# Properties

 Label 322.2.g.b Level $322$ Weight $2$ Character orbit 322.g Analytic conductor $2.571$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(45,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("322.45");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.g (of order $$6$$, degree $$2$$, 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: $$2.57118294509$$ 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: $$x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561$$ x^16 + 12*x^14 + 73*x^12 + 312*x^10 + 1045*x^8 + 2808*x^6 + 5913*x^4 + 8748*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} - 1) q^{4} + (\beta_{7} - \beta_{4} + \beta_1) q^{5} + (\beta_{11} - \beta_{8}) q^{6} + ( - \beta_{15} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{13} - 2 \beta_{6} + \beta_{3}) q^{9}+O(q^{10})$$ q - b6 * q^2 + b11 * q^3 + (-b6 - 1) * q^4 + (b7 - b4 + b1) * q^5 + (b11 - b8) * q^6 + (-b15 + b10 + b9 - b7 - b5 - b4 + b2 + b1) * q^7 - q^8 + (2*b13 - 2*b6 + b3) * q^9 $$q - \beta_{6} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} - 1) q^{4} + (\beta_{7} - \beta_{4} + \beta_1) q^{5} + (\beta_{11} - \beta_{8}) q^{6} + ( - \beta_{15} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{13} - 2 \beta_{6} + \beta_{3}) q^{9} + (\beta_{15} - \beta_{9} + \beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{10} + (\beta_{10} + 2 \beta_{9} - \beta_{7} + 2 \beta_{4} + \beta_{2}) q^{11} - \beta_{8} q^{12} + ( - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{3}) q^{13} + ( - \beta_{15} + \beta_{10} - \beta_{7} - \beta_{4} + \beta_1) q^{14} + ( - \beta_{15} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{2} + \beta_1) q^{15} + \beta_{6} q^{16} + ( - 2 \beta_{15} + \beta_{9} - 2 \beta_{7} + \beta_{2}) q^{17} + (\beta_{13} - 2 \beta_{6} + 2 \beta_{3} - 2) q^{18} + (2 \beta_{9} - \beta_{7} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{19} + (\beta_{15} - \beta_{9} - \beta_{2}) q^{20} + (3 \beta_{15} - 2 \beta_{9} + \beta_{7} + \beta_{5} - \beta_{2} - 2 \beta_1) q^{21} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \beta_1) q^{22} + (2 \beta_{14} - \beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{23} - \beta_{11} q^{24} + ( - 2 \beta_{12} - \beta_{11} + 2 \beta_{8} + \beta_{6} - 2 \beta_{3} + 1) q^{25} + (\beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} - 2 \beta_{8}) q^{26} + ( - \beta_{14} - 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} + 2 \beta_{6} - 2 \beta_{3} + \cdots + 1) q^{27}+ \cdots + ( - 5 \beta_{15} + 5 \beta_{9} - 6 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100})$$ q - b6 * q^2 + b11 * q^3 + (-b6 - 1) * q^4 + (b7 - b4 + b1) * q^5 + (b11 - b8) * q^6 + (-b15 + b10 + b9 - b7 - b5 - b4 + b2 + b1) * q^7 - q^8 + (2*b13 - 2*b6 + b3) * q^9 + (b15 - b9 + b7 - b4 - b2 + b1) * q^10 + (b10 + 2*b9 - b7 + 2*b4 + b2) * q^11 - b8 * q^12 + (-b14 - b13 + b12 - b3) * q^13 + (-b15 + b10 - b7 - b4 + b1) * q^14 + (-b15 + b9 - b7 - b5 + b2 + b1) * q^15 + b6 * q^16 + (-2*b15 + b9 - 2*b7 + b2) * q^17 + (b13 - 2*b6 + 2*b3 - 2) * q^18 + (2*b9 - b7 + b4 + 2*b2 + b1) * q^19 + (b15 - b9 - b2) * q^20 + (3*b15 - 2*b9 + b7 + b5 - b2 - 2*b1) * q^21 + (-b7 + b5 + 2*b4 + 2*b2 + b1) * q^22 + (2*b14 - b10 - b8 - b6 + b5 + b3 - b2 - b1) * q^23 - b11 * q^24 + (-2*b12 - b11 + 2*b8 + b6 - 2*b3 + 1) * q^25 + (b14 - b13 + 2*b12 + b11 - 2*b8) * q^26 + (-b14 - 2*b13 + b12 + b11 - b8 + 2*b6 - 2*b3 + 1) * q^27 + (-b9 + b5 - b2) * q^28 + (b14 - b13 + b12 + b11 - b8 + b3 - 1) * q^29 + (-2*b15 + b10 - b7 - b5 - b4 + b2 + 2*b1) * q^30 + (-b13 - b11) * q^31 + (b6 + 1) * q^32 + (-b10 - 3*b9 + 4*b7 - b5 - 3*b4 - 4*b2) * q^33 + (-2*b15 - b4 + b1) * q^34 + (-b14 - 2*b12 - 2*b11 + 2*b8 - 3*b3) * q^35 + (-b13 + b3 - 2) * q^36 + (-2*b15 + b10 + 3*b9 - b7 - b5 + b4 + 2*b2 - b1) * q^37 + (-b15 - b9 - b7 - b4 + b2 + 3*b1) * q^38 + (2*b13 - 2*b12 - b11 + 2*b8 - b6 + 2*b3 - 1) * q^39 + (-b7 + b4 - b1) * q^40 + (-b14 + b13 + b12 + 4*b6 + b3 + 2) * q^41 + (2*b15 - b10 + b9 - b7 + b5 + b4 - b2 - 4*b1) * q^42 + (b15 - b9 + 3*b7 - b5 - b4 - b2 + 2*b1) * q^43 + (-b10 - 2*b9 + b5 + b2 + b1) * q^44 + (2*b15 + b9 + 2*b7 + 4*b4 + b2 - 4*b1) * q^45 + (b13 - 2*b12 - b11 - b10 + 2*b8 + b7 - b6 - b2 - 1) * q^46 + (b14 - b13 + 2*b12 + b11 - 2*b8 - b6 + 4*b3 - 2) * q^47 + (-b11 + b8) * q^48 + (-b14 + 2*b13 - 3*b12 + b11 + 3*b8 - b3) * q^49 + (-2*b14 + 2*b13 - 2*b12 - b11 + 3*b8 - 2*b3 + 1) * q^50 + (6*b15 - 2*b10 + b9 + 3*b7 + 2*b5 + 3*b4 - 3*b2 - 6*b1) * q^51 + (2*b14 + b12 + b11 - 2*b8 + b3) * q^52 + (2*b15 - b10 - 5*b9 - b7 - b4 - 3*b2 - 2*b1) * q^53 + (b14 - b13 + 2*b12 + b11 - 3*b8 + b6 - b3 + 2) * q^54 + (-2*b14 - b13 + 2*b12 + b11 - b8 + 2*b6 - b3 + 1) * q^55 + (b15 - b10 - b9 + b7 + b5 + b4 - b2 - b1) * q^56 + (3*b15 - 3*b9 + 7*b7 - b5 - 2*b4 - 5*b2 - b1) * q^57 + (b14 - b13 + b11 - b8 + b6) * q^58 + (3*b13 - b11 - 2*b6 + 2) * q^59 + (-b15 + b10 - b9 - b4 + b1) * q^60 + (-2*b10 + b9 - 2*b7 - 2*b5 + b4 - b2 - 3*b1) * q^61 + (-b13 - b11 + b8 - b3) * q^62 + (-5*b15 - b9 - 4*b7 + b5 - b4 + 3*b2) * q^63 + q^64 + (-2*b10 - 2*b9 + 2*b5 + b4 + 2*b2) * q^65 + (2*b15 + b10 + b9 + 3*b7 - 2*b5 - 2*b2 - 2*b1) * q^66 + (-b15 + 3*b10 + b9 - 2*b7 - 2*b4 + 5*b2 + b1) * q^67 + (-b9 + 2*b7 - b4 - b2 + b1) * q^68 + (-b14 - 3*b13 + b12 + b11 - 2*b10 + 2*b9 - b8 + b7 + 4*b6 + b5 + b4 - 3*b3 - 2*b2 - 2*b1 + 2) * q^69 + (-2*b14 + 2*b13 - b12 - 2*b11 + 3*b8 - 2*b3) * q^70 + (b14 - b13 + b12 - 2*b8 + b3 - 4) * q^71 + (-2*b13 + 2*b6 - b3) * q^72 + (4*b14 + b13 + 2*b12 + 3*b11 - 4*b8 - 2*b6 + 2*b3 + 2) * q^73 + (-b15 + b10 + 2*b9 + 2*b2 + b1) * q^74 + (3*b8 + b6 - b3 + 2) * q^75 + (-b15 - 3*b9 - 2*b4 - b2 + 2*b1) * q^76 + (-3*b14 + 2*b13 - 3*b12 - 4*b11 + 5*b8 + 2*b3) * q^77 + (-2*b14 - 2*b12 - b11 + 3*b8 - 1) * q^78 + (-4*b15 + 2*b10 + 3*b9 - 2*b7 - 2*b5 - 3*b4 - 3*b2 + b1) * q^79 + (-b15 + b9 - b7 + b4 + b2 - b1) * q^80 + (b13 - 3*b12 - 3*b11 + 6*b8 - b6 - b3 - 1) * q^81 + (b14 - b13 + 2*b12 + b11 - 2*b8 + 2*b6 + 2*b3 + 4) * q^82 + (-b15 - 5*b9 - 3*b4 - 2*b2 + 3*b1) * q^83 + (-b15 - b10 + 3*b9 - 2*b7 + b4 - 2*b1) * q^84 + (b14 + b12 - 2*b8 + 5) * q^85 + (2*b15 + b10 - b9 + b7 - b5 + b1) * q^86 + (b13 - 2*b11 - 2*b6 + 2) * q^87 + (-b10 - 2*b9 + b7 - 2*b4 - b2) * q^88 + (3*b7 + 3*b4 + 3*b1) * q^89 + (2*b15 + 4*b9 + 3*b4 + 4*b2 - 3*b1) * q^90 + (b15 + 2*b10 + 4*b9 - b7 + 2*b5 + 5*b4 + 5*b2) * q^91 + (-2*b14 + b13 - 2*b12 - b11 + 3*b8 + b7 - b5 - b3 + b1 - 1) * q^92 + (b14 - b13 - b11 + 4*b6) * q^93 + (2*b14 - 4*b13 + b12 + b11 - 2*b8 + b6 + b3 - 1) * q^94 + (-2*b13 + b11 - 2*b8 - 4*b3) * q^95 + b8 * q^96 + (6*b15 - 8*b10 + 2*b9 + 4*b7 + 4*b5 + 4*b4 - b2 - 8*b1) * q^97 + (-3*b14 + 2*b13 - 2*b12 + b11 + b8 - b3) * q^98 + (-5*b15 + 5*b9 - 6*b7 - 4*b5 - 2*b4 + 4*b2 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8} + 10 q^{9}+O(q^{10})$$ 16 * q + 8 * q^2 - 6 * q^3 - 8 * q^4 - 16 * q^8 + 10 * q^9 $$16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8} + 10 q^{9} + 6 q^{12} - 8 q^{16} - 10 q^{18} + 8 q^{23} + 6 q^{24} + 2 q^{25} - 6 q^{26} - 16 q^{29} + 12 q^{31} + 8 q^{32} - 20 q^{36} - 2 q^{39} - 8 q^{46} - 6 q^{47} - 18 q^{49} + 4 q^{50} - 6 q^{52} + 18 q^{54} - 8 q^{58} + 36 q^{59} + 16 q^{64} - 12 q^{70} - 52 q^{71} - 10 q^{72} + 24 q^{73} + 30 q^{77} - 4 q^{78} - 20 q^{81} + 54 q^{82} + 80 q^{85} + 54 q^{87} - 16 q^{92} - 26 q^{93} - 6 q^{94} - 6 q^{95} - 6 q^{96}+O(q^{100})$$ 16 * q + 8 * q^2 - 6 * q^3 - 8 * q^4 - 16 * q^8 + 10 * q^9 + 6 * q^12 - 8 * q^16 - 10 * q^18 + 8 * q^23 + 6 * q^24 + 2 * q^25 - 6 * q^26 - 16 * q^29 + 12 * q^31 + 8 * q^32 - 20 * q^36 - 2 * q^39 - 8 * q^46 - 6 * q^47 - 18 * q^49 + 4 * q^50 - 6 * q^52 + 18 * q^54 - 8 * q^58 + 36 * q^59 + 16 * q^64 - 12 * q^70 - 52 * q^71 - 10 * q^72 + 24 * q^73 + 30 * q^77 - 4 * q^78 - 20 * q^81 + 54 * q^82 + 80 * q^85 + 54 * q^87 - 16 * q^92 - 26 * q^93 - 6 * q^94 - 6 * q^95 - 6 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{15} + 12\nu^{13} + 73\nu^{11} + 312\nu^{9} + 1045\nu^{7} + 2808\nu^{5} + 5913\nu^{3} + 6561\nu ) / 2187$$ (v^15 + 12*v^13 + 73*v^11 + 312*v^9 + 1045*v^7 + 2808*v^5 + 5913*v^3 + 6561*v) / 2187 $$\beta_{3}$$ $$=$$ $$( - 100 \nu^{14} - 1713 \nu^{12} - 9244 \nu^{10} - 39975 \nu^{8} - 129853 \nu^{6} - 323595 \nu^{4} - 601668 \nu^{2} - 661203 ) / 103518$$ (-100*v^14 - 1713*v^12 - 9244*v^10 - 39975*v^8 - 129853*v^6 - 323595*v^4 - 601668*v^2 - 661203) / 103518 $$\beta_{4}$$ $$=$$ $$( - 311 \nu^{15} - 3174 \nu^{13} - 16250 \nu^{11} - 59943 \nu^{9} - 179573 \nu^{7} - 452016 \nu^{5} - 856251 \nu^{3} - 769095 \nu ) / 310554$$ (-311*v^15 - 3174*v^13 - 16250*v^11 - 59943*v^9 - 179573*v^7 - 452016*v^5 - 856251*v^3 - 769095*v) / 310554 $$\beta_{5}$$ $$=$$ $$( - 452 \nu^{15} - 4650 \nu^{13} - 23303 \nu^{11} - 95700 \nu^{9} - 286580 \nu^{7} - 673740 \nu^{5} - 1285470 \nu^{3} - 1237113 \nu ) / 310554$$ (-452*v^15 - 4650*v^13 - 23303*v^11 - 95700*v^9 - 286580*v^7 - 673740*v^5 - 1285470*v^3 - 1237113*v) / 310554 $$\beta_{6}$$ $$=$$ $$( - 386 \nu^{14} - 3660 \nu^{12} - 19349 \nu^{10} - 74748 \nu^{8} - 227600 \nu^{6} - 541512 \nu^{4} - 985446 \nu^{2} - 1070901 ) / 103518$$ (-386*v^14 - 3660*v^12 - 19349*v^10 - 74748*v^8 - 227600*v^6 - 541512*v^4 - 985446*v^2 - 1070901) / 103518 $$\beta_{7}$$ $$=$$ $$( - 199 \nu^{15} - 2346 \nu^{13} - 12619 \nu^{11} - 49464 \nu^{9} - 149950 \nu^{7} - 367725 \nu^{5} - 668304 \nu^{3} - 737991 \nu ) / 103518$$ (-199*v^15 - 2346*v^13 - 12619*v^11 - 49464*v^9 - 149950*v^7 - 367725*v^5 - 668304*v^3 - 737991*v) / 103518 $$\beta_{8}$$ $$=$$ $$( 148 \nu^{14} + 1155 \nu^{12} + 5323 \nu^{10} + 18906 \nu^{8} + 55462 \nu^{6} + 118908 \nu^{4} + 194751 \nu^{2} + 142155 ) / 34506$$ (148*v^14 + 1155*v^12 + 5323*v^10 + 18906*v^8 + 55462*v^6 + 118908*v^4 + 194751*v^2 + 142155) / 34506 $$\beta_{9}$$ $$=$$ $$( - 386 \nu^{15} - 3660 \nu^{13} - 19349 \nu^{11} - 74748 \nu^{9} - 227600 \nu^{7} - 541512 \nu^{5} - 985446 \nu^{3} - 967383 \nu ) / 103518$$ (-386*v^15 - 3660*v^13 - 19349*v^11 - 74748*v^9 - 227600*v^7 - 541512*v^5 - 985446*v^3 - 967383*v) / 103518 $$\beta_{10}$$ $$=$$ $$( - 149 \nu^{15} - 1525 \nu^{13} - 7784 \nu^{11} - 30151 \nu^{9} - 92585 \nu^{7} - 221938 \nu^{5} - 403254 \nu^{3} - 433269 \nu ) / 34506$$ (-149*v^15 - 1525*v^13 - 7784*v^11 - 30151*v^9 - 92585*v^7 - 221938*v^5 - 403254*v^3 - 433269*v) / 34506 $$\beta_{11}$$ $$=$$ $$( - 361 \nu^{14} - 3498 \nu^{12} - 18316 \nu^{10} - 69813 \nu^{8} - 211591 \nu^{6} - 511680 \nu^{4} - 907875 \nu^{2} - 935793 ) / 34506$$ (-361*v^14 - 3498*v^12 - 18316*v^10 - 69813*v^8 - 211591*v^6 - 511680*v^4 - 907875*v^2 - 935793) / 34506 $$\beta_{12}$$ $$=$$ $$( 1241 \nu^{14} + 12012 \nu^{12} + 60812 \nu^{10} + 231384 \nu^{8} + 703412 \nu^{6} + 1673847 \nu^{4} + 3027618 \nu^{2} + 3031182 ) / 103518$$ (1241*v^14 + 12012*v^12 + 60812*v^10 + 231384*v^8 + 703412*v^6 + 1673847*v^4 + 3027618*v^2 + 3031182) / 103518 $$\beta_{13}$$ $$=$$ $$( 1249 \nu^{14} + 11280 \nu^{12} + 56644 \nu^{10} + 216690 \nu^{8} + 651076 \nu^{6} + 1557621 \nu^{4} + 2756916 \nu^{2} + 2703132 ) / 103518$$ (1249*v^14 + 11280*v^12 + 56644*v^10 + 216690*v^8 + 651076*v^6 + 1557621*v^4 + 2756916*v^2 + 2703132) / 103518 $$\beta_{14}$$ $$=$$ $$( 1493 \nu^{14} + 13236 \nu^{12} + 65627 \nu^{10} + 247134 \nu^{8} + 730286 \nu^{6} + 1700397 \nu^{4} + 2902716 \nu^{2} + 2635335 ) / 103518$$ (1493*v^14 + 13236*v^12 + 65627*v^10 + 247134*v^8 + 730286*v^6 + 1700397*v^4 + 2902716*v^2 + 2635335) / 103518 $$\beta_{15}$$ $$=$$ $$( - 2365 \nu^{15} - 20352 \nu^{13} - 102067 \nu^{11} - 382929 \nu^{9} - 1142809 \nu^{7} - 2673774 \nu^{5} - 4606875 \nu^{3} - 4299642 \nu ) / 310554$$ (-2365*v^15 - 20352*v^13 - 102067*v^11 - 382929*v^9 - 1142809*v^7 - 2673774*v^5 - 4606875*v^3 - 4299642*v) / 310554
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{11} - \beta_{8} - \beta_{6} + \beta_{3} - 2$$ b12 + b11 - b8 - b6 + b3 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{15} - 2\beta_{9} + 2\beta_{7} - \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1$$ b15 - 2*b9 + 2*b7 - b5 - 3*b4 - b2 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{14} + 2\beta_{13} - 2\beta_{12} - 3\beta_{11} + \beta_{8} + 6\beta_{6} - \beta_{3} + 2$$ -b14 + 2*b13 - 2*b12 - 3*b11 + b8 + 6*b6 - b3 + 2 $$\nu^{5}$$ $$=$$ $$-5\beta_{15} + 2\beta_{10} + 11\beta_{9} - 8\beta_{7} + 2\beta_{5} + 5\beta_{4} + 9\beta_{2} - 5\beta_1$$ -5*b15 + 2*b10 + 11*b9 - 8*b7 + 2*b5 + 5*b4 + 9*b2 - 5*b1 $$\nu^{6}$$ $$=$$ $$-6\beta_{14} - \beta_{13} - 5\beta_{12} - 4\beta_{11} + 18\beta_{8} - 8\beta_{6} - 10\beta_{3} - 4$$ -6*b14 - b13 - 5*b12 - 4*b11 + 18*b8 - 8*b6 - 10*b3 - 4 $$\nu^{7}$$ $$=$$ $$10\beta_{15} - 18\beta_{10} - 19\beta_{9} + 20\beta_{7} + 11\beta_{5} + 12\beta_{4} - 13\beta_{2} - 10\beta_1$$ 10*b15 - 18*b10 - 19*b9 + 20*b7 + 11*b5 + 12*b4 - 13*b2 - 10*b1 $$\nu^{8}$$ $$=$$ $$31\beta_{14} - 3\beta_{13} + 8\beta_{12} + 34\beta_{11} - 56\beta_{8} - 27\beta_{6} + 12\beta_{3} + 5$$ 31*b14 - 3*b13 + 8*b12 + 34*b11 - 56*b8 - 27*b6 + 12*b3 + 5 $$\nu^{9}$$ $$=$$ $$-22\beta_{15} + 52\beta_{10} + 34\beta_{9} - 52\beta_{7} - 62\beta_{5} + 16\beta_{4} + 21\beta_{2} + 25\beta_1$$ -22*b15 + 52*b10 + 34*b9 - 52*b7 - 62*b5 + 16*b4 + 21*b2 + 25*b1 $$\nu^{10}$$ $$=$$ $$-18\beta_{14} - 56\beta_{13} + 10\beta_{12} - 72\beta_{11} + 28\beta_{8} + 60\beta_{3} - 57$$ -18*b14 - 56*b13 + 10*b12 - 72*b11 + 28*b8 + 60*b3 - 57 $$\nu^{11}$$ $$=$$ $$76\beta_{15} + 12\beta_{10} - 228\beta_{9} + 56\beta_{7} + 146\beta_{5} - 168\beta_{4} - 148\beta_{2} + 81\beta_1$$ 76*b15 + 12*b10 - 228*b9 + 56*b7 + 146*b5 - 168*b4 - 148*b2 + 81*b1 $$\nu^{12}$$ $$=$$ $$60\beta_{14} + 4\beta_{13} + 219\beta_{12} + 119\beta_{11} - 101\beta_{8} + 519\beta_{6} - 77\beta_{3} + 476$$ 60*b14 + 4*b13 + 219*b12 + 119*b11 - 101*b8 + 519*b6 - 77*b3 + 476 $$\nu^{13}$$ $$=$$ $$-9\beta_{15} - 414\beta_{10} + 546\beta_{9} + 114\beta_{7} - 183\beta_{5} - 183\beta_{4} - 111\beta_{2} - 257\beta_1$$ -9*b15 - 414*b10 + 546*b9 + 114*b7 - 183*b5 - 183*b4 - 111*b2 - 257*b1 $$\nu^{14}$$ $$=$$ $$-729\beta_{14} + 1134\beta_{13} - 926\beta_{12} - 89\beta_{11} + 935\beta_{8} - 1108\beta_{6} + 145\beta_{3} - 740$$ -729*b14 + 1134*b13 - 926*b12 - 89*b11 + 935*b8 - 1108*b6 + 145*b3 - 740 $$\nu^{15}$$ $$=$$ $$- 899 \beta_{15} + 1062 \beta_{10} + 277 \beta_{9} + 506 \beta_{7} - 316 \beta_{5} + 627 \beta_{4} + 1997 \beta_{2} + 1387 \beta_1$$ -899*b15 + 1062*b10 + 277*b9 + 506*b7 - 316*b5 + 627*b4 + 1997*b2 + 1387*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 0.105715 + 1.72882i −0.105715 − 1.72882i −0.452119 − 1.67200i 0.452119 + 1.67200i 1.36749 − 1.06300i −1.36749 + 1.06300i −0.956239 − 1.44416i 0.956239 + 1.44416i 0.105715 − 1.72882i −0.105715 + 1.72882i −0.452119 + 1.67200i 0.452119 − 1.67200i 1.36749 + 1.06300i −1.36749 − 1.06300i −0.956239 + 1.44416i 0.956239 − 1.44416i
0.500000 + 0.866025i −2.64992 1.52993i −0.500000 + 0.866025i −0.920495 1.59434i 3.05986i −0.254505 + 2.63348i −1.00000 3.18138 + 5.51031i 0.920495 1.59434i
45.2 0.500000 + 0.866025i −2.64992 1.52993i −0.500000 + 0.866025i 0.920495 + 1.59434i 3.05986i 0.254505 2.63348i −1.00000 3.18138 + 5.51031i −0.920495 + 1.59434i
45.3 0.500000 + 0.866025i −1.07743 0.622057i −0.500000 + 0.866025i −0.668984 1.15871i 1.24411i −2.09536 + 1.61539i −1.00000 −0.726090 1.25762i 0.668984 1.15871i
45.4 0.500000 + 0.866025i −1.07743 0.622057i −0.500000 + 0.866025i 0.668984 + 1.15871i 1.24411i 2.09536 1.61539i −1.00000 −0.726090 1.25762i −0.668984 + 1.15871i
45.5 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i −1.78715 3.09543i 0.106550i −0.672033 2.55898i −1.00000 −1.49432 2.58824i 1.78715 3.09543i
45.6 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i 1.78715 + 3.09543i 0.106550i 0.672033 + 2.55898i −1.00000 −1.49432 2.58824i −1.78715 + 3.09543i
45.7 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i −0.511122 0.885289i 2.46537i 2.61593 + 0.396147i −1.00000 1.53904 + 2.66569i 0.511122 0.885289i
45.8 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i 0.511122 + 0.885289i 2.46537i −2.61593 0.396147i −1.00000 1.53904 + 2.66569i −0.511122 + 0.885289i
229.1 0.500000 0.866025i −2.64992 + 1.52993i −0.500000 0.866025i −0.920495 + 1.59434i 3.05986i −0.254505 2.63348i −1.00000 3.18138 5.51031i 0.920495 + 1.59434i
229.2 0.500000 0.866025i −2.64992 + 1.52993i −0.500000 0.866025i 0.920495 1.59434i 3.05986i 0.254505 + 2.63348i −1.00000 3.18138 5.51031i −0.920495 1.59434i
229.3 0.500000 0.866025i −1.07743 + 0.622057i −0.500000 0.866025i −0.668984 + 1.15871i 1.24411i −2.09536 1.61539i −1.00000 −0.726090 + 1.25762i 0.668984 + 1.15871i
229.4 0.500000 0.866025i −1.07743 + 0.622057i −0.500000 0.866025i 0.668984 1.15871i 1.24411i 2.09536 + 1.61539i −1.00000 −0.726090 + 1.25762i −0.668984 1.15871i
229.5 0.500000 0.866025i 0.0922753 0.0532752i −0.500000 0.866025i −1.78715 + 3.09543i 0.106550i −0.672033 + 2.55898i −1.00000 −1.49432 + 2.58824i 1.78715 + 3.09543i
229.6 0.500000 0.866025i 0.0922753 0.0532752i −0.500000 0.866025i 1.78715 3.09543i 0.106550i 0.672033 2.55898i −1.00000 −1.49432 + 2.58824i −1.78715 3.09543i
229.7 0.500000 0.866025i 2.13508 1.23269i −0.500000 0.866025i −0.511122 + 0.885289i 2.46537i 2.61593 0.396147i −1.00000 1.53904 2.66569i 0.511122 + 0.885289i
229.8 0.500000 0.866025i 2.13508 1.23269i −0.500000 0.866025i 0.511122 0.885289i 2.46537i −2.61593 + 0.396147i −1.00000 1.53904 2.66569i −0.511122 0.885289i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.8 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.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.g.b 16
7.c even 3 1 2254.2.c.b 16
7.d odd 6 1 inner 322.2.g.b 16
7.d odd 6 1 2254.2.c.b 16
23.b odd 2 1 inner 322.2.g.b 16
161.f odd 6 1 2254.2.c.b 16
161.g even 6 1 inner 322.2.g.b 16
161.g even 6 1 2254.2.c.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.b 16 1.a even 1 1 trivial
322.2.g.b 16 7.d odd 6 1 inner
322.2.g.b 16 23.b odd 2 1 inner
322.2.g.b 16 161.g even 6 1 inner
2254.2.c.b 16 7.c even 3 1
2254.2.c.b 16 7.d odd 6 1
2254.2.c.b 16 161.f odd 6 1
2254.2.c.b 16 161.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 3T_{3}^{7} - 4T_{3}^{6} - 21T_{3}^{5} + 33T_{3}^{4} + 105T_{3}^{3} + 68T_{3}^{2} - 15T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{8}$$
$3$ $$(T^{8} + 3 T^{7} - 4 T^{6} - 21 T^{5} + 33 T^{4} + \cdots + 1)^{2}$$
$5$ $$T^{16} + 19 T^{14} + 270 T^{12} + \cdots + 6561$$
$7$ $$T^{16} + 9 T^{14} - 28 T^{12} + \cdots + 5764801$$
$11$ $$T^{16} - 64 T^{14} + 2928 T^{12} + \cdots + 74805201$$
$13$ $$(T^{8} + 55 T^{6} + 562 T^{4} + 1356 T^{2} + \cdots + 9)^{2}$$
$17$ $$T^{16} + 49 T^{14} + 2037 T^{12} + \cdots + 194481$$
$19$ $$T^{16} + 83 T^{14} + \cdots + 639128961$$
$23$ $$T^{16} - 8 T^{15} + \cdots + 78310985281$$
$29$ $$(T^{4} + 4 T^{3} - 11 T^{2} - 48 T - 15)^{4}$$
$31$ $$(T^{8} - 6 T^{7} + 4 T^{6} + 48 T^{5} + \cdots + 225)^{2}$$
$37$ $$T^{16} - 62 T^{14} + 2886 T^{12} + \cdots + 81$$
$41$ $$(T^{8} + 131 T^{6} + 930 T^{4} + 1820 T^{2} + \cdots + 961)^{2}$$
$43$ $$(T^{8} + 80 T^{6} + 1756 T^{4} + 8073 T^{2} + \cdots + 729)^{2}$$
$47$ $$(T^{8} + 3 T^{7} - 112 T^{6} - 345 T^{5} + \cdots + 15625)^{2}$$
$53$ $$T^{16} - 315 T^{14} + 69572 T^{12} + \cdots + 625$$
$59$ $$(T^{8} - 18 T^{7} + 44 T^{6} + 1152 T^{5} + \cdots + 73441)^{2}$$
$61$ $$T^{16} + 345 T^{14} + \cdots + 10756569837841$$
$67$ $$T^{16} - 405 T^{14} + \cdots + 2311278643521$$
$71$ $$(T^{4} + 13 T^{3} + 46 T^{2} + 36 T - 15)^{4}$$
$73$ $$(T^{8} - 12 T^{7} - 80 T^{6} + \cdots + 10595025)^{2}$$
$79$ $$T^{16} + \cdots + 415557398300625$$
$83$ $$(T^{8} - 196 T^{6} + 11713 T^{4} + \cdots + 1896129)^{2}$$
$89$ $$T^{16} + \cdots + 121639745379681$$
$97$ $$(T^{8} - 900 T^{6} + 302197 T^{4} + \cdots + 2486119321)^{2}$$