 344*b11 + 4348*b10 + 5132*b9 - 5132*b8 + 1656*b5 - 1656*b4 - 1252*b3 - 1068*b2 $$\nu^{16}$$ $$=$$ $$36 \beta_{19} - 36 \beta_{16} - 1940 \beta_{14} - 1940 \beta_{12} + 1152 \beta_{11} + 2416 \beta_{9} + 2416 \beta_{8} - 5832 \beta_{7} - 1904 \beta_{6} + 11192 \beta_{5} + 11192 \beta_{4} + 264 \beta_{3} + \cdots - 21584$$ 36*b19 - 36*b16 - 1940*b14 - 1940*b12 + 1152*b11 + 2416*b9 + 2416*b8 - 5832*b7 - 1904*b6 + 11192*b5 + 11192*b4 + 264*b3 + 13232*b1 - 21584 $$\nu^{17}$$ $$=$$ $$340 \beta_{19} + 9200 \beta_{18} - 2164 \beta_{17} + 340 \beta_{16} + 7096 \beta_{15} + 1100 \beta_{14} + 1636 \beta_{13} - 1100 \beta_{12} + 1052 \beta_{11} + 7748 \beta_{10} + 15716 \beta_{9} - 15716 \beta_{8} + \cdots + 708 \beta_{2}$$ 340*b19 + 9200*b18 - 2164*b17 + 340*b16 + 7096*b15 + 1100*b14 + 1636*b13 - 1100*b12 + 1052*b11 + 7748*b10 + 15716*b9 - 15716*b8 + 6708*b5 - 6708*b4 + 8148*b3 + 708*b2 $$\nu^{18}$$ $$=$$ $$7096 \beta_{19} - 7096 \beta_{16} - 16912 \beta_{14} - 16912 \beta_{12} - 6824 \beta_{11} - 9576 \beta_{9} - 9576 \beta_{8} - 6936 \beta_{7} + 10104 \beta_{6} - 21112 \beta_{5} - 21112 \beta_{4} + \cdots - 28416$$ 7096*b19 - 7096*b16 - 16912*b14 - 16912*b12 - 6824*b11 - 9576*b9 - 9576*b8 - 6936*b7 + 10104*b6 - 21112*b5 - 21112*b4 + 6888*b3 - 61472*b1 - 28416 $$\nu^{19}$$ $$=$$ $$- 10944 \beta_{19} + 10200 \beta_{18} - 15480 \beta_{17} - 10944 \beta_{16} - 10040 \beta_{15} - 6664 \beta_{14} + 18600 \beta_{13} + 6664 \beta_{12} + 10120 \beta_{11} - 63952 \beta_{10} - 62448 \beta_{9} + \cdots - 9600 \beta_{2}$$ -10944*b19 + 10200*b18 - 15480*b17 - 10944*b16 - 10040*b15 - 6664*b14 + 18600*b13 + 6664*b12 + 10120*b11 - 63952*b10 - 62448*b9 + 62448*b8 + 97368*b5 - 97368*b4 + 80*b3 - 9600*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.21144 − 1.59136i 0.312316 + 1.97546i −1.85381 + 0.750590i −1.21144 + 1.59136i 1.96139 − 0.391068i −0.312316 − 1.97546i −1.28499 − 1.53258i 1.28499 + 1.53258i 1.85381 − 0.750590i −1.96139 + 0.391068i 1.21144 + 1.59136i 0.312316 − 1.97546i −1.85381 − 0.750590i −1.21144 − 1.59136i 1.96139 + 0.391068i −0.312316 + 1.97546i −1.28499 + 1.53258i 1.28499 − 1.53258i 1.85381 + 0.750590i −1.96139 − 0.391068i
0 −2.77106 1.14944i 0 4.80434 + 4.80434i 0 7.36187i 0 6.35757 + 6.37035i 0
161.2 0 −2.75602 + 1.18505i 0 −0.00985921 0.00985921i 0 6.42277i 0 6.19134 6.53203i 0
161.3 0 −1.50491 + 2.59524i 0 2.59897 + 2.59897i 0 7.30027i 0 −4.47050 7.81118i 0
161.4 0 −1.14944 2.77106i 0 −4.80434 4.80434i 0 7.36187i 0 −6.35757 + 6.37035i 0
161.5 0 0.164573 + 2.99548i 0 3.61305 + 3.61305i 0 12.2792i 0 −8.94583 + 0.985948i 0
161.6 0 1.18505 2.75602i 0 0.00985921 + 0.00985921i 0 6.42277i 0 −6.19134 6.53203i 0
161.7 0 2.06336 + 2.17774i 0 3.17955 + 3.17955i 0 6.03979i 0 −0.485128 + 8.98692i 0
161.8 0 2.17774 + 2.06336i 0 −3.17955 3.17955i 0 6.03979i 0 0.485128 + 8.98692i 0
161.9 0 2.59524 1.50491i 0 −2.59897 2.59897i 0 7.30027i 0 4.47050 7.81118i 0
161.10 0 2.99548 + 0.164573i 0 −3.61305 3.61305i 0 12.2792i 0 8.94583 + 0.985948i 0
353.1 0 −2.77106 + 1.14944i 0 4.80434 4.80434i 0 7.36187i 0 6.35757 6.37035i 0
353.2 0 −2.75602 1.18505i 0 −0.00985921 + 0.00985921i 0 6.42277i 0 6.19134 + 6.53203i 0
353.3 0 −1.50491 2.59524i 0 2.59897 2.59897i 0 7.30027i 0 −4.47050 + 7.81118i 0
353.4 0 −1.14944 + 2.77106i 0 −4.80434 + 4.80434i 0 7.36187i 0 −6.35757 6.37035i 0
353.5 0 0.164573 2.99548i 0 3.61305 3.61305i 0 12.2792i 0 −8.94583 0.985948i 0
353.6 0 1.18505 + 2.75602i 0 0.00985921 0.00985921i 0 6.42277i 0 −6.19134 + 6.53203i 0
353.7 0 2.06336 2.17774i 0 3.17955 3.17955i 0 6.03979i 0 −0.485128 8.98692i 0
353.8 0 2.17774 2.06336i 0 −3.17955 + 3.17955i 0 6.03979i 0 0.485128 8.98692i 0
353.9 0 2.59524 + 1.50491i 0 −2.59897 + 2.59897i 0 7.30027i 0 4.47050 + 7.81118i 0
353.10 0 2.99548 0.164573i 0 −3.61305 + 3.61305i 0 12.2792i 0 8.94583 0.985948i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.10 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.i.d 20
3.b odd 2 1 inner 384.3.i.d 20
4.b odd 2 1 384.3.i.c 20
8.b even 2 1 48.3.i.b 20
8.d odd 2 1 192.3.i.b 20
12.b even 2 1 384.3.i.c 20
16.e even 4 1 48.3.i.b 20
16.e even 4 1 inner 384.3.i.d 20
16.f odd 4 1 192.3.i.b 20
16.f odd 4 1 384.3.i.c 20
24.f even 2 1 192.3.i.b 20
24.h odd 2 1 48.3.i.b 20
48.i odd 4 1 48.3.i.b 20
48.i odd 4 1 inner 384.3.i.d 20
48.k even 4 1 192.3.i.b 20
48.k even 4 1 384.3.i.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.b 20 8.b even 2 1
48.3.i.b 20 16.e even 4 1
48.3.i.b 20 24.h odd 2 1
48.3.i.b 20 48.i odd 4 1
192.3.i.b 20 8.d odd 2 1
192.3.i.b 20 16.f odd 4 1
192.3.i.b 20 24.f even 2 1
192.3.i.b 20 48.k even 4 1
384.3.i.c 20 4.b odd 2 1
384.3.i.c 20 12.b even 2 1
384.3.i.c 20 16.f odd 4 1
384.3.i.c 20 48.k even 4 1
384.3.i.d 20 1.a even 1 1 trivial
384.3.i.d 20 3.b odd 2 1 inner
384.3.i.d 20 16.e even 4 1 inner
384.3.i.d 20 48.i odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{20} + 3404T_{5}^{16} + 3190384T_{5}^{12} + 1068787520T_{5}^{8} + 108375444480T_{5}^{4} + 4096$$ T5^20 + 3404*T5^16 + 3190384*T5^12 + 1068787520*T5^8 + 108375444480*T5^4 + 4096 $$T_{19}^{10} - 26 T_{19}^{9} + 338 T_{19}^{8} - 3952 T_{19}^{7} + 942404 T_{19}^{6} - 25536424 T_{19}^{5} + 353223624 T_{19}^{4} - 2439206208 T_{19}^{3} + 9936102400 T_{19}^{2} + \cdots + 23975244288$$ T19^10 - 26*T19^9 + 338*T19^8 - 3952*T19^7 + 942404*T19^6 - 25536424*T19^5 + 353223624*T19^4 - 2439206208*T19^3 + 9936102400*T19^2 - 21827527680*T19 + 23975244288

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} - 6 T^{19} + \cdots + 3486784401$$
$5$ $$T^{20} + 3404 T^{16} + 3190384 T^{12} + \cdots + 4096$$
$7$ $$(T^{10} + 336 T^{8} + 40676 T^{6} + \cdots + 655360000)^{2}$$
$11$ $$T^{20} + 207308 T^{16} + \cdots + 22\!\cdots\!00$$
$13$ $$(T^{10} + 46 T^{9} + 1058 T^{8} + \cdots + 33620000)^{2}$$
$17$ $$(T^{10} + 952 T^{8} + \cdots + 15510536192)^{2}$$
$19$ $$(T^{10} - 26 T^{9} + 338 T^{8} + \cdots + 23975244288)^{2}$$
$23$ $$(T^{10} - 2236 T^{8} + \cdots - 2157878476800)^{2}$$
$29$ $$T^{20} + 8250700 T^{16} + \cdots + 14\!\cdots\!00$$
$31$ $$(T^{5} + 20 T^{4} - 2750 T^{3} + \cdots + 6473680)^{4}$$
$37$ $$(T^{10} - 58 T^{9} + \cdots + 93878430976800)^{2}$$
$41$ $$(T^{10} - 8644 T^{8} + \cdots - 89172136396800)^{2}$$
$43$ $$(T^{10} + 86 T^{9} + \cdots + 398518394892800)^{2}$$
$47$ $$(T^{10} + 4944 T^{8} + \cdots + 2199023255552)^{2}$$
$53$ $$T^{20} + 33387084 T^{16} + \cdots + 51\!\cdots\!00$$
$59$ $$T^{20} + 96029644 T^{16} + \cdots + 55\!\cdots\!00$$
$61$ $$(T^{10} - 122 T^{9} + \cdots + 79\!\cdots\!68)^{2}$$
$67$ $$(T^{10} + 178 T^{9} + \cdots + 14\!\cdots\!00)^{2}$$
$71$ $$(T^{10} - 12876 T^{8} + \cdots - 63\!\cdots\!00)^{2}$$
$73$ $$(T^{10} + 16160 T^{8} + \cdots + 900192010240000)^{2}$$
$79$ $$(T^{5} - 96 T^{4} - 3534 T^{3} + \cdots - 147403248)^{4}$$
$83$ $$T^{20} + 433330892 T^{16} + \cdots + 53\!\cdots\!00$$
$89$ $$(T^{10} - 49740 T^{8} + \cdots - 15\!\cdots\!00)^{2}$$
$97$ $$(T^{5} - 118 T^{4} - 11780 T^{3} + \cdots - 2657552000)^{4}$$