# Properties

 Label 378.2.m.a Level $378$ Weight $2$ Character orbit 378.m Analytic conductor $3.018$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.m (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561$$ x^16 - 6*x^14 + 9*x^12 + 54*x^10 - 288*x^8 + 486*x^6 + 729*x^4 - 4374*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{15} + \beta_{8} + \beta_{6}) q^{5} + (\beta_{13} - \beta_{10} + \beta_{2} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b7 * q^2 + (b5 + 1) * q^4 + (-b15 + b8 + b6) * q^5 + (b13 - b10 + b2 - 1) * q^7 + b3 * q^8 $$q + \beta_{7} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{15} + \beta_{8} + \beta_{6}) q^{5} + (\beta_{13} - \beta_{10} + \beta_{2} - 1) q^{7} + \beta_{3} q^{8} + ( - \beta_{15} + \beta_{8}) q^{10} + ( - \beta_{12} + \beta_{10} - \beta_{7} + \beta_{5} + \beta_{3} - \beta_1 + 2) q^{11} + ( - \beta_{15} + \beta_{4}) q^{13} + (\beta_{11} - \beta_{9} + \beta_{3} - \beta_{2} + 1) q^{14} + \beta_{5} q^{16} + ( - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{15} + \beta_{8} + \beta_{4}) q^{20} + ( - \beta_{12} + \beta_{10} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{22} + (\beta_{12} - \beta_{10} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 1) q^{23} + ( - \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{7} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{25} + (\beta_{15} + \beta_{8} - \beta_{6} + \beta_{4}) q^{26} + ( - \beta_{10} + \beta_{7}) q^{28} + (\beta_{14} + \beta_{13} - \beta_{10} - 2 \beta_{7} + \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{3} + \cdots + 1) q^{31}+ \cdots + ( - 2 \beta_{15} + 2 \beta_{14} - \beta_{12} + \beta_{11} - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{98}+O(q^{100})$$ q + b7 * q^2 + (b5 + 1) * q^4 + (-b15 + b8 + b6) * q^5 + (b13 - b10 + b2 - 1) * q^7 + b3 * q^8 + (-b15 + b8) * q^10 + (-b12 + b10 - b7 + b5 + b3 - b1 + 2) * q^11 + (-b15 + b4) * q^13 + (b11 - b9 + b3 - b2 + 1) * q^14 + b5 * q^16 + (-b14 + b13 - b11 - b3 + b2 + b1 - 1) * q^17 + (b14 + b13 - b12 - b11 - b10 + 2*b9 + b6 + b4 + b2 - b1 - 1) * q^19 + (-b15 + b8 + b4) * q^20 + (-b12 + b10 + b5 + 2*b3 - b2 - 2*b1 + 2) * q^22 + (b12 - b10 + 2*b7 - 2*b5 - 2*b3 + b2 + 1) * q^23 + (-b14 - b13 + b12 - 2*b7 + 3*b3 + b2 + 2*b1 - 1) * q^25 + (b15 + b8 - b6 + b4) * q^26 + (-b10 + b7) * q^28 + (b14 + b13 - b10 - 2*b7 + b2 + b1 - 1) * q^29 + (-b14 - b13 - b12 - 2*b11 + 2*b10 - 2*b9 - b3 - b2 + b1 + 1) * q^31 + (-b7 + b3) * q^32 + (-b13 + b12 + 2*b11 - b9 + b3 - b2 + 1) * q^34 + (-b15 - 2*b14 + 2*b12 + b11 - b10 - b8 + b7 + 2*b6 - 2*b5 - 2*b4 - 2*b3 + b2 + 3*b1 - 2) * q^35 + (2*b14 + 2*b13 - b12 - b10 - 4*b7 + 3*b3 - 2) * q^37 + (b15 - b14 + b12 + b10 - b9 - b6 - b4 - b3 + b1) * q^38 + (b8 - b6 + b4) * q^40 + (2*b15 + 2*b14 + b13 - b12 - 3*b11 + b10 + 3*b9 + 2*b4 - 2*b3 + b2 - 2*b1 - 1) * q^41 + (b12 - b10 + b7 - b5 - 3*b3 + 2*b2 + b1 - 2) * q^43 + (b14 + b13 - b12 - b7 + 2*b5 + 2*b3 - 2*b1 + 1) * q^44 + (b14 + b13 - b12 + 3*b7 - b3 + 1) * q^46 + (b14 - b13 - 2*b12 - b10 - 2*b8 + 2*b6 + 2*b3 - b2 - b1 + 1) * q^47 + (2*b15 - b14 - b12 - b11 + 2*b10 - b9 - 2*b4 + b3 - b2 - b1 + 1) * q^49 + (-b14 - b13 + b10 - b7 + 2*b5 + b1 - 2) * q^50 + (-b8 - b6) * q^52 + (2*b15 + 2*b14 + 2*b13 + b12 - 2*b11 + b10 + 4*b9 - 2*b8 - 3*b3 + 2*b2 - 2*b1 - 2) * q^55 + (-b9 + b1) * q^56 + (-b14 - b13 + b12 + 2*b7 - 2*b5 - b3 - b2 + b1 - 1) * q^58 + (-b15 - b13 + b12 + b11 + b10 - b9 - 2*b8 - 2*b6 - 3*b4 - b2 + 1) * q^59 + (b15 - b14 - b13 + 2*b12 + 4*b11 - b10 - 2*b9 + b8 + b4 + 2*b3 - b2 + b1 + 1) * q^61 + (2*b14 - 2*b13 - b12 + b10 + b3 - 2*b2 - 2*b1 + 2) * q^62 - q^64 + (-b14 - b13 + b10 - b7 - 3*b5 - b2 - b1 - 5) * q^65 + (-b14 - b13 + b10 - 4*b7 - 3*b5 - 5*b3 - 2*b2 - b1 - 1) * q^67 + (b13 - b12 - b11 - b10 + b9 + b2 - 1) * q^68 + (-b15 - 2*b14 - b13 + 2*b12 + 2*b11 - b10 - b9 + b8 + b7 + 2*b6 - b5 - b4 - 2*b3 + b2 + 3*b1 - 1) * q^70 + (-b14 - b13 + b10 + b7 - 2*b5 - 2*b3 - 2*b2 + 1) * q^71 + (-b14 - b13 + b11 - 2*b9 - 2*b6 - 2*b4 + b3 - b2 + b1 + 1) * q^73 + (-b14 - b13 + 2*b12 - b10 + 2*b7 - 2*b5 - 2*b3 - b2 + b1 - 3) * q^74 + (-b15 + b14 + b11 - b10 + b9 - b8 + b6 + b3 - b1) * q^76 + (2*b15 - 2*b14 - b13 + b12 + 2*b11 - b10 - 2*b9 - 2*b8 - 2*b7 - 2*b6 + 3*b5 + 3*b3 + 2*b1 - 3) * q^77 + (-2*b14 - 2*b13 + b12 + b10 + 4*b7 - b5 - 9*b3 + 3*b1) * q^79 + (-b6 + b4) * q^80 + (2*b15 + b14 + b13 + b12 - b11 + b10 + 2*b9 - 2*b8 - 2*b6 - 2*b4 - 2*b3 + b2 - b1 - 1) * q^82 + (-2*b15 + 4*b14 + 2*b13 - 2*b12 - 4*b10 + 6*b9 + 3*b8 - b6 + 2*b4 + 2*b3 + 2*b2 - 4*b1 - 2) * q^83 + (-b14 - b13 + b12 + 2*b7 - 2*b5 - b3 - b2 + b1 - 1) * q^85 + (-b12 + b10 - b5 - b2 + 2) * q^86 + (b14 + b13 - b12 - b7 + b5 + 3*b3 - b2 - 2*b1 + 1) * q^88 + (2*b14 - 2*b13 + 2*b11 - 2*b6 + 2*b4 + 2*b3 - 2*b2 - 2*b1 + 2) * q^89 + (-b15 - 2*b12 + b8 - 2*b7 - b6 - b4 + 2*b3 + 2) * q^91 + (-b14 - b13 + b12 + 3*b7 + 2*b5 - b3 - b2 + 5) * q^92 + (2*b15 - b14 - b13 - b12 - 2*b11 + 2*b10 - 2*b9 - 2*b4 - b3 - b2 + b1 + 1) * q^94 + (b14 + b13 - b10 + 3*b7 + b5 - 2*b3 - b1 - 1) * q^95 + (2*b15 + 3*b14 - 4*b13 + b12 + 2*b11 + 3*b10 - b9 - 2*b8 - 4*b6 + 2*b4 + b3 - 4*b2 - 3*b1 + 4) * q^97 + (-2*b15 + 2*b14 - b12 + b11 - 2*b8 - b7 + 2*b6 + 2*b5 - 2*b4 + 2*b3 - b2 - 3*b1 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + 2 q^{7}+O(q^{10})$$ 16 * q + 8 * q^4 + 2 * q^7 $$16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100})$$ 16 * q + 8 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^14 - 8 * q^16 + 48 * q^23 - 8 * q^25 + 4 * q^28 + 12 * q^29 - 8 * q^37 + 4 * q^43 + 24 * q^46 - 8 * q^49 - 60 * q^50 + 6 * q^56 - 12 * q^58 - 16 * q^64 - 84 * q^65 - 28 * q^67 - 36 * q^74 - 78 * q^77 - 4 * q^79 - 12 * q^85 + 24 * q^86 + 24 * q^91 + 48 * q^92 - 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 72\nu^{6} + 234\nu^{4} + 729\nu^{2} - 243 ) / 1944$$ (-v^14 - v^12 + 6*v^10 - 36*v^8 + 72*v^6 + 234*v^4 + 729*v^2 - 243) / 1944 $$\beta_{2}$$ $$=$$ $$( -2\nu^{14} + \nu^{12} - 6\nu^{10} + 36\nu^{8} - 180\nu^{6} - 396\nu^{4} + 972\nu^{2} - 4131 ) / 1944$$ (-2*v^14 + v^12 - 6*v^10 + 36*v^8 - 180*v^6 - 396*v^4 + 972*v^2 - 4131) / 1944 $$\beta_{3}$$ $$=$$ $$( \nu^{14} - 3\nu^{12} - 9\nu^{10} + 81\nu^{8} - 126\nu^{6} - 135\nu^{4} + 1458\nu^{2} - 2187 ) / 1458$$ (v^14 - 3*v^12 - 9*v^10 + 81*v^8 - 126*v^6 - 135*v^4 + 1458*v^2 - 2187) / 1458 $$\beta_{4}$$ $$=$$ $$( -2\nu^{15} - 9\nu^{13} + 18\nu^{11} - 396\nu^{7} + 216\nu^{5} - 324\nu^{3} - 9477\nu ) / 5832$$ (-2*v^15 - 9*v^13 + 18*v^11 - 396*v^7 + 216*v^5 - 324*v^3 - 9477*v) / 5832 $$\beta_{5}$$ $$=$$ $$( 2\nu^{14} - 21\nu^{12} + 18\nu^{10} + 108\nu^{8} - 576\nu^{6} + 648\nu^{4} + 972\nu^{2} - 9477 ) / 5832$$ (2*v^14 - 21*v^12 + 18*v^10 + 108*v^8 - 576*v^6 + 648*v^4 + 972*v^2 - 9477) / 5832 $$\beta_{6}$$ $$=$$ $$( -7\nu^{15} + 24\nu^{13} - 36\nu^{11} - 540\nu^{9} + 1044\nu^{7} - 1134\nu^{5} - 8019\nu^{3} + 13122\nu ) / 17496$$ (-7*v^15 + 24*v^13 - 36*v^11 - 540*v^9 + 1044*v^7 - 1134*v^5 - 8019*v^3 + 13122*v) / 17496 $$\beta_{7}$$ $$=$$ $$( 3\nu^{14} - 10\nu^{12} - 12\nu^{10} + 180\nu^{8} - 432\nu^{6} - 198\nu^{4} + 3483\nu^{2} - 5832 ) / 1944$$ (3*v^14 - 10*v^12 - 12*v^10 + 180*v^8 - 432*v^6 - 198*v^4 + 3483*v^2 - 5832) / 1944 $$\beta_{8}$$ $$=$$ $$( -4\nu^{15} + 9\nu^{13} + 18\nu^{11} - 216\nu^{9} + 504\nu^{7} + 432\nu^{5} - 2754\nu^{3} + 9477\nu ) / 5832$$ (-4*v^15 + 9*v^13 + 18*v^11 - 216*v^9 + 504*v^7 + 432*v^5 - 2754*v^3 + 9477*v) / 5832 $$\beta_{9}$$ $$=$$ $$( - 8 \nu^{15} - 3 \nu^{14} + 48 \nu^{13} - 36 \nu^{12} - 18 \nu^{11} + 216 \nu^{10} - 270 \nu^{9} - 162 \nu^{8} + 1332 \nu^{7} - 594 \nu^{6} - 2430 \nu^{5} + 5346 \nu^{4} + 486 \nu^{3} + \cdots - 8748 ) / 17496$$ (-8*v^15 - 3*v^14 + 48*v^13 - 36*v^12 - 18*v^11 + 216*v^10 - 270*v^9 - 162*v^8 + 1332*v^7 - 594*v^6 - 2430*v^5 + 5346*v^4 + 486*v^3 - 729*v^2 + 17496*v - 8748) / 17496 $$\beta_{10}$$ $$=$$ $$( - 8 \nu^{15} - 27 \nu^{14} + 12 \nu^{13} + 72 \nu^{12} + 90 \nu^{11} + 54 \nu^{10} - 432 \nu^{9} - 810 \nu^{8} - 126 \nu^{7} + 2916 \nu^{6} + 2106 \nu^{5} - 3240 \nu^{4} - 7776 \nu^{3} + \cdots + 21870 ) / 17496$$ (-8*v^15 - 27*v^14 + 12*v^13 + 72*v^12 + 90*v^11 + 54*v^10 - 432*v^9 - 810*v^8 - 126*v^7 + 2916*v^6 + 2106*v^5 - 3240*v^4 - 7776*v^3 - 6561*v^2 + 4374*v + 21870) / 17496 $$\beta_{11}$$ $$=$$ $$( - 4 \nu^{15} - 39 \nu^{14} + 24 \nu^{13} + 108 \nu^{12} - 90 \nu^{11} + 162 \nu^{10} + 108 \nu^{9} - 1782 \nu^{8} + 666 \nu^{7} + 4428 \nu^{6} - 3402 \nu^{5} - 1620 \nu^{4} + 3888 \nu^{3} + \cdots + 48114 ) / 17496$$ (-4*v^15 - 39*v^14 + 24*v^13 + 108*v^12 - 90*v^11 + 162*v^10 + 108*v^9 - 1782*v^8 + 666*v^7 + 4428*v^6 - 3402*v^5 - 1620*v^4 + 3888*v^3 - 24057*v^2 + 13122*v + 48114) / 17496 $$\beta_{12}$$ $$=$$ $$( - 8 \nu^{15} + 39 \nu^{14} + 12 \nu^{13} - 108 \nu^{12} + 90 \nu^{11} - 162 \nu^{10} - 432 \nu^{9} + 1782 \nu^{8} - 126 \nu^{7} - 4428 \nu^{6} + 2106 \nu^{5} + 1620 \nu^{4} - 7776 \nu^{3} + \cdots - 48114 ) / 17496$$ (-8*v^15 + 39*v^14 + 12*v^13 - 108*v^12 + 90*v^11 - 162*v^10 - 432*v^9 + 1782*v^8 - 126*v^7 - 4428*v^6 + 2106*v^5 + 1620*v^4 - 7776*v^3 + 24057*v^2 + 4374*v - 48114) / 17496 $$\beta_{13}$$ $$=$$ $$( 14 \nu^{15} - 6 \nu^{14} - 66 \nu^{13} + 99 \nu^{12} + 72 \nu^{11} - 108 \nu^{10} + 594 \nu^{9} - 972 \nu^{8} - 2574 \nu^{7} + 5130 \nu^{6} + 1620 \nu^{5} - 5022 \nu^{4} + 7776 \nu^{3} + \cdots + 85293 ) / 17496$$ (14*v^15 - 6*v^14 - 66*v^13 + 99*v^12 + 72*v^11 - 108*v^10 + 594*v^9 - 972*v^8 - 2574*v^7 + 5130*v^6 + 1620*v^5 - 5022*v^4 + 7776*v^3 - 14580*v^2 - 17496*v + 85293) / 17496 $$\beta_{14}$$ $$=$$ $$( - 22 \nu^{15} - 6 \nu^{14} + 78 \nu^{13} + 27 \nu^{12} + 18 \nu^{11} - 162 \nu^{10} - 1026 \nu^{9} - 162 \nu^{8} + 2448 \nu^{7} + 1242 \nu^{6} + 486 \nu^{5} - 3240 \nu^{4} - 15552 \nu^{3} + \cdots + 6561 ) / 17496$$ (-22*v^15 - 6*v^14 + 78*v^13 + 27*v^12 + 18*v^11 - 162*v^10 - 1026*v^9 - 162*v^8 + 2448*v^7 + 1242*v^6 + 486*v^5 - 3240*v^4 - 15552*v^3 + 7290*v^2 + 21870*v + 6561) / 17496 $$\beta_{15}$$ $$=$$ $$( -35\nu^{15} + 138\nu^{13} + 36\nu^{11} - 2052\nu^{9} + 6192\nu^{7} - 2106\nu^{5} - 37179\nu^{3} + 87480\nu ) / 17496$$ (-35*v^15 + 138*v^13 + 36*v^11 - 2052*v^9 + 6192*v^7 - 2106*v^5 - 37179*v^3 + 87480*v) / 17496
 $$\nu$$ $$=$$ $$( -\beta_{14} + 2\beta_{12} + 2\beta_{11} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_1 ) / 3$$ (-b14 + 2*b12 + 2*b11 - b9 + b8 - b4 + b1) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} - \beta_{3} + \beta_1$$ b7 - b5 - b3 + b1 $$\nu^{3}$$ $$=$$ $$- \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{6} + \beta_{2} - \beta _1 - 1$$ -b15 + b14 + b13 - b12 - b11 - b10 + 2*b9 + 2*b8 + b6 + b2 - b1 - 1 $$\nu^{4}$$ $$=$$ $$-\beta_{14} - \beta_{13} + 2\beta_{12} - \beta_{10} - 4\beta_{7} - 3\beta_{5} + 3\beta_{3} + 3\beta_1$$ -b14 - b13 + 2*b12 - b10 - 4*b7 - 3*b5 + 3*b3 + 3*b1 $$\nu^{5}$$ $$=$$ $$- 2 \beta_{15} - 5 \beta_{13} + \beta_{11} + 4 \beta_{10} - 5 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 5 \beta_{2} + 5$$ -2*b15 - 5*b13 + b11 + 4*b10 - 5*b9 + 3*b8 - 2*b6 - 3*b4 + b3 - 5*b2 + 5 $$\nu^{6}$$ $$=$$ $$-3\beta_{12} + 3\beta_{10} - 6\beta_{7} + 6\beta_{5} + 18\beta_{3} - 6\beta_{2} - 6$$ -3*b12 + 3*b10 - 6*b7 + 6*b5 + 18*b3 - 6*b2 - 6 $$\nu^{7}$$ $$=$$ $$3 \beta_{15} + 3 \beta_{14} - 18 \beta_{12} - 12 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 9 \beta_{6} + 6 \beta_{4} + 6 \beta_{3} - 3 \beta_1$$ 3*b15 + 3*b14 - 18*b12 - 12*b11 - 6*b10 + 3*b9 + 6*b8 - 9*b6 + 6*b4 + 6*b3 - 3*b1 $$\nu^{8}$$ $$=$$ $$- 12 \beta_{14} - 12 \beta_{13} - 3 \beta_{12} + 15 \beta_{10} - 33 \beta_{7} + 3 \beta_{5} + 75 \beta_{3} - 12 \beta_{2} - 15 \beta _1 + 27$$ -12*b14 - 12*b13 - 3*b12 + 15*b10 - 33*b7 + 3*b5 + 75*b3 - 12*b2 - 15*b1 + 27 $$\nu^{9}$$ $$=$$ $$18 \beta_{15} - 15 \beta_{14} - 27 \beta_{13} + 21 \beta_{12} + 39 \beta_{11} + 9 \beta_{10} - 42 \beta_{9} - 21 \beta_{8} - 72 \beta_{6} + 3 \beta_{4} + 18 \beta_{3} - 27 \beta_{2} + 15 \beta _1 + 27$$ 18*b15 - 15*b14 - 27*b13 + 21*b12 + 39*b11 + 9*b10 - 42*b9 - 21*b8 - 72*b6 + 3*b4 + 18*b3 - 27*b2 + 15*b1 + 27 $$\nu^{10}$$ $$=$$ $$- 45 \beta_{14} - 45 \beta_{13} - 45 \beta_{12} + 90 \beta_{10} + 90 \beta_{7} - 27 \beta_{5} - 27 \beta_{3} - 72 \beta_{2} - 45 \beta _1 + 27$$ -45*b14 - 45*b13 - 45*b12 + 90*b10 + 90*b7 - 27*b5 - 27*b3 - 72*b2 - 45*b1 + 27 $$\nu^{11}$$ $$=$$ $$63 \beta_{15} + 36 \beta_{14} + 153 \beta_{13} - 54 \beta_{12} - 135 \beta_{11} - 72 \beta_{10} + 189 \beta_{9} - 36 \beta_{8} - 117 \beta_{6} + 72 \beta_{4} - 81 \beta_{3} + 153 \beta_{2} - 36 \beta _1 - 153$$ 63*b15 + 36*b14 + 153*b13 - 54*b12 - 135*b11 - 72*b10 + 189*b9 - 36*b8 - 117*b6 + 72*b4 - 81*b3 + 153*b2 - 36*b1 - 153 $$\nu^{12}$$ $$=$$ $$-126\beta_{14} - 126\beta_{13} + 90\beta_{12} + 36\beta_{10} + 90\beta_{7} - 648\beta_{5} - 216\beta_{3} - 243$$ -126*b14 - 126*b13 + 90*b12 + 36*b10 + 90*b7 - 648*b5 - 216*b3 - 243 $$\nu^{13}$$ $$=$$ $$- 36 \beta_{15} + 189 \beta_{14} - 36 \beta_{13} + 216 \beta_{12} + 18 \beta_{11} + 234 \beta_{10} + 153 \beta_{9} - 459 \beta_{8} - 144 \beta_{6} - 297 \beta_{4} - 198 \beta_{3} - 36 \beta_{2} - 189 \beta _1 + 36$$ -36*b15 + 189*b14 - 36*b13 + 216*b12 + 18*b11 + 234*b10 + 153*b9 - 459*b8 - 144*b6 - 297*b4 - 198*b3 - 36*b2 - 189*b1 + 36 $$\nu^{14}$$ $$=$$ $$54 \beta_{14} + 54 \beta_{13} - 54 \beta_{10} + 999 \beta_{7} - 621 \beta_{5} - 1377 \beta_{3} - 432 \beta_{2} - 243 \beta _1 - 1242$$ 54*b14 + 54*b13 - 54*b10 + 999*b7 - 621*b5 - 1377*b3 - 432*b2 - 243*b1 - 1242 $$\nu^{15}$$ $$=$$ $$81 \beta_{15} + 297 \beta_{14} + 837 \beta_{13} - 891 \beta_{12} - 1809 \beta_{11} + 81 \beta_{10} + 1134 \beta_{9} - 1026 \beta_{8} + 999 \beta_{6} - 864 \beta_{4} - 918 \beta_{3} + 837 \beta_{2} - 297 \beta _1 - 837$$ 81*b15 + 297*b14 + 837*b13 - 891*b12 - 1809*b11 + 81*b10 + 1134*b9 - 1026*b8 + 999*b6 - 864*b4 - 918*b3 + 837*b2 - 297*b1 - 837

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1 + \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 −1.69547 − 0.354107i 0.0967785 − 1.72934i −0.0967785 + 1.72934i 1.69547 + 0.354107i −1.62181 + 0.608059i 1.40917 + 1.00709i −1.40917 − 1.00709i 1.62181 − 0.608059i −1.69547 + 0.354107i 0.0967785 + 1.72934i −0.0967785 − 1.72934i 1.69547 − 0.354107i −1.62181 − 0.608059i 1.40917 − 1.00709i −1.40917 + 1.00709i 1.62181 + 0.608059i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.895175 + 1.55049i 0 0.0213944 2.64566i 1.00000i 0 1.79035i
125.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.183299 + 0.317483i 0 −0.624224 + 2.57106i 1.00000i 0 0.366598i
125.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.183299 0.317483i 0 2.53871 + 0.744936i 1.00000i 0 0.366598i
125.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.895175 1.55049i 0 −2.30191 1.30430i 1.00000i 0 1.79035i
125.5 0.866025 0.500000i 0 0.500000 0.866025i −1.94556 + 3.36980i 0 0.343982 + 2.62329i 1.00000i 0 3.89111i
125.6 0.866025 0.500000i 0 0.500000 0.866025i −1.17468 + 2.03460i 0 1.55364 2.14154i 1.00000i 0 2.34936i
125.7 0.866025 0.500000i 0 0.500000 0.866025i 1.17468 2.03460i 0 −2.63145 + 0.274725i 1.00000i 0 2.34936i
125.8 0.866025 0.500000i 0 0.500000 0.866025i 1.94556 3.36980i 0 2.09985 + 1.60954i 1.00000i 0 3.89111i
251.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.895175 1.55049i 0 0.0213944 + 2.64566i 1.00000i 0 1.79035i
251.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.183299 0.317483i 0 −0.624224 2.57106i 1.00000i 0 0.366598i
251.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.183299 + 0.317483i 0 2.53871 0.744936i 1.00000i 0 0.366598i
251.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.895175 + 1.55049i 0 −2.30191 + 1.30430i 1.00000i 0 1.79035i
251.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.94556 3.36980i 0 0.343982 2.62329i 1.00000i 0 3.89111i
251.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.17468 2.03460i 0 1.55364 + 2.14154i 1.00000i 0 2.34936i
251.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.17468 + 2.03460i 0 −2.63145 0.274725i 1.00000i 0 2.34936i
251.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.94556 + 3.36980i 0 2.09985 1.60954i 1.00000i 0 3.89111i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.m.a 16
3.b odd 2 1 126.2.m.a 16
4.b odd 2 1 3024.2.cc.b 16
7.b odd 2 1 inner 378.2.m.a 16
7.c even 3 1 2646.2.l.b 16
7.c even 3 1 2646.2.t.a 16
7.d odd 6 1 2646.2.l.b 16
7.d odd 6 1 2646.2.t.a 16
9.c even 3 1 126.2.m.a 16
9.c even 3 1 1134.2.d.a 16
9.d odd 6 1 inner 378.2.m.a 16
9.d odd 6 1 1134.2.d.a 16
12.b even 2 1 1008.2.cc.b 16
21.c even 2 1 126.2.m.a 16
21.g even 6 1 882.2.l.a 16
21.g even 6 1 882.2.t.b 16
21.h odd 6 1 882.2.l.a 16
21.h odd 6 1 882.2.t.b 16
28.d even 2 1 3024.2.cc.b 16
36.f odd 6 1 1008.2.cc.b 16
36.h even 6 1 3024.2.cc.b 16
63.g even 3 1 882.2.l.a 16
63.h even 3 1 882.2.t.b 16
63.i even 6 1 2646.2.t.a 16
63.j odd 6 1 2646.2.t.a 16
63.k odd 6 1 882.2.l.a 16
63.l odd 6 1 126.2.m.a 16
63.l odd 6 1 1134.2.d.a 16
63.n odd 6 1 2646.2.l.b 16
63.o even 6 1 inner 378.2.m.a 16
63.o even 6 1 1134.2.d.a 16
63.s even 6 1 2646.2.l.b 16
63.t odd 6 1 882.2.t.b 16
84.h odd 2 1 1008.2.cc.b 16
252.s odd 6 1 3024.2.cc.b 16
252.bi even 6 1 1008.2.cc.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 3.b odd 2 1
126.2.m.a 16 9.c even 3 1
126.2.m.a 16 21.c even 2 1
126.2.m.a 16 63.l odd 6 1
378.2.m.a 16 1.a even 1 1 trivial
378.2.m.a 16 7.b odd 2 1 inner
378.2.m.a 16 9.d odd 6 1 inner
378.2.m.a 16 63.o even 6 1 inner
882.2.l.a 16 21.g even 6 1
882.2.l.a 16 21.h odd 6 1
882.2.l.a 16 63.g even 3 1
882.2.l.a 16 63.k odd 6 1
882.2.t.b 16 21.g even 6 1
882.2.t.b 16 21.h odd 6 1
882.2.t.b 16 63.h even 3 1
882.2.t.b 16 63.t odd 6 1
1008.2.cc.b 16 12.b even 2 1
1008.2.cc.b 16 36.f odd 6 1
1008.2.cc.b 16 84.h odd 2 1
1008.2.cc.b 16 252.bi even 6 1
1134.2.d.a 16 9.c even 3 1
1134.2.d.a 16 9.d odd 6 1
1134.2.d.a 16 63.l odd 6 1
1134.2.d.a 16 63.o even 6 1
2646.2.l.b 16 7.c even 3 1
2646.2.l.b 16 7.d odd 6 1
2646.2.l.b 16 63.n odd 6 1
2646.2.l.b 16 63.s even 6 1
2646.2.t.a 16 7.c even 3 1
2646.2.t.a 16 7.d odd 6 1
2646.2.t.a 16 63.i even 6 1
2646.2.t.a 16 63.j odd 6 1
3024.2.cc.b 16 4.b odd 2 1
3024.2.cc.b 16 28.d even 2 1
3024.2.cc.b 16 36.h even 6 1
3024.2.cc.b 16 252.s odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 24 T^{14} + 423 T^{12} + \cdots + 1296$$
$7$ $$T^{16} - 2 T^{15} + 6 T^{14} + \cdots + 5764801$$
$11$ $$(T^{8} - 6 T^{7} - 9 T^{6} + 126 T^{5} + \cdots + 1296)^{2}$$
$13$ $$T^{16} - 36 T^{14} + 972 T^{12} + \cdots + 331776$$
$17$ $$(T^{8} - 42 T^{6} + 477 T^{4} - 1296 T^{2} + \cdots + 576)^{2}$$
$19$ $$(T^{8} + 54 T^{6} + 594 T^{4} + 1854 T^{2} + \cdots + 1521)^{2}$$
$23$ $$(T^{8} - 24 T^{7} + 225 T^{6} + \cdots + 443556)^{2}$$
$29$ $$(T^{8} - 6 T^{7} - 18 T^{6} + 180 T^{5} + \cdots + 20736)^{2}$$
$31$ $$T^{16} - 144 T^{14} + \cdots + 557256278016$$
$37$ $$(T^{4} + 2 T^{3} - 102 T^{2} - 184 T + 1336)^{4}$$
$41$ $$T^{16} + 258 T^{14} + \cdots + 73499483897856$$
$43$ $$(T^{8} - 2 T^{7} + 43 T^{6} - 218 T^{5} + \cdots + 10816)^{2}$$
$47$ $$T^{16} + 240 T^{14} + \cdots + 1485512441856$$
$53$ $$T^{16}$$
$59$ $$T^{16} + 294 T^{14} + 68787 T^{12} + \cdots + 1296$$
$61$ $$T^{16} - 240 T^{14} + \cdots + 2425818710016$$
$67$ $$(T^{8} + 14 T^{7} + 307 T^{6} + \cdots + 824464)^{2}$$
$71$ $$(T^{8} + 90 T^{6} + 2745 T^{4} + \cdots + 82944)^{2}$$
$73$ $$(T^{8} + 222 T^{6} + 12069 T^{4} + \cdots + 1710864)^{2}$$
$79$ $$(T^{8} + 2 T^{7} + 133 T^{6} + \cdots + 1444804)^{2}$$
$83$ $$T^{16} + 708 T^{14} + \cdots + 33\!\cdots\!56$$
$89$ $$(T^{8} - 216 T^{6} + 12960 T^{4} + \cdots + 186624)^{2}$$
$97$ $$T^{16} - 702 T^{14} + \cdots + 45\!\cdots\!56$$