# Properties

 Label 855.2.k.k Level $855$ Weight $2$ Character orbit 855.k Analytic conductor $6.827$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.k (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: $$6.82720937282$$ 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: $$x^{12} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + 14 x^{3} + 46 x^{2} + 12 x + 4$$ x^12 - 3*x^11 + 13*x^10 - 10*x^9 + 44*x^8 - 20*x^7 + 119*x^6 + 13*x^5 + 83*x^4 + 14*x^3 + 46*x^2 + 12*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + 14 x^{3} + 46 x^{2} + 12 x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 9467 \nu^{11} - 219232 \nu^{10} + 319009 \nu^{9} - 2066475 \nu^{8} - 2131461 \nu^{7} - 6476307 \nu^{6} - 8355646 \nu^{5} - 19407929 \nu^{4} + \cdots - 31535764 ) / 24043688$$ (-9467*v^11 - 219232*v^10 + 319009*v^9 - 2066475*v^8 - 2131461*v^7 - 6476307*v^6 - 8355646*v^5 - 19407929*v^4 - 41624620*v^3 - 12566274*v^2 - 3413944*v - 31535764) / 24043688 $$\beta_{3}$$ $$=$$ $$( 119083 \nu^{11} - 330656 \nu^{10} + 1240077 \nu^{9} - 175781 \nu^{8} + 2267093 \nu^{7} + 376383 \nu^{6} + 5464606 \nu^{5} + 10715465 \nu^{4} + \cdots + 20278716 ) / 12021844$$ (119083*v^11 - 330656*v^10 + 1240077*v^9 - 175781*v^8 + 2267093*v^7 + 376383*v^6 + 5464606*v^5 + 10715465*v^4 - 14595442*v^3 + 7227938*v^2 + 1982264*v + 20278716) / 12021844 $$\beta_{4}$$ $$=$$ $$( - 1873019 \nu^{11} + 5628524 \nu^{10} - 24130015 \nu^{9} + 18411181 \nu^{8} - 80346361 \nu^{7} + 39591841 \nu^{6} - 216412954 \nu^{5} + \cdots + 4981404 ) / 24043688$$ (-1873019*v^11 + 5628524*v^10 - 24130015*v^9 + 18411181*v^8 - 80346361*v^7 + 39591841*v^6 - 216412954*v^5 - 15993601*v^4 - 136052648*v^3 + 15402354*v^2 - 73592600*v + 4981404) / 24043688 $$\beta_{5}$$ $$=$$ $$( 489392 \nu^{11} - 929709 \nu^{10} + 4518956 \nu^{9} + 2914321 \nu^{8} + 12727847 \nu^{7} + 17628827 \nu^{6} + 36304389 \nu^{5} + 76473900 \nu^{4} + \cdots + 12751576 ) / 6010922$$ (489392*v^11 - 929709*v^10 + 4518956*v^9 + 2914321*v^8 + 12727847*v^7 + 17628827*v^6 + 36304389*v^5 + 76473900*v^4 + 17848301*v^3 + 50666432*v^2 + 13838716*v + 12751576) / 6010922 $$\beta_{6}$$ $$=$$ $$( 2233513 \nu^{11} - 9651444 \nu^{10} + 38772881 \nu^{9} - 62266695 \nu^{8} + 135956727 \nu^{7} - 168306583 \nu^{6} + 348687850 \nu^{5} + \cdots - 11971084 ) / 24043688$$ (2233513*v^11 - 9651444*v^10 + 38772881*v^9 - 62266695*v^8 + 135956727*v^7 - 168306583*v^6 + 348687850*v^5 - 283097945*v^4 + 216199176*v^3 - 67090322*v^2 + 96458496*v - 11971084) / 24043688 $$\beta_{7}$$ $$=$$ $$( 2481235 \nu^{11} - 3945300 \nu^{10} + 21441087 \nu^{9} + 21286535 \nu^{8} + 70637065 \nu^{7} + 104402375 \nu^{6} + 208854210 \nu^{5} + \cdots + 121494172 ) / 24043688$$ (2481235*v^11 - 3945300*v^10 + 21441087*v^9 + 21286535*v^8 + 70637065*v^7 + 104402375*v^6 + 208854210*v^5 + 445797105*v^4 + 197090848*v^3 + 294408850*v^2 + 80353640*v + 121494172) / 24043688 $$\beta_{8}$$ $$=$$ $$( 2992771 \nu^{11} - 6744800 \nu^{10} + 29254579 \nu^{9} + 8845171 \nu^{8} + 69415229 \nu^{7} + 76101307 \nu^{6} + 187833166 \nu^{5} + \cdots + 108328060 ) / 24043688$$ (2992771*v^11 - 6744800*v^10 + 29254579*v^9 + 8845171*v^8 + 69415229*v^7 + 76101307*v^6 + 187833166*v^5 + 387593873*v^4 - 34697952*v^3 + 258097970*v^2 + 70577144*v + 108328060) / 24043688 $$\beta_{9}$$ $$=$$ $$( 33067 \nu^{11} - 101560 \nu^{10} + 429903 \nu^{9} - 344149 \nu^{8} + 1403197 \nu^{7} - 758053 \nu^{6} + 3756374 \nu^{5} + 111321 \nu^{4} + 2039652 \nu^{3} + \cdots + 271084 ) / 212776$$ (33067*v^11 - 101560*v^10 + 429903*v^9 - 344149*v^8 + 1403197*v^7 - 758053*v^6 + 3756374*v^5 + 111321*v^4 + 2039652*v^3 - 171038*v^2 + 1059536*v + 271084) / 212776 $$\beta_{10}$$ $$=$$ $$( - 723138 \nu^{11} + 2635302 \nu^{10} - 10862733 \nu^{9} + 13343636 \nu^{8} - 36937486 \nu^{7} + 34016485 \nu^{6} - 97570459 \nu^{5} + 42613533 \nu^{4} + \cdots + 2856086 ) / 3005461$$ (-723138*v^11 + 2635302*v^10 - 10862733*v^9 + 13343636*v^8 - 36937486*v^7 + 34016485*v^6 - 97570459*v^5 + 42613533*v^4 - 60874036*v^3 + 13483075*v^2 - 37251044*v + 2856086) / 3005461 $$\beta_{11}$$ $$=$$ $$( - 3942893 \nu^{11} + 13919486 \nu^{10} - 58052365 \nu^{9} + 68012787 \nu^{8} - 199825235 \nu^{7} + 171226187 \nu^{6} - 521338058 \nu^{5} + \cdots + 14863148 ) / 12021844$$ (-3942893*v^11 + 13919486*v^10 - 58052365*v^9 + 68012787*v^8 - 199825235*v^7 + 171226187*v^6 - 521338058*v^5 + 191517271*v^4 - 325565256*v^3 + 67783378*v^2 - 119868512*v + 14863148) / 12021844
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + 2\beta_{4} - \beta_{2} + \beta _1 - 3$$ b9 + 2*b4 - b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{5} + 2\beta_{3} - 5\beta_{2} - 7$$ -b7 + b5 + 2*b3 - 5*b2 - 7 $$\nu^{4}$$ $$=$$ $$-\beta_{11} - 2\beta_{10} - 9\beta_{9} - 3\beta_{6} - 10\beta_{4} + 9\beta_{3} - 10\beta_1$$ -b11 - 2*b10 - 9*b9 - 3*b6 - 10*b4 + 9*b3 - 10*b1 $$\nu^{5}$$ $$=$$ $$- 3 \beta_{11} - 9 \beta_{10} - 24 \beta_{9} + 3 \beta_{8} + 9 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 20 \beta_{4} + 35 \beta_{2} - 35 \beta _1 + 55$$ -3*b11 - 9*b10 - 24*b9 + 3*b8 + 9*b7 - 13*b6 - 13*b5 - 20*b4 + 35*b2 - 35*b1 + 55 $$\nu^{6}$$ $$=$$ $$13\beta_{8} + 24\beta_{7} - 40\beta_{5} - 81\beta_{3} + 91\beta_{2} + 161$$ 13*b8 + 24*b7 - 40*b5 - 81*b3 + 91*b2 + 161 $$\nu^{7}$$ $$=$$ $$40\beta_{11} + 81\beta_{10} + 236\beta_{9} + 134\beta_{6} + 182\beta_{4} - 236\beta_{3} + 286\beta_1$$ 40*b11 + 81*b10 + 236*b9 + 134*b6 + 182*b4 - 236*b3 + 286*b1 $$\nu^{8}$$ $$=$$ $$134 \beta_{11} + 236 \beta_{10} + 737 \beta_{9} - 134 \beta_{8} - 236 \beta_{7} + 410 \beta_{6} + 410 \beta_{5} + 572 \beta_{4} - 819 \beta_{2} + 819 \beta _1 - 1391$$ 134*b11 + 236*b10 + 737*b9 - 134*b8 - 236*b7 + 410*b6 + 410*b5 + 572*b4 - 819*b2 + 819*b1 - 1391 $$\nu^{9}$$ $$=$$ $$-410\beta_{8} - 737\beta_{7} + 1281\beta_{5} + 2202\beta_{3} - 2495\beta_{2} - 4133$$ -410*b8 - 737*b7 + 1281*b5 + 2202*b3 - 2495*b2 - 4133 $$\nu^{10}$$ $$=$$ $$-1281\beta_{11} - 2202\beta_{10} - 6715\beta_{9} - 3893\beta_{6} - 4990\beta_{4} + 6715\beta_{3} - 7390\beta_1$$ -1281*b11 - 2202*b10 - 6715*b9 - 3893*b6 - 4990*b4 + 6715*b3 - 7390*b1 $$\nu^{11}$$ $$=$$ $$- 3893 \beta_{11} - 6715 \beta_{10} - 20200 \beta_{9} + 3893 \beta_{8} + 6715 \beta_{7} - 11889 \beta_{6} - 11889 \beta_{5} - 14780 \beta_{4} + 22327 \beta_{2} - 22327 \beta _1 + 37107$$ -3893*b11 - 6715*b10 - 20200*b9 + 3893*b8 + 6715*b7 - 11889*b6 - 11889*b5 - 14780*b4 + 22327*b2 - 22327*b1 + 37107

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 1.50733 + 2.61078i 0.964458 + 1.67049i 0.414953 + 0.718719i −0.156312 − 0.270740i −0.398236 − 0.689765i −0.832197 − 1.44141i 1.50733 − 2.61078i 0.964458 − 1.67049i 0.414953 − 0.718719i −0.156312 + 0.270740i −0.398236 + 0.689765i −0.832197 + 1.44141i
−1.00733 1.74475i 0 −1.02944 + 1.78305i 0.500000 + 0.866025i 0 1.66469 0.118641 0 1.00733 1.74475i
406.2 −0.464458 0.804466i 0 0.568557 0.984769i 0.500000 + 0.866025i 0 −3.66299 −2.91412 0 0.464458 0.804466i
406.3 0.0850473 + 0.147306i 0 0.985534 1.70699i 0.500000 + 0.866025i 0 −0.218469 0.675457 0 −0.0850473 + 0.147306i
406.4 0.656312 + 1.13677i 0 0.138510 0.239907i 0.500000 + 0.866025i 0 3.11486 2.98887 0 −0.656312 + 1.13677i
406.5 0.898236 + 1.55579i 0 −0.613656 + 1.06288i 0.500000 + 0.866025i 0 −3.41478 1.38811 0 −0.898236 + 1.55579i
406.6 1.33220 + 2.30743i 0 −2.54950 + 4.41586i 0.500000 + 0.866025i 0 4.51669 −8.25696 0 −1.33220 + 2.30743i
676.1 −1.00733 + 1.74475i 0 −1.02944 1.78305i 0.500000 0.866025i 0 1.66469 0.118641 0 1.00733 + 1.74475i
676.2 −0.464458 + 0.804466i 0 0.568557 + 0.984769i 0.500000 0.866025i 0 −3.66299 −2.91412 0 0.464458 + 0.804466i
676.3 0.0850473 0.147306i 0 0.985534 + 1.70699i 0.500000 0.866025i 0 −0.218469 0.675457 0 −0.0850473 0.147306i
676.4 0.656312 1.13677i 0 0.138510 + 0.239907i 0.500000 0.866025i 0 3.11486 2.98887 0 −0.656312 1.13677i
676.5 0.898236 1.55579i 0 −0.613656 1.06288i 0.500000 0.866025i 0 −3.41478 1.38811 0 −0.898236 1.55579i
676.6 1.33220 2.30743i 0 −2.54950 4.41586i 0.500000 0.866025i 0 4.51669 −8.25696 0 −1.33220 2.30743i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 406.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.k.k yes 12
3.b odd 2 1 855.2.k.j 12
19.c even 3 1 inner 855.2.k.k yes 12
57.h odd 6 1 855.2.k.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.k.j 12 3.b odd 2 1
855.2.k.j 12 57.h odd 6 1
855.2.k.k yes 12 1.a even 1 1 trivial
855.2.k.k yes 12 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}^{12} - 3 T_{2}^{11} + 13 T_{2}^{10} - 18 T_{2}^{9} + 62 T_{2}^{8} - 78 T_{2}^{7} + 189 T_{2}^{6} - 117 T_{2}^{5} + 189 T_{2}^{4} - 48 T_{2}^{3} + 146 T_{2}^{2} - 24 T_{2} + 4$$ T2^12 - 3*T2^11 + 13*T2^10 - 18*T2^9 + 62*T2^8 - 78*T2^7 + 189*T2^6 - 117*T2^5 + 189*T2^4 - 48*T2^3 + 146*T2^2 - 24*T2 + 4 $$T_{7}^{6} - 2T_{7}^{5} - 27T_{7}^{4} + 44T_{7}^{3} + 180T_{7}^{2} - 256T_{7} - 64$$ T7^6 - 2*T7^5 - 27*T7^4 + 44*T7^3 + 180*T7^2 - 256*T7 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 3 T^{11} + 13 T^{10} - 18 T^{9} + \cdots + 4$$
$3$ $$T^{12}$$
$5$ $$(T^{2} - T + 1)^{6}$$
$7$ $$(T^{6} - 2 T^{5} - 27 T^{4} + 44 T^{3} + \cdots - 64)^{2}$$
$11$ $$(T^{6} - 27 T^{4} - 24 T^{3} + 68 T^{2} + \cdots + 32)^{2}$$
$13$ $$T^{12} + 8 T^{11} + 85 T^{10} + \cdots + 678976$$
$17$ $$T^{12} - 4 T^{11} + 41 T^{10} + 48 T^{9} + \cdots + 64$$
$19$ $$T^{12} + 6 T^{11} + 42 T^{10} + \cdots + 47045881$$
$23$ $$T^{12} + 2 T^{11} + 74 T^{10} + \cdots + 65536$$
$29$ $$T^{12} - 4 T^{11} + 83 T^{10} + \cdots + 4194304$$
$31$ $$(T^{6} - 12 T^{5} + 29 T^{4} + 72 T^{3} + \cdots + 47)^{2}$$
$37$ $$(T^{6} - 125 T^{4} + 102 T^{3} + \cdots - 14104)^{2}$$
$41$ $$T^{12} - 12 T^{11} + 188 T^{10} + \cdots + 1183744$$
$43$ $$T^{12} + 4 T^{11} + 101 T^{10} + \cdots + 130690624$$
$47$ $$T^{12} - 6 T^{11} + 239 T^{10} + \cdots + 66455104$$
$53$ $$T^{12} - 26 T^{11} + \cdots + 254466304$$
$59$ $$T^{12} - 16 T^{11} + 235 T^{10} + \cdots + 14622976$$
$61$ $$T^{12} - 20 T^{11} + \cdots + 253064464$$
$67$ $$T^{12} + 12 T^{11} + 257 T^{10} + \cdots + 27541504$$
$71$ $$T^{12} + 8 T^{11} + \cdots + 19394461696$$
$73$ $$T^{12} + 4 T^{11} + 113 T^{10} + \cdots + 984064$$
$79$ $$T^{12} + 12 T^{11} + \cdots + 3586572544$$
$83$ $$(T^{6} + 22 T^{5} - 35 T^{4} - 3350 T^{3} + \cdots - 15872)^{2}$$
$89$ $$T^{12} + 8 T^{11} + \cdots + 1185562624$$
$97$ $$T^{12} - 30 T^{11} + 790 T^{10} + \cdots + 16384$$
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