Properties

 Label 912.2.bn.n Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(65,912)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("912.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.bn (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561$$ x^16 - x^15 - 6*x^14 + 5*x^13 + 21*x^12 - 4*x^11 - 94*x^10 - 6*x^9 + 364*x^8 - 18*x^7 - 846*x^6 - 108*x^5 + 1701*x^4 + 1215*x^3 - 4374*x^2 - 2187*x + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 456) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{3} + (\beta_{14} - \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{12} - \beta_{3} - 1) q^{9}+O(q^{10})$$ q - b4 * q^3 + (b14 - b11) * q^5 + (b13 - b5) * q^7 + (b12 - b3 - 1) * q^9 $$q - \beta_{4} q^{3} + (\beta_{14} - \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{12} - \beta_{3} - 1) q^{9} + (\beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{4} - \beta_{2}) q^{13} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{15} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_1) q^{17} + ( - \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{19}+ \cdots + ( - \beta_{13} - \beta_{12} - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots - \beta_1) q^{99}+O(q^{100})$$ q - b4 * q^3 + (b14 - b11) * q^5 + (b13 - b5) * q^7 + (b12 - b3 - 1) * q^9 + (b14 - b12 + b11 + b10 + b8 - b7 - b5 - 2*b3 - b2 - 1) * q^11 + (b13 - b12 - b10 + b9 - b5 - b4 - b2) * q^13 + (b8 + b7 + b6 - b3 - 1) * q^15 + (-b15 - b14 - b13 - b10 + b7 + b6 + b5 + b1) * q^17 + (-b14 - b13 + b12 - 2*b9 + b7 + b6 + b5 + 2*b4 + 2*b1 - 1) * q^19 + (b15 + 2*b14 + b13 - b12 - b11 + b10 + b9 - b8 - b6 - b5 - b4 - 2*b1 - 1) * q^21 + (-b14 - 2*b9 + 2*b7 + 2*b4) * q^23 + (-b13 + 2*b11 + b10 - b9 + b8 - 2*b7 - b5 + b4 - b3 - b2 + 2*b1) * q^25 + (b14 + 2*b13 - b12 - 2*b11 - b10 + b9 + b7 - b5 - b4 - 2*b1) * q^27 + (-b15 - b14 - 2*b13 + b12 - b10 + b8 + 2*b6 + b5 + 2*b4 - b3 + 2*b1) * q^29 + (b14 - b13 - b12 + b11 + b10 - b9 + 2*b8 + b7 + b6 - b5 - 2*b2 - b1) * q^31 + (-b15 - 2*b14 + b12 - b8 + 2*b7 + b5 + b2 - 2*b1 + 1) * q^33 + (b15 + b14 + b13 - 2*b12 - b11 + 2*b9 - 2*b8 + b7 - b6 - 3*b4 + b3 - b2 - 3*b1 - 1) * q^35 + (-b9 + b7 - b4 + 2*b3 - b1 + 1) * q^37 + (b15 + b14 + b13 - b11 + b10 - b9 - 2*b8 - b1) * q^39 + (b14 + b10 + 2*b9 - 3*b7 - b5 + b3 + b2 + 2*b1 + 1) * q^41 + (2*b14 - b12 + b11 + b10 + b9 + b8 - 4*b7 - 3*b5 - 2*b4 - b3 + b1 - 1) * q^43 + (b15 - b13 + b11 + 2*b10 - 2*b9 + b7 - b6 - 2*b5 + b4 - b3 - b2 - b1 + 1) * q^45 + (-b15 - b14 + b12 + b10 + b8 - b5 + 2*b4 - b3 + 2*b2 + 2*b1 - 2) * q^47 + (-2*b15 - 2*b14 + 2*b12 + b11 + b9 + b8 - 2*b7 + b6 - b5 + b2 + b1 + 1) * q^49 + (-2*b14 - 3*b13 + 2*b12 - b9 - b7 + b6 + 2*b5 + 2*b4 - 2*b3 + 2*b1) * q^51 + (b14 + 2*b13 - b12 - 2*b11 + b9 - b8 + b3 - b1) * q^53 + (-b15 - b14 + b13 - b12 - b11 + b9 + b8 - 3*b7 - b6 - b5 - 2*b4 + 3*b3 + 3) * q^55 + (b15 - b14 - b6 - b5 + b4 + b2 - 2) * q^57 + (-b15 - 3*b14 + b13 + 2*b12 - b11 - 2*b10 - 2*b8 + 2*b7 - b6 + 2*b5 + b3 + b2 - b1 + 1) * q^59 + (2*b14 + b13 - 2*b11 + b10 + b8 - 2*b7 - b5 + b4 + b3 - b2 + b1) * q^61 + (b14 + b13 - b12 + 2*b11 + b10 + 3*b9 + b8 - 2*b7 - b5 - b4 - 2*b3 - b2 - b1 + 1) * q^63 + (-b14 + 4*b13 - 3*b12 - b11 - 3*b10 + 3*b9 - b8 + 2*b7 - b5 - 3*b4 - b2 - b1 - 1) * q^65 + (-2*b15 - 2*b14 + b13 + b12 - 2*b10 + 2*b9 - b8 + b3 + 2*b1 + 2) * q^67 + (b15 - 2*b12 - b9 - b8 + b7 - b6 + b5 - 3*b3 - b1 - 4) * q^69 + (b15 + b14 - b13 + b10 + b7 + b6 + b5 + 2*b4 - 4*b3 + b1 - 4) * q^71 + (2*b14 - 2*b12 + 2*b10 + 2*b8 - 2*b7 - 2*b5 + b3 + 1) * q^73 + (-b15 - 2*b14 + b13 + b12 - b11 - 2*b10 + b9 - 2*b8 + 2*b7 + b5 - b4 + 2*b3 + 2*b2 - b1) * q^75 + (-2*b14 + b13 + 2*b12 - b11 - 2*b10 - b9 + 3*b7 + b6 + 4*b5 - 4*b3 - 2) * q^77 + (-2*b14 + b12 + b11 - b10 - 3*b9 + b8 + 2*b7 + b5 + 2*b4 - b3 + 2*b2 + 3*b1 + 1) * q^79 + (2*b15 + 3*b14 - 2*b12 + 3*b10 - 2*b9 + b7 - b5 - b3 - 2*b2 - 2*b1 - 2) * q^81 + (-b13 + 2*b11 + 2*b9 - 2*b7 - b6 - 2*b5 + b4 + b1) * q^83 + (-b15 - 3*b14 + 2*b12 - 3*b10 - 3*b9 + 4*b7 + 2*b6 + 3*b5 + 2*b2 + 3*b1) * q^85 + (b15 - 2*b14 - 3*b13 - 2*b9 + b7 + b6 + b4 + 3*b3 - b2 + b1 + 3) * q^87 + (-b14 - 2*b13 + b12 + 2*b11 - 3*b9 + b8 + 2*b7 - 2*b4 - 3*b3 + b1) * q^89 + (-b15 + b14 + b12 + b9 + b4 + 3*b3 + b2 + 2*b1 + 6) * q^91 + (-4*b14 - 2*b9 - 2*b8 + 5*b7 + b5 + 2*b4 - 4*b3 - b2 - 1) * q^93 + (b14 - 2*b13 + b12 + b11 + 2*b10 - 3*b9 - b8 - 2*b7 - 2*b5 - b3 - b1 - 2) * q^95 + (b15 + b14 - b13 - b12 + b11 + 2*b10 - b9 - b8 - 2*b7 - b6 - b5 + b4 + 3*b3 - 2*b2 - 3) * q^97 + (-b13 - b12 - 2*b11 - b10 + 2*b9 - b7 - b6 + 3*b5 - b4 - 3*b3 - b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - q^{3} + 3 q^{5} - 5 q^{9}+O(q^{10})$$ 16 * q - q^3 + 3 * q^5 - 5 * q^9 $$16 q - q^{3} + 3 q^{5} - 5 q^{9} - 3 q^{13} - 12 q^{15} - 3 q^{17} - 11 q^{19} - 12 q^{21} + 3 q^{23} + 11 q^{25} - 4 q^{27} + 5 q^{29} + 14 q^{33} - 24 q^{35} + 9 q^{39} + 6 q^{41} - 13 q^{43} + 33 q^{45} - 27 q^{47} + 8 q^{49} + 18 q^{51} - 7 q^{53} + 12 q^{55} - 36 q^{57} + 10 q^{59} - q^{61} + 26 q^{63} - 30 q^{65} + 24 q^{67} - 41 q^{69} - 27 q^{71} + 2 q^{73} - 21 q^{75} + 21 q^{79} - 13 q^{81} - 5 q^{85} + 23 q^{87} + 25 q^{89} + 78 q^{91} + 22 q^{93} - 13 q^{95} - 60 q^{97} + 20 q^{99}+O(q^{100})$$ 16 * q - q^3 + 3 * q^5 - 5 * q^9 - 3 * q^13 - 12 * q^15 - 3 * q^17 - 11 * q^19 - 12 * q^21 + 3 * q^23 + 11 * q^25 - 4 * q^27 + 5 * q^29 + 14 * q^33 - 24 * q^35 + 9 * q^39 + 6 * q^41 - 13 * q^43 + 33 * q^45 - 27 * q^47 + 8 * q^49 + 18 * q^51 - 7 * q^53 + 12 * q^55 - 36 * q^57 + 10 * q^59 - q^61 + 26 * q^63 - 30 * q^65 + 24 * q^67 - 41 * q^69 - 27 * q^71 + 2 * q^73 - 21 * q^75 + 21 * q^79 - 13 * q^81 - 5 * q^85 + 23 * q^87 + 25 * q^89 + 78 * q^91 + 22 * q^93 - 13 * q^95 - 60 * q^97 + 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ v^2 - 1 $$\beta_{3}$$ $$=$$ $$( 3 \nu^{15} - 4 \nu^{14} + 16 \nu^{13} + 33 \nu^{12} - 50 \nu^{11} - 30 \nu^{10} + 316 \nu^{9} + 106 \nu^{8} - 1050 \nu^{7} - 1012 \nu^{6} + 2850 \nu^{5} + 2862 \nu^{4} - 7911 \nu^{3} - 1782 \nu^{2} + \cdots + 2187 ) / 23328$$ (3*v^15 - 4*v^14 + 16*v^13 + 33*v^12 - 50*v^11 - 30*v^10 + 316*v^9 + 106*v^8 - 1050*v^7 - 1012*v^6 + 2850*v^5 + 2862*v^4 - 7911*v^3 - 1782*v^2 + 18954*v + 2187) / 23328 $$\beta_{4}$$ $$=$$ $$( - \nu^{15} + 34 \nu^{14} + 18 \nu^{13} - 113 \nu^{12} - 18 \nu^{11} + 598 \nu^{10} + 124 \nu^{9} - 2142 \nu^{8} - 958 \nu^{7} + 5388 \nu^{6} + 3186 \nu^{5} - 13014 \nu^{4} - 5427 \nu^{3} + \cdots - 19683 ) / 23328$$ (-v^15 + 34*v^14 + 18*v^13 - 113*v^12 - 18*v^11 + 598*v^10 + 124*v^9 - 2142*v^8 - 958*v^7 + 5388*v^6 + 3186*v^5 - 13014*v^4 - 5427*v^3 + 32076*v^2 + 8748*v - 19683) / 23328 $$\beta_{5}$$ $$=$$ $$( - 4 \nu^{15} - 53 \nu^{14} + 153 \nu^{13} + 385 \nu^{12} - 612 \nu^{11} - 1226 \nu^{10} + 1576 \nu^{9} + 5904 \nu^{8} - 4534 \nu^{7} - 19056 \nu^{6} + 13932 \nu^{5} + 42066 \nu^{4} + \cdots + 203391 ) / 34992$$ (-4*v^15 - 53*v^14 + 153*v^13 + 385*v^12 - 612*v^11 - 1226*v^10 + 1576*v^9 + 5904*v^8 - 4534*v^7 - 19056*v^6 + 13932*v^5 + 42066*v^4 - 8748*v^3 - 95499*v^2 + 15309*v + 203391) / 34992 $$\beta_{6}$$ $$=$$ $$( 25 \nu^{15} - 76 \nu^{14} + 467 \nu^{12} - 54 \nu^{11} - 1162 \nu^{10} - 364 \nu^{9} + 4734 \nu^{8} + 2962 \nu^{7} - 20796 \nu^{6} - 4122 \nu^{5} + 12474 \nu^{4} + 8667 \nu^{3} + \cdots + 111537 ) / 69984$$ (25*v^15 - 76*v^14 + 467*v^12 - 54*v^11 - 1162*v^10 - 364*v^9 + 4734*v^8 + 2962*v^7 - 20796*v^6 - 4122*v^5 + 12474*v^4 + 8667*v^3 + 2430*v^2 - 71442*v + 111537) / 69984 $$\beta_{7}$$ $$=$$ $$( - \nu^{15} + \nu^{14} + 6 \nu^{13} - 5 \nu^{12} - 21 \nu^{11} + 4 \nu^{10} + 94 \nu^{9} + 6 \nu^{8} - 364 \nu^{7} + 18 \nu^{6} + 846 \nu^{5} + 108 \nu^{4} - 1701 \nu^{3} - 1215 \nu^{2} + 4374 \nu + 2187 ) / 2187$$ (-v^15 + v^14 + 6*v^13 - 5*v^12 - 21*v^11 + 4*v^10 + 94*v^9 + 6*v^8 - 364*v^7 + 18*v^6 + 846*v^5 + 108*v^4 - 1701*v^3 - 1215*v^2 + 4374*v + 2187) / 2187 $$\beta_{8}$$ $$=$$ $$( - \nu^{15} - 2 \nu^{14} + 9 \nu^{13} + 13 \nu^{12} - 36 \nu^{11} - 59 \nu^{10} + 106 \nu^{9} + 288 \nu^{8} - 346 \nu^{7} - 1074 \nu^{6} + 900 \nu^{5} + 2646 \nu^{4} - 1377 \nu^{3} - 6318 \nu^{2} + \cdots + 13122 ) / 2187$$ (-v^15 - 2*v^14 + 9*v^13 + 13*v^12 - 36*v^11 - 59*v^10 + 106*v^9 + 288*v^8 - 346*v^7 - 1074*v^6 + 900*v^5 + 2646*v^4 - 1377*v^3 - 6318*v^2 + 729*v + 13122) / 2187 $$\beta_{9}$$ $$=$$ $$( - 35 \nu^{15} + 62 \nu^{14} + 174 \nu^{13} - 31 \nu^{12} - 438 \nu^{11} - 310 \nu^{10} + 3020 \nu^{9} + 3054 \nu^{8} - 11786 \nu^{7} - 8820 \nu^{6} + 20502 \nu^{5} + 29430 \nu^{4} + \cdots + 247131 ) / 69984$$ (-35*v^15 + 62*v^14 + 174*v^13 - 31*v^12 - 438*v^11 - 310*v^10 + 3020*v^9 + 3054*v^8 - 11786*v^7 - 8820*v^6 + 20502*v^5 + 29430*v^4 - 33777*v^3 - 113724*v^2 + 137052*v + 247131) / 69984 $$\beta_{10}$$ $$=$$ $$( - 2 \nu^{15} + 13 \nu^{14} + 31 \nu^{13} - 25 \nu^{12} - 86 \nu^{11} - 16 \nu^{10} + 450 \nu^{9} + 154 \nu^{8} - 1508 \nu^{7} - 514 \nu^{6} + 3018 \nu^{5} + 2844 \nu^{4} - 3564 \nu^{3} + \cdots + 22599 ) / 5832$$ (-2*v^15 + 13*v^14 + 31*v^13 - 25*v^12 - 86*v^11 - 16*v^10 + 450*v^9 + 154*v^8 - 1508*v^7 - 514*v^6 + 3018*v^5 + 2844*v^4 - 3564*v^3 - 9315*v^2 + 18225*v + 22599) / 5832 $$\beta_{11}$$ $$=$$ $$( 5 \nu^{15} - 2 \nu^{14} - 24 \nu^{13} - 2 \nu^{12} + 66 \nu^{11} + 88 \nu^{10} - 293 \nu^{9} - 348 \nu^{8} + 956 \nu^{7} + 948 \nu^{6} - 1008 \nu^{5} - 3240 \nu^{4} + 2754 \nu^{3} + 10206 \nu^{2} + \cdots - 13122 ) / 4374$$ (5*v^15 - 2*v^14 - 24*v^13 - 2*v^12 + 66*v^11 + 88*v^10 - 293*v^9 - 348*v^8 + 956*v^7 + 948*v^6 - 1008*v^5 - 3240*v^4 + 2754*v^3 + 10206*v^2 - 7290*v - 13122) / 4374 $$\beta_{12}$$ $$=$$ $$( - 15 \nu^{15} - 8 \nu^{14} + 62 \nu^{13} + 15 \nu^{12} - 322 \nu^{11} - 30 \nu^{10} + 1232 \nu^{9} + 350 \nu^{8} - 3210 \nu^{7} - 1676 \nu^{6} + 7986 \nu^{5} + 3294 \nu^{4} - 20601 \nu^{3} + \cdots + 9477 ) / 11664$$ (-15*v^15 - 8*v^14 + 62*v^13 + 15*v^12 - 322*v^11 - 30*v^10 + 1232*v^9 + 350*v^8 - 3210*v^7 - 1676*v^6 + 7986*v^5 + 3294*v^4 - 20601*v^3 - 14742*v^2 + 20412*v + 9477) / 11664 $$\beta_{13}$$ $$=$$ $$( 52 \nu^{15} + 23 \nu^{14} - 171 \nu^{13} - \nu^{12} + 252 \nu^{11} + 746 \nu^{10} - 1300 \nu^{9} - 3852 \nu^{8} + 6274 \nu^{7} + 6924 \nu^{6} - 2196 \nu^{5} - 17874 \nu^{4} - 7128 \nu^{3} + \cdots - 107163 ) / 34992$$ (52*v^15 + 23*v^14 - 171*v^13 - v^12 + 252*v^11 + 746*v^10 - 1300*v^9 - 3852*v^8 + 6274*v^7 + 6924*v^6 - 2196*v^5 - 17874*v^4 - 7128*v^3 + 83349*v^2 + 729*v - 107163) / 34992 $$\beta_{14}$$ $$=$$ $$( - 53 \nu^{15} - 73 \nu^{14} + 327 \nu^{13} + 176 \nu^{12} - 1014 \nu^{11} - 508 \nu^{10} + 3056 \nu^{9} + 5178 \nu^{8} - 10184 \nu^{7} - 13644 \nu^{6} + 25254 \nu^{5} + 16956 \nu^{4} + \cdots + 183708 ) / 34992$$ (-53*v^15 - 73*v^14 + 327*v^13 + 176*v^12 - 1014*v^11 - 508*v^10 + 3056*v^9 + 5178*v^8 - 10184*v^7 - 13644*v^6 + 25254*v^5 + 16956*v^4 - 32643*v^3 - 74601*v^2 + 12393*v + 183708) / 34992 $$\beta_{15}$$ $$=$$ $$( 289 \nu^{15} + 86 \nu^{14} - 1398 \nu^{13} - 679 \nu^{12} + 3570 \nu^{11} + 5738 \nu^{10} - 15868 \nu^{9} - 27354 \nu^{8} + 49774 \nu^{7} + 67308 \nu^{6} - 83970 \nu^{5} + \cdots - 925101 ) / 69984$$ (289*v^15 + 86*v^14 - 1398*v^13 - 679*v^12 + 3570*v^11 + 5738*v^10 - 15868*v^9 - 27354*v^8 + 49774*v^7 + 67308*v^6 - 83970*v^5 - 176634*v^4 + 112995*v^3 + 593892*v^2 - 239112*v - 925101) / 69984
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ b2 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{14} - 2\beta_{13} + \beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 2\beta_1$$ -b14 - 2*b13 + b12 + 2*b11 + b10 - b9 - b7 + b5 + b4 + 2*b1 $$\nu^{4}$$ $$=$$ $$\beta_{8} - 2\beta_{7} - 2\beta_{6} + 2\beta_{3} + 2\beta_{2} + 1$$ b8 - 2*b7 - 2*b6 + 2*b3 + 2*b2 + 1 $$\nu^{5}$$ $$=$$ $$- 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta _1 - 4$$ -2*b15 - 2*b14 + 6*b11 - 2*b9 + 2*b8 + 3*b7 + 2*b4 - 2*b3 + 2*b1 - 4 $$\nu^{6}$$ $$=$$ $$2 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 10 \beta _1 + 11$$ 2*b14 + 4*b13 - 2*b12 - 4*b11 - 2*b10 + 2*b9 - 6*b8 + 2*b7 - 6*b6 + 4*b5 - 4*b4 + 2*b2 - 10*b1 + 11 $$\nu^{7}$$ $$=$$ $$- 6 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 2 \beta_{6} - 10 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 13 \beta _1 - 2$$ -6*b15 + 4*b14 + 14*b13 - 2*b12 + 4*b11 + 2*b10 - 2*b9 + 4*b8 - 10*b7 - 2*b6 - 10*b5 - 4*b4 - 6*b3 - 2*b2 + 13*b1 - 2 $$\nu^{8}$$ $$=$$ $$4 \beta_{15} + 24 \beta_{14} + 26 \beta_{13} - 12 \beta_{12} - 20 \beta_{11} - 4 \beta_{10} + 34 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 30 \beta_{4} - 22 \beta_{3} + 17 \beta_{2} - 32 \beta _1 - 17$$ 4*b15 + 24*b14 + 26*b13 - 12*b12 - 20*b11 - 4*b10 + 34*b9 - 18*b8 - 2*b7 - 10*b6 - 2*b5 - 30*b4 - 22*b3 + 17*b2 - 32*b1 - 17 $$\nu^{9}$$ $$=$$ $$- 8 \beta_{15} + \beta_{14} - 2 \beta_{13} + 23 \beta_{12} + 32 \beta_{11} + 31 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} - 69 \beta_{7} - 21 \beta_{5} - 21 \beta_{4} + 40 \beta_{3} + 4 \beta _1 + 40$$ -8*b15 + b14 - 2*b13 + 23*b12 + 32*b11 + 31*b10 + 5*b9 - 8*b8 - 69*b7 - 21*b5 - 21*b4 + 40*b3 + 4*b1 + 40 $$\nu^{10}$$ $$=$$ $$20 \beta_{15} + 60 \beta_{14} + 24 \beta_{13} - 24 \beta_{11} + 100 \beta_{9} - 7 \beta_{8} - 120 \beta_{7} - 40 \beta_{6} - 16 \beta_{5} - 20 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 4 \beta _1 - 83$$ 20*b15 + 60*b14 + 24*b13 - 24*b11 + 100*b9 - 7*b8 - 120*b7 - 40*b6 - 16*b5 - 20*b4 + 28*b3 + 20*b2 + 4*b1 - 83 $$\nu^{11}$$ $$=$$ $$- 56 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} + 112 \beta_{11} + 8 \beta_{10} + 24 \beta_{9} + 40 \beta_{8} - 77 \beta_{7} + 16 \beta_{6} - 108 \beta_{5} + 4 \beta_{4} + 220 \beta_{3} - 12 \beta_{2} - 8 \beta _1 - 24$$ -56*b15 - 8*b14 + 8*b13 - 24*b12 + 112*b11 + 8*b10 + 24*b9 + 40*b8 - 77*b7 + 16*b6 - 108*b5 + 4*b4 + 220*b3 - 12*b2 - 8*b1 - 24 $$\nu^{12}$$ $$=$$ $$- 16 \beta_{15} - 64 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 60 \beta_{11} - 144 \beta_{10} + 116 \beta_{9} - 160 \beta_{8} - 20 \beta_{7} - 20 \beta_{6} + 216 \beta_{5} + 176 \beta_{4} + 48 \beta_{3} - 32 \beta_{2} - 64 \beta _1 - 15$$ -16*b15 - 64*b14 + 12*b13 + 16*b12 - 60*b11 - 144*b10 + 116*b9 - 160*b8 - 20*b7 - 20*b6 + 216*b5 + 176*b4 + 48*b3 - 32*b2 - 64*b1 - 15 $$\nu^{13}$$ $$=$$ $$- 104 \beta_{15} + 176 \beta_{14} + 220 \beta_{13} - 256 \beta_{12} - 160 \beta_{11} + 16 \beta_{10} - 260 \beta_{9} + 84 \beta_{8} - 128 \beta_{7} - 24 \beta_{6} - 232 \beta_{5} + 96 \beta_{4} + 500 \beta_{3} - 168 \beta_{2} + \cdots - 164$$ -104*b15 + 176*b14 + 220*b13 - 256*b12 - 160*b11 + 16*b10 - 260*b9 + 84*b8 - 128*b7 - 24*b6 - 232*b5 + 96*b4 + 500*b3 - 168*b2 + 81*b1 - 164 $$\nu^{14}$$ $$=$$ $$- 68 \beta_{15} - 16 \beta_{14} + 596 \beta_{13} - 252 \beta_{12} - 524 \beta_{11} - 292 \beta_{10} + 548 \beta_{9} - 292 \beta_{8} + 388 \beta_{7} + 156 \beta_{6} + 260 \beta_{5} + 224 \beta_{4} - 1188 \beta_{3} + 129 \beta_{2} + \cdots - 1127$$ -68*b15 - 16*b14 + 596*b13 - 252*b12 - 524*b11 - 292*b10 + 548*b9 - 292*b8 + 388*b7 + 156*b6 + 260*b5 + 224*b4 - 1188*b3 + 129*b2 - 584*b1 - 1127 $$\nu^{15}$$ $$=$$ $$408 \beta_{15} + 595 \beta_{14} + 130 \beta_{13} - 283 \beta_{12} - 598 \beta_{11} + 781 \beta_{10} - 249 \beta_{9} - 480 \beta_{8} - 129 \beta_{7} - 40 \beta_{6} + 17 \beta_{5} - 1023 \beta_{4} + 1400 \beta_{3} - 520 \beta_{2} + \cdots + 416$$ 408*b15 + 595*b14 + 130*b13 - 283*b12 - 598*b11 + 781*b10 - 249*b9 - 480*b8 - 129*b7 - 40*b6 + 17*b5 - 1023*b4 + 1400*b3 - 520*b2 - 1658*b1 + 416

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$ $$1$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −0.809132 + 1.53144i −1.63023 + 0.585107i 0.956703 + 1.44386i −1.72340 − 0.172852i 1.66415 + 0.480229i −0.909329 − 1.47415i 1.70042 − 0.329508i 1.25083 − 1.19809i −0.809132 − 1.53144i −1.63023 − 0.585107i 0.956703 − 1.44386i −1.72340 + 0.172852i 1.66415 − 0.480229i −0.909329 + 1.47415i 1.70042 + 0.329508i 1.25083 + 1.19809i
0 −1.73083 + 0.0649909i 0 3.15813 + 1.82335i 0 −1.32651 0 2.99155 0.224976i 0
65.2 0 −1.32183 1.11927i 0 −2.85312 1.64725i 0 4.86833 0 0.494481 + 2.95897i 0
65.3 0 −0.772064 + 1.55046i 0 1.67415 + 0.966572i 0 1.30064 0 −1.80783 2.39410i 0
65.4 0 −0.712008 1.57894i 0 1.02997 + 0.594652i 0 −1.04533 0 −1.98609 + 2.24843i 0
65.5 0 0.416182 + 1.68131i 0 −2.52758 1.45930i 0 −0.106684 0 −2.65359 + 1.39946i 0
65.6 0 0.821988 1.52458i 0 −2.16072 1.24749i 0 −3.30744 0 −1.64867 2.50637i 0
65.7 0 1.13557 + 1.30785i 0 1.87360 + 1.08173i 0 −3.41109 0 −0.420952 + 2.97032i 0
65.8 0 1.66299 + 0.484201i 0 1.30557 + 0.753769i 0 3.02808 0 2.53110 + 1.61045i 0
449.1 0 −1.73083 0.0649909i 0 3.15813 1.82335i 0 −1.32651 0 2.99155 + 0.224976i 0
449.2 0 −1.32183 + 1.11927i 0 −2.85312 + 1.64725i 0 4.86833 0 0.494481 2.95897i 0
449.3 0 −0.772064 1.55046i 0 1.67415 0.966572i 0 1.30064 0 −1.80783 + 2.39410i 0
449.4 0 −0.712008 + 1.57894i 0 1.02997 0.594652i 0 −1.04533 0 −1.98609 2.24843i 0
449.5 0 0.416182 1.68131i 0 −2.52758 + 1.45930i 0 −0.106684 0 −2.65359 1.39946i 0
449.6 0 0.821988 + 1.52458i 0 −2.16072 + 1.24749i 0 −3.30744 0 −1.64867 + 2.50637i 0
449.7 0 1.13557 1.30785i 0 1.87360 1.08173i 0 −3.41109 0 −0.420952 2.97032i 0
449.8 0 1.66299 0.484201i 0 1.30557 0.753769i 0 3.02808 0 2.53110 1.61045i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.n 16
3.b odd 2 1 912.2.bn.o 16
4.b odd 2 1 456.2.bf.d yes 16
12.b even 2 1 456.2.bf.c 16
19.d odd 6 1 912.2.bn.o 16
57.f even 6 1 inner 912.2.bn.n 16
76.f even 6 1 456.2.bf.c 16
228.n odd 6 1 456.2.bf.d yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.c 16 12.b even 2 1
456.2.bf.c 16 76.f even 6 1
456.2.bf.d yes 16 4.b odd 2 1
456.2.bf.d yes 16 228.n odd 6 1
912.2.bn.n 16 1.a even 1 1 trivial
912.2.bn.n 16 57.f even 6 1 inner
912.2.bn.o 16 3.b odd 2 1
912.2.bn.o 16 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{16} - 3 T_{5}^{15} - 21 T_{5}^{14} + 72 T_{5}^{13} + 326 T_{5}^{12} - 1188 T_{5}^{11} - 2224 T_{5}^{10} + 10632 T_{5}^{9} + 9624 T_{5}^{8} - 72960 T_{5}^{7} + 25760 T_{5}^{6} + 242688 T_{5}^{5} - 252000 T_{5}^{4} + \cdots + 430336$$ T5^16 - 3*T5^15 - 21*T5^14 + 72*T5^13 + 326*T5^12 - 1188*T5^11 - 2224*T5^10 + 10632*T5^9 + 9624*T5^8 - 72960*T5^7 + 25760*T5^6 + 242688*T5^5 - 252000*T5^4 - 544320*T5^3 + 1443392*T5^2 - 1275264*T5 + 430336 $$T_{7}^{8} - 30T_{7}^{6} - 22T_{7}^{5} + 223T_{7}^{4} + 234T_{7}^{3} - 278T_{7}^{2} - 332T_{7} - 32$$ T7^8 - 30*T7^6 - 22*T7^5 + 223*T7^4 + 234*T7^3 - 278*T7^2 - 332*T7 - 32 $$T_{17}^{16} + 3 T_{17}^{15} - 69 T_{17}^{14} - 216 T_{17}^{13} + 3260 T_{17}^{12} + 13344 T_{17}^{11} - 73808 T_{17}^{10} - 402432 T_{17}^{9} + 1128960 T_{17}^{8} + 9556992 T_{17}^{7} + 2434048 T_{17}^{6} + \cdots + 2945449984$$ T17^16 + 3*T17^15 - 69*T17^14 - 216*T17^13 + 3260*T17^12 + 13344*T17^11 - 73808*T17^10 - 402432*T17^9 + 1128960*T17^8 + 9556992*T17^7 + 2434048*T17^6 - 99373056*T17^5 - 150208512*T17^4 + 748683264*T17^3 + 3228762112*T17^2 + 4960026624*T17 + 2945449984

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} + T^{15} + 3 T^{14} + 4 T^{13} + \cdots + 6561$$
$5$ $$T^{16} - 3 T^{15} - 21 T^{14} + \cdots + 430336$$
$7$ $$(T^{8} - 30 T^{6} - 22 T^{5} + 223 T^{4} + \cdots - 32)^{2}$$
$11$ $$T^{16} + 121 T^{14} + 5883 T^{12} + \cdots + 2637376$$
$13$ $$T^{16} + 3 T^{15} - 61 T^{14} - 192 T^{13} + \cdots + 64$$
$17$ $$T^{16} + 3 T^{15} + \cdots + 2945449984$$
$19$ $$T^{16} + 11 T^{15} + \cdots + 16983563041$$
$23$ $$T^{16} - 3 T^{15} + \cdots + 1328456704$$
$29$ $$T^{16} - 5 T^{15} + 153 T^{14} + \cdots + 554696704$$
$31$ $$T^{16} + 300 T^{14} + \cdots + 5554422784$$
$37$ $$T^{16} + 80 T^{14} + 2386 T^{12} + \cdots + 262144$$
$41$ $$T^{16} - 6 T^{15} + 188 T^{14} + \cdots + 27541504$$
$43$ $$T^{16} + 13 T^{15} + \cdots + 334805733376$$
$47$ $$T^{16} + 27 T^{15} + \cdots + 18653553688576$$
$53$ $$T^{16} + 7 T^{15} + 139 T^{14} + \cdots + 50176$$
$59$ $$T^{16} - 10 T^{15} + \cdots + 933720229264$$
$61$ $$T^{16} + T^{15} + \cdots + 215225477776$$
$67$ $$T^{16} - 24 T^{15} + \cdots + 5605723904881$$
$71$ $$T^{16} + 27 T^{15} + \cdots + 59895709696$$
$73$ $$T^{16} - 2 T^{15} + \cdots + 24812542725961$$
$79$ $$T^{16} - 21 T^{15} + \cdots + 3046449086464$$
$83$ $$T^{16} + 585 T^{14} + \cdots + 8981961048064$$
$89$ $$T^{16} - 25 T^{15} + 627 T^{14} + \cdots + 6885376$$
$97$ $$T^{16} + 60 T^{15} + \cdots + 14736876810496$$