# Properties

 Label 819.2.s.d Level $819$ Weight $2$ Character orbit 819.s Analytic conductor $6.540$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(289,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

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

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

chi := DirichletCharacter("819.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.s (of order $$3$$, 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: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1$$ x^12 - x^11 + 7*x^10 - 2*x^9 + 33*x^8 - 11*x^7 + 55*x^6 + 17*x^5 + 47*x^4 + x^3 + 8*x^2 + x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{10} - \beta_{6} + 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{10} - \beta_{6} + \beta_{5} + 1) q^{8}+O(q^{10})$$ q + (b11 - b5 - b3) * q^2 + (-b10 - b6 + 1) * q^4 - b2 * q^5 + (-b9 + b8 + b7 - b5 + b2 - 1) * q^7 + (-b10 - b6 + b5 + 1) * q^8 $$q + (\beta_{11} - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{10} - \beta_{6} + 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - 4 \beta_{11} + \beta_{8} + \cdots + 3) q^{98}+O(q^{100})$$ q + (b11 - b5 - b3) * q^2 + (-b10 - b6 + 1) * q^4 - b2 * q^5 + (-b9 + b8 + b7 - b5 + b2 - 1) * q^7 + (-b10 - b6 + b5 + 1) * q^8 + (b7 - b2 - b1) * q^10 + (-b7 + b6 - b3 - b1) * q^11 + (-b10 - b6 + b4 - b3 - b1) * q^13 + (-b11 - b9 + b4 - b3 - b1) * q^14 + (2*b11 + b10 + b6 - b5 - b4 - 2*b3 - 2) * q^16 + (b11 - b10 - b6 - b3 + 1) * q^17 + (2*b11 + b10 - b9 - 2*b8 + b7 - 2*b5 - 2*b4 - 2*b1 - 1) * q^19 + (b8 - b3 + b2) * q^20 + (2*b8 + 2*b3 - b1) * q^22 + (-b11 + b10 - 3*b9 + b6 - 2*b5 - b4 + b3 + 3*b2 - 1) * q^23 + (-b11 - 2*b10 - b9 - 2*b7 + b5 + b1 + 2) * q^25 + (-b10 - b9 + 2*b8 + b7 - 2*b6 - 2*b5 + b2 + 1) * q^26 + (-b11 + b10 + b8 + 2*b7 - b6 - b5 + b4 + b3 - b2 - b1 - 1) * q^28 + (b11 + b10 + 2*b8 + b7 + 2*b5 + 2*b4 + 2*b1 - 1) * q^29 + (b10 + 2*b9 - 3*b7 + 2*b5 + 2*b1 + 3) * q^31 + (-2*b11 + b10 + b9 + b6 + 2*b5 - b4 + 2*b3 - b2 + 2) * q^32 + (2*b11 - 2*b10 - 2*b6 - b5 - b4 - 2*b3 + 3) * q^34 + (b10 - b9 - b8 - b7 - b6 - b4 + b3 + b2 - 2*b1 - 1) * q^35 + (-b10 - 3*b9 - b6 + 3*b2 + 2) * q^37 + (-b10 + b9 - 4*b7 + 4*b5 + 4*b1 + 4) * q^38 + (b8 - b7 - b6 - b3 + 2*b2 + b1) * q^40 + (-b11 - 2*b10 - b9 - 2*b7 + 2) * q^41 + (-3*b7 + 2*b6 - b3 + 2*b1) * q^43 + (-b8 - 3*b7 - 2*b2 - 2*b1) * q^44 + (-b11 + 2*b10 - 2*b9 + 2*b6 + 3*b4 + b3 + 2*b2 - 4) * q^46 + (-2*b8 - 2*b7 + 2*b6 - 3*b3 + 4*b1) * q^47 + (b11 + b10 + 2*b9 + b8 + b7 + 2*b6 + 2*b4 + b3 - b2 - 3*b1 - 1) * q^49 + (4*b11 - b10 - b9 + 2*b7 - 3*b5 - 3*b1 - 2) * q^50 + (2*b11 + b10 - b9 + b7 - 2*b5 - 4*b3 - b2 - b1 + 2) * q^52 + (-3*b9 - 2*b8 - b5 - 2*b4 - b1) * q^53 + (b11 + 2*b10 - b8 - b7 + b5 - b4 + b1 + 1) * q^55 + (-b11 + 2*b10 + b9 + b8 + 2*b7 - b5 - b2 + b1 - 2) * q^56 + (-4*b11 - 2*b9 - 3*b8 + b7 - 3*b5 - 3*b4 - 3*b1 - 1) * q^58 + (b11 + 2*b10 - b9 + 2*b6 + b5 + 2*b4 - b3 + b2 + 1) * q^59 + (-5*b11 - 2*b10 + 2*b9 - 2*b7 + 6*b5 + 6*b1 + 2) * q^61 + (2*b11 + b10 + 2*b9 - 2*b8 + b7 - b5 - 2*b4 - b1 - 1) * q^62 + (-2*b11 + b10 + 2*b9 + b6 + 3*b5 + 2*b4 + 2*b3 - 2*b2 - 2) * q^64 + (2*b10 + b9 + 2*b8 - 2*b7 + b6 + b5 - b3 + 2*b2 + b1 + 1) * q^65 + (-2*b8 - 3*b7 + 3*b6 - 2*b3 - 2*b2 - b1) * q^67 + (4*b11 - 2*b10 + b9 - 2*b6 - b5 - b4 - 4*b3 - b2 + 5) * q^68 + (-b11 + b10 + b8 - 2*b7 + 2*b5 + b3 + b2 + b1 - 2) * q^70 + (2*b8 - 2*b7 + 2*b6 - 3*b3 + b2) * q^71 + (-b11 + 4*b7 + 5*b5 + 5*b1 - 4) * q^73 + (3*b11 - b10 - 3*b9 - b6 - 5*b5 - 3*b3 + 3*b2 - 2) * q^74 + (b11 - 3*b10 + 3*b9 + b5 + b1) * q^76 + (-4*b11 - 2*b10 + 2*b9 + b8 - 3*b7 + b6 + 4*b5 - b4 - b2 + 3*b1 + 2) * q^77 + (b8 + 2*b7 + 3*b3 - 2*b2 + b1) * q^79 + (-2*b8 + 2*b7 - 2*b6 + b3 - b2 - b1) * q^80 + (4*b11 - b10 - b9 + b8 + b7 - 3*b5 + b4 - 3*b1 - 1) * q^82 + (b11 - b10 - b9 - b6 + 3*b4 - b3 + b2 + 4) * q^83 + (b8 - b6 - b3 + b2) * q^85 + (-b8 + 2*b7 + b6 + 5*b3 - 3*b1) * q^86 + (-2*b8 - b2 - b1) * q^88 + (-5*b11 - 4*b10 - 4*b6 + 8*b5 + b4 + 5*b3 + 4) * q^89 + (-4*b11 + b9 + b8 - 5*b7 + b6 + b4 + b3 - 3*b2 + b1 + 1) * q^91 + (-7*b11 + b10 + b9 + b6 + 3*b4 + 7*b3 - b2 - 4) * q^92 + (-b8 + 8*b7 - b6 + 6*b3 + 2*b2 + 2*b1) * q^94 + (-2*b11 - 4*b10 + 2*b9 - 4*b6 + 3*b5 + 3*b4 + 2*b3 - 2*b2 + 1) * q^95 + (-b8 - 8*b7 + 3*b6 - 4*b3 + 2*b1) * q^97 + (-4*b11 + b8 - 9*b7 + 3*b6 - b5 - b4 + 3*b3 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{2} + 8 q^{4} - q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10})$$ 12 * q + 4 * q^2 + 8 * q^4 - q^5 - 3 * q^7 + 6 * q^8 $$12 q + 4 q^{2} + 8 q^{4} - q^{5} - 3 q^{7} + 6 q^{8} + 4 q^{10} - 4 q^{11} - 2 q^{13} + 2 q^{14} - 16 q^{16} + 10 q^{17} - q^{19} + q^{20} - 5 q^{22} - 2 q^{23} + 7 q^{25} + 16 q^{26} - q^{28} - 3 q^{29} + 16 q^{31} + 16 q^{32} + 32 q^{34} - 20 q^{35} + 26 q^{37} + 17 q^{38} - 5 q^{40} + 8 q^{41} - 11 q^{43} - 21 q^{44} - 32 q^{46} + q^{47} - 3 q^{49} - 6 q^{50} + 41 q^{52} + 2 q^{53} + 9 q^{55} - 9 q^{56} - 8 q^{58} + 26 q^{59} - 5 q^{61} - 5 q^{62} - 30 q^{64} + 5 q^{65} - 11 q^{67} + 58 q^{68} - 39 q^{70} - 6 q^{71} - 30 q^{73} - 6 q^{74} - 9 q^{76} - 11 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 54 q^{83} - q^{85} + 7 q^{86} + 8 q^{89} - 23 q^{91} - 54 q^{92} + 45 q^{94} - 12 q^{95} - 35 q^{97} - 20 q^{98}+O(q^{100})$$ 12 * q + 4 * q^2 + 8 * q^4 - q^5 - 3 * q^7 + 6 * q^8 + 4 * q^10 - 4 * q^11 - 2 * q^13 + 2 * q^14 - 16 * q^16 + 10 * q^17 - q^19 + q^20 - 5 * q^22 - 2 * q^23 + 7 * q^25 + 16 * q^26 - q^28 - 3 * q^29 + 16 * q^31 + 16 * q^32 + 32 * q^34 - 20 * q^35 + 26 * q^37 + 17 * q^38 - 5 * q^40 + 8 * q^41 - 11 * q^43 - 21 * q^44 - 32 * q^46 + q^47 - 3 * q^49 - 6 * q^50 + 41 * q^52 + 2 * q^53 + 9 * q^55 - 9 * q^56 - 8 * q^58 + 26 * q^59 - 5 * q^61 - 5 * q^62 - 30 * q^64 + 5 * q^65 - 11 * q^67 + 58 * q^68 - 39 * q^70 - 6 * q^71 - 30 * q^73 - 6 * q^74 - 9 * q^76 - 11 * q^77 + 7 * q^79 + 7 * q^80 + q^82 + 54 * q^83 - q^85 + 7 * q^86 + 8 * q^89 - 23 * q^91 - 54 * q^92 + 45 * q^94 - 12 * q^95 - 35 * q^97 - 20 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 29696 \nu^{11} - 478424 \nu^{10} + 682506 \nu^{9} - 3846008 \nu^{8} + 2684563 \nu^{7} + \cdots - 2119374 ) / 3318773$$ (-29696*v^11 - 478424*v^10 + 682506*v^9 - 3846008*v^8 + 2684563*v^7 - 16878368*v^6 + 16008568*v^5 - 31119861*v^4 + 8363982*v^3 - 14058754*v^2 + 5624108*v - 2119374) / 3318773 $$\beta_{3}$$ $$=$$ $$( - 73788 \nu^{11} - 498559 \nu^{10} + 495146 \nu^{9} - 4188508 \nu^{8} + 1631143 \nu^{7} + \cdots - 2229034 ) / 3318773$$ (-73788*v^11 - 498559*v^10 + 495146*v^9 - 4188508*v^8 + 1631143*v^7 - 18206928*v^6 + 16328192*v^5 - 34289666*v^4 + 8704710*v^3 - 14803002*v^2 + 21668998*v - 2229034) / 3318773 $$\beta_{4}$$ $$=$$ $$( - 109660 \nu^{11} + 153752 \nu^{10} - 747485 \nu^{9} + 406680 \nu^{8} - 3276280 \nu^{7} + \cdots - 6198231 ) / 3318773$$ (-109660*v^11 + 153752*v^10 - 747485*v^9 + 406680*v^8 - 3276280*v^7 + 2259680*v^6 - 4702740*v^5 - 2183844*v^4 - 1984215*v^3 - 450388*v^2 - 133032*v - 6198231) / 3318773 $$\beta_{5}$$ $$=$$ $$( 439315 \nu^{11} - 329655 \nu^{10} + 2921453 \nu^{9} - 131145 \nu^{8} + 14090715 \nu^{7} + \cdots + 572347 ) / 3318773$$ (439315*v^11 - 329655*v^10 + 2921453*v^9 - 131145*v^8 + 14090715*v^7 - 1556185*v^6 + 21902645*v^5 + 12171095*v^4 + 22831649*v^3 + 2423530*v^2 + 646135*v + 572347) / 3318773 $$\beta_{6}$$ $$=$$ $$( 566698 \nu^{11} - 1732988 \nu^{10} + 5617249 \nu^{9} - 9944902 \nu^{8} + 24340355 \nu^{7} + \cdots - 6707921 ) / 3318773$$ (566698*v^11 - 1732988*v^10 + 5617249*v^9 - 9944902*v^8 + 24340355*v^7 - 46353032*v^6 + 58565408*v^5 - 63065800*v^4 + 27901335*v^3 - 44235433*v^2 + 12588213*v - 6707921) / 3318773 $$\beta_{7}$$ $$=$$ $$( - 572347 \nu^{11} + 1011662 \nu^{10} - 4336084 \nu^{9} + 4066147 \nu^{8} - 19018596 \nu^{7} + \cdots + 3392561 ) / 3318773$$ (-572347*v^11 + 1011662*v^10 - 4336084*v^9 + 4066147*v^8 - 19018596*v^7 + 20386532*v^6 - 33035270*v^5 + 12172746*v^4 - 14729214*v^3 + 22259302*v^2 - 2155246*v + 3392561) / 3318773 $$\beta_{8}$$ $$=$$ $$( - 1035034 \nu^{11} + 1869572 \nu^{10} - 7924683 \nu^{9} + 7725614 \nu^{8} - 34760912 \nu^{7} + \cdots + 6345807 ) / 3318773$$ (-1035034*v^11 + 1869572*v^10 - 7924683*v^9 + 7725614*v^8 - 34760912*v^7 + 38513384*v^6 - 61367800*v^5 + 26529336*v^4 - 27474213*v^3 + 41650219*v^2 - 4177460*v + 6345807) / 3318773 $$\beta_{9}$$ $$=$$ $$( 1166290 \nu^{11} - 1650363 \nu^{10} + 8811506 \nu^{9} - 5639321 \nu^{8} + 40119354 \nu^{7} + \cdots + 566698 ) / 3318773$$ (1166290*v^11 - 1650363*v^10 + 8811506*v^9 - 5639321*v^8 + 40119354*v^7 - 27397018*v^6 + 72699666*v^5 - 1266529*v^4 + 44802131*v^3 - 8054629*v^2 + 7274619*v + 566698) / 3318773 $$\beta_{10}$$ $$=$$ $$( - 2686072 \nu^{11} + 3882058 \nu^{10} - 19974443 \nu^{9} + 13501144 \nu^{8} - 90433689 \nu^{7} + \cdots - 1035561 ) / 3318773$$ (-2686072*v^11 + 3882058*v^10 - 19974443*v^9 + 13501144*v^8 - 90433689*v^7 + 66981583*v^6 - 158252610*v^5 + 11027874*v^4 - 96392052*v^3 + 33752077*v^2 - 15484451*v - 1035561) / 3318773 $$\beta_{11}$$ $$=$$ $$\nu^{11} - \nu^{10} + 7\nu^{9} - 2\nu^{8} + 33\nu^{7} - 11\nu^{6} + 55\nu^{5} + 17\nu^{4} + 47\nu^{3} + \nu^{2} + 8\nu + 1$$ v^11 - v^10 + 7*v^9 - 2*v^8 + 33*v^7 - 11*v^6 + 55*v^5 + 17*v^4 + 47*v^3 + v^2 + 8*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{8} + 2\beta_{7} - \beta_{4} - 2$$ -b8 + 2*b7 - b4 - 2 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} + 5\beta_{5} + \beta_{3} - \beta_{2}$$ -b11 + b9 + 5*b5 + b3 - b2 $$\nu^{4}$$ $$=$$ $$5\beta_{8} - 8\beta_{7} + \beta_{6} - \beta_{2} - \beta_1$$ 5*b8 - 8*b7 + b6 - b2 - b1 $$\nu^{5}$$ $$=$$ $$5\beta_{11} - \beta_{10} - 7\beta_{9} + \beta_{8} - \beta_{7} - 24\beta_{5} + \beta_{4} - 24\beta _1 + 1$$ 5*b11 - b10 - 7*b9 + b8 - b7 - 24*b5 + b4 - 24*b1 + 1 $$\nu^{6}$$ $$=$$ $$\beta_{11} - 7\beta_{10} - 9\beta_{9} - 7\beta_{6} - 11\beta_{5} + 24\beta_{4} - \beta_{3} + 9\beta_{2} + 36$$ b11 - 7*b10 - 9*b9 - 7*b6 - 11*b5 + 24*b4 - b3 + 9*b2 + 36 $$\nu^{7}$$ $$=$$ $$-11\beta_{8} + 12\beta_{7} - 9\beta_{6} - 24\beta_{3} + 40\beta_{2} + 117\beta_1$$ -11*b8 + 12*b7 - 9*b6 - 24*b3 + 40*b2 + 117*b1 $$\nu^{8}$$ $$=$$ $$- 11 \beta_{11} + 40 \beta_{10} + 60 \beta_{9} - 117 \beta_{8} + 170 \beta_{7} + 85 \beta_{5} + \cdots - 170$$ -11*b11 + 40*b10 + 60*b9 - 117*b8 + 170*b7 + 85*b5 - 117*b4 + 85*b1 - 170 $$\nu^{9}$$ $$=$$ $$- 117 \beta_{11} + 60 \beta_{10} + 217 \beta_{9} + 60 \beta_{6} + 581 \beta_{5} - 85 \beta_{4} + \cdots - 99$$ -117*b11 + 60*b10 + 217*b9 + 60*b6 + 581*b5 - 85*b4 + 117*b3 - 217*b2 - 99 $$\nu^{10}$$ $$=$$ $$581\beta_{8} - 828\beta_{7} + 217\beta_{6} + 85\beta_{3} - 362\beta_{2} - 571\beta_1$$ 581*b8 - 828*b7 + 217*b6 + 85*b3 - 362*b2 - 571*b1 $$\nu^{11}$$ $$=$$ $$581 \beta_{11} - 362 \beta_{10} - 1160 \beta_{9} + 571 \beta_{8} - 695 \beta_{7} - 2933 \beta_{5} + \cdots + 695$$ 581*b11 - 362*b10 - 1160*b9 + 571*b8 - 695*b7 - 2933*b5 + 571*b4 - 2933*b1 + 695

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\beta_{7}$$ $$-\beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.217953 − 0.377506i −1.02197 + 1.77010i −0.437442 + 0.757672i 0.756174 − 1.30973i 1.16700 − 2.02131i −0.181721 + 0.314749i 0.217953 + 0.377506i −1.02197 − 1.77010i −0.437442 − 0.757672i 0.756174 + 1.30973i 1.16700 + 2.02131i −0.181721 − 0.314749i
−1.85816 0 1.45276 −0.0986811 + 0.170921i 0 1.03826 2.43352i 1.01686 0 0.183365 0.317598i
289.2 −1.55469 0 0.417051 −0.595756 + 1.03188i 0 −2.44127 + 1.01990i 2.46099 0 0.926214 1.60425i
289.3 0.268125 0 −1.92811 −1.28088 + 2.21854i 0 1.80416 1.93520i −1.05323 0 −0.343436 + 0.594848i
289.4 0.851125 0 −1.27559 1.72074 2.98041i 0 −2.57273 0.617304i −2.78793 0 1.46456 2.53670i
289.5 1.90556 0 1.63116 −0.736565 + 1.27577i 0 1.58334 + 2.11968i −0.702849 0 −1.40357 + 2.43105i
289.6 2.38804 0 3.70272 0.491140 0.850679i 0 −0.911766 2.48368i 4.06616 0 1.17286 2.03145i
802.1 −1.85816 0 1.45276 −0.0986811 0.170921i 0 1.03826 + 2.43352i 1.01686 0 0.183365 + 0.317598i
802.2 −1.55469 0 0.417051 −0.595756 1.03188i 0 −2.44127 1.01990i 2.46099 0 0.926214 + 1.60425i
802.3 0.268125 0 −1.92811 −1.28088 2.21854i 0 1.80416 + 1.93520i −1.05323 0 −0.343436 0.594848i
802.4 0.851125 0 −1.27559 1.72074 + 2.98041i 0 −2.57273 + 0.617304i −2.78793 0 1.46456 + 2.53670i
802.5 1.90556 0 1.63116 −0.736565 1.27577i 0 1.58334 2.11968i −0.702849 0 −1.40357 2.43105i
802.6 2.38804 0 3.70272 0.491140 + 0.850679i 0 −0.911766 + 2.48368i 4.06616 0 1.17286 + 2.03145i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.s.d 12
3.b odd 2 1 91.2.h.b yes 12
7.c even 3 1 819.2.n.d 12
13.c even 3 1 819.2.n.d 12
21.c even 2 1 637.2.h.l 12
21.g even 6 1 637.2.f.j 12
21.g even 6 1 637.2.g.l 12
21.h odd 6 1 91.2.g.b 12
21.h odd 6 1 637.2.f.k 12
39.h odd 6 1 1183.2.e.g 12
39.i odd 6 1 91.2.g.b 12
39.i odd 6 1 1183.2.e.h 12
91.h even 3 1 inner 819.2.s.d 12
273.r even 6 1 637.2.h.l 12
273.r even 6 1 8281.2.a.ca 6
273.s odd 6 1 91.2.h.b yes 12
273.s odd 6 1 8281.2.a.bz 6
273.x odd 6 1 1183.2.e.g 12
273.bf even 6 1 637.2.f.j 12
273.bm odd 6 1 637.2.f.k 12
273.bm odd 6 1 1183.2.e.h 12
273.bn even 6 1 637.2.g.l 12
273.bp odd 6 1 8281.2.a.ce 6
273.br even 6 1 8281.2.a.cf 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 21.h odd 6 1
91.2.g.b 12 39.i odd 6 1
91.2.h.b yes 12 3.b odd 2 1
91.2.h.b yes 12 273.s odd 6 1
637.2.f.j 12 21.g even 6 1
637.2.f.j 12 273.bf even 6 1
637.2.f.k 12 21.h odd 6 1
637.2.f.k 12 273.bm odd 6 1
637.2.g.l 12 21.g even 6 1
637.2.g.l 12 273.bn even 6 1
637.2.h.l 12 21.c even 2 1
637.2.h.l 12 273.r even 6 1
819.2.n.d 12 7.c even 3 1
819.2.n.d 12 13.c even 3 1
819.2.s.d 12 1.a even 1 1 trivial
819.2.s.d 12 91.h even 3 1 inner
1183.2.e.g 12 39.h odd 6 1
1183.2.e.g 12 273.x odd 6 1
1183.2.e.h 12 39.i odd 6 1
1183.2.e.h 12 273.bm odd 6 1
8281.2.a.bz 6 273.s odd 6 1
8281.2.a.ca 6 273.r even 6 1
8281.2.a.ce 6 273.bp odd 6 1
8281.2.a.cf 6 273.br even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 11T_{2}^{3} + 8T_{2}^{2} - 14T_{2} + 3$$ T2^6 - 2*T2^5 - 6*T2^4 + 11*T2^3 + 8*T2^2 - 14*T2 + 3 $$T_{11}^{12} + 4 T_{11}^{11} + 37 T_{11}^{10} + 68 T_{11}^{9} + 664 T_{11}^{8} + 1155 T_{11}^{7} + \cdots + 6561$$ T11^12 + 4*T11^11 + 37*T11^10 + 68*T11^9 + 664*T11^8 + 1155*T11^7 + 6811*T11^6 + 2862*T11^5 + 23994*T11^4 + 29079*T11^3 + 36288*T11^2 + 16767*T11 + 6561

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - 2 T^{5} - 6 T^{4} + \cdots + 3)^{2}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + T^{11} + \cdots + 9$$
$7$ $$T^{12} + 3 T^{11} + \cdots + 117649$$
$11$ $$T^{12} + 4 T^{11} + \cdots + 6561$$
$13$ $$T^{12} + 2 T^{11} + \cdots + 4826809$$
$17$ $$(T^{6} - 5 T^{5} - 12 T^{4} + \cdots - 9)^{2}$$
$19$ $$T^{12} + T^{11} + \cdots + 762129$$
$23$ $$(T^{6} + T^{5} + \cdots - 24387)^{2}$$
$29$ $$T^{12} + 3 T^{11} + \cdots + 40401$$
$31$ $$T^{12} - 16 T^{11} + \cdots + 6135529$$
$37$ $$(T^{6} - 13 T^{5} + \cdots - 13477)^{2}$$
$41$ $$T^{12} - 8 T^{11} + \cdots + 4173849$$
$43$ $$T^{12} + 11 T^{11} + \cdots + 1369$$
$47$ $$T^{12} + \cdots + 318515409$$
$53$ $$T^{12} - 2 T^{11} + \cdots + 4761$$
$59$ $$(T^{6} - 13 T^{5} + \cdots + 9123)^{2}$$
$61$ $$T^{12} + \cdots + 1055015361$$
$67$ $$T^{12} + \cdots + 276324129$$
$71$ $$T^{12} + \cdots + 530979849$$
$73$ $$T^{12} + \cdots + 196812841$$
$79$ $$T^{12} + \cdots + 110859841$$
$83$ $$(T^{6} - 27 T^{5} + \cdots + 2673)^{2}$$
$89$ $$(T^{6} - 4 T^{5} + \cdots - 304479)^{2}$$
$97$ $$T^{12} + 35 T^{11} + \cdots + 15202201$$