# Properties

 Label 1183.2.e.f Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(170,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1183.170");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.e (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: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4$$ x^10 - x^9 + 8*x^8 + 7*x^7 + 41*x^6 + 18*x^5 + 58*x^4 + 28*x^3 + 64*x^2 + 16*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + \cdots + 30518 ) / 118350$$ (-364*v^9 + 176*v^8 - 220*v^7 - 5913*v^6 + 880*v^5 + 6908*v^4 + 84549*v^3 + 9416*v^2 + 2376*v + 30518) / 118350 $$\beta_{3}$$ $$=$$ $$( - 983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} + \cdots - 22604 ) / 118350$$ (-983*v^9 - 7328*v^8 + 9160*v^7 - 87336*v^6 - 36640*v^5 - 287624*v^4 - 39747*v^3 - 392048*v^2 - 98928*v - 22604) / 118350 $$\beta_{4}$$ $$=$$ $$( - 1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + \cdots - 348592 ) / 118350$$ (-1159*v^9 - 9844*v^8 + 12305*v^7 - 109053*v^6 - 49220*v^5 - 386377*v^4 + 25194*v^3 - 526654*v^2 - 132894*v - 348592) / 118350 $$\beta_{5}$$ $$=$$ $$( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} + \cdots - 180842 ) / 59175$$ (916*v^9 - 3044*v^8 + 3805*v^7 - 3978*v^6 - 15220*v^5 - 119477*v^4 - 129531*v^3 - 162854*v^2 - 41094*v - 180842) / 59175 $$\beta_{6}$$ $$=$$ $$( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + \cdots - 3988 ) / 78900$$ (2249*v^9 - 9541*v^8 + 26720*v^7 - 34617*v^6 + 31195*v^5 - 152578*v^4 + 39066*v^3 - 46906*v^2 - 20316*v - 3988) / 78900 $$\beta_{7}$$ $$=$$ $$( - 15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} + \cdots - 2692 ) / 236700$$ (-15259*v^9 + 14531*v^8 - 121720*v^7 - 107253*v^6 - 637445*v^5 - 272902*v^4 - 871206*v^3 - 258154*v^2 - 957744*v - 2692) / 236700 $$\beta_{8}$$ $$=$$ $$( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + \cdots + 532 ) / 118350$$ (18139*v^9 - 21776*v^8 + 145570*v^7 + 96813*v^6 + 719570*v^5 + 92092*v^4 + 981276*v^3 + 255184*v^2 + 1362924*v + 532) / 118350 $$\beta_{9}$$ $$=$$ $$( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + \cdots - 136 ) / 4734$$ (1058*v^9 - 1552*v^8 + 9041*v^7 + 3726*v^6 + 39580*v^5 + 2993*v^4 + 53832*v^3 + 11648*v^2 + 50058*v - 136) / 4734
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + 3\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3$$ b9 + 3*b7 - b6 - b4 + b3 + b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} - 4$$ b5 - 2*b4 + 2*b3 + 6*b2 - 4 $$\nu^{4}$$ $$=$$ $$-8\beta_{9} + 2\beta_{8} - 19\beta_{7} + 9\beta_{6} - 13\beta_1$$ -8*b9 + 2*b8 - 19*b7 + 9*b6 - 13*b1 $$\nu^{5}$$ $$=$$ $$- 22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} + \cdots + 45$$ -22*b9 + 9*b8 - 45*b7 + 23*b6 - 9*b5 + 22*b4 - 23*b3 - 47*b2 - 47*b1 + 45 $$\nu^{6}$$ $$=$$ $$-23\beta_{5} + 70\beta_{4} - 78\beta_{3} - 128\beta_{2} + 154$$ -23*b5 + 70*b4 - 78*b3 - 128*b2 + 154 $$\nu^{7}$$ $$=$$ $$206\beta_{9} - 78\beta_{8} + 431\beta_{7} - 221\beta_{6} + 407\beta_1$$ 206*b9 - 78*b8 + 431*b7 - 221*b6 + 407*b1 $$\nu^{8}$$ $$=$$ $$628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + \cdots - 1349$$ 628*b9 - 221*b8 + 1349*b7 - 691*b6 + 221*b5 - 628*b4 + 691*b3 + 1187*b2 + 1187*b1 - 1349 $$\nu^{9}$$ $$=$$ $$691\beta_{5} - 1878\beta_{4} + 2036\beta_{3} + 3634\beta_{2} - 3968$$ 691*b5 - 1878*b4 + 2036*b3 + 3634*b2 - 3968

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1
 1.50426 − 2.60546i 0.597828 − 1.03547i −0.132804 + 0.230024i −0.606661 + 1.05077i −0.862625 + 1.49411i 1.50426 + 2.60546i 0.597828 + 1.03547i −0.132804 − 0.230024i −0.606661 − 1.05077i −0.862625 − 1.49411i
−1.00426 + 1.73943i −0.879528 1.52339i −1.01709 1.76164i 0.452861 0.784378i 3.53311 −0.237709 2.63505i 0.0686323 −0.0471392 + 0.0816475i 0.909582 + 1.57544i
170.2 −0.0978281 + 0.169443i 0.129894 + 0.224983i 0.980859 + 1.69890i 1.96625 3.40565i −0.0508292 −1.12324 + 2.39548i −0.775135 1.46625 2.53963i 0.384710 + 0.666337i
170.3 0.632804 1.09605i 1.31364 + 2.27529i 0.199118 + 0.344882i −1.45130 + 2.51373i 3.32511 1.29536 + 2.30696i 3.03523 −1.95130 + 3.37975i 1.83678 + 3.18139i
170.4 1.10666 1.91679i −1.23721 2.14292i −1.44940 2.51043i −1.06140 + 1.83839i −5.47671 −2.63169 + 0.272389i −1.98932 −1.56140 + 2.70442i 2.34921 + 4.06896i
170.5 1.36263 2.36014i 0.673208 + 1.16603i −2.71349 4.69991i 1.09358 1.89414i 3.66932 2.19729 1.47375i −9.33940 0.593582 1.02811i −2.98028 5.16200i
508.1 −1.00426 1.73943i −0.879528 + 1.52339i −1.01709 + 1.76164i 0.452861 + 0.784378i 3.53311 −0.237709 + 2.63505i 0.0686323 −0.0471392 0.0816475i 0.909582 1.57544i
508.2 −0.0978281 0.169443i 0.129894 0.224983i 0.980859 1.69890i 1.96625 + 3.40565i −0.0508292 −1.12324 2.39548i −0.775135 1.46625 + 2.53963i 0.384710 0.666337i
508.3 0.632804 + 1.09605i 1.31364 2.27529i 0.199118 0.344882i −1.45130 2.51373i 3.32511 1.29536 2.30696i 3.03523 −1.95130 3.37975i 1.83678 3.18139i
508.4 1.10666 + 1.91679i −1.23721 + 2.14292i −1.44940 + 2.51043i −1.06140 1.83839i −5.47671 −2.63169 0.272389i −1.98932 −1.56140 2.70442i 2.34921 4.06896i
508.5 1.36263 + 2.36014i 0.673208 1.16603i −2.71349 + 4.69991i 1.09358 + 1.89414i 3.66932 2.19729 + 1.47375i −9.33940 0.593582 + 1.02811i −2.98028 + 5.16200i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 170.5 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.f 10
7.c even 3 1 inner 1183.2.e.f 10
7.c even 3 1 8281.2.a.bw 5
7.d odd 6 1 8281.2.a.bx 5
13.b even 2 1 91.2.e.c 10
39.d odd 2 1 819.2.j.h 10
52.b odd 2 1 1456.2.r.p 10
91.b odd 2 1 637.2.e.m 10
91.r even 6 1 91.2.e.c 10
91.r even 6 1 637.2.a.l 5
91.s odd 6 1 637.2.a.k 5
91.s odd 6 1 637.2.e.m 10
273.w odd 6 1 819.2.j.h 10
273.w odd 6 1 5733.2.a.bl 5
273.ba even 6 1 5733.2.a.bm 5
364.bl odd 6 1 1456.2.r.p 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 13.b even 2 1
91.2.e.c 10 91.r even 6 1
637.2.a.k 5 91.s odd 6 1
637.2.a.l 5 91.r even 6 1
637.2.e.m 10 91.b odd 2 1
637.2.e.m 10 91.s odd 6 1
819.2.j.h 10 39.d odd 2 1
819.2.j.h 10 273.w odd 6 1
1183.2.e.f 10 1.a even 1 1 trivial
1183.2.e.f 10 7.c even 3 1 inner
1456.2.r.p 10 52.b odd 2 1
1456.2.r.p 10 364.bl odd 6 1
5733.2.a.bl 5 273.w odd 6 1
5733.2.a.bm 5 273.ba even 6 1
8281.2.a.bw 5 7.c even 3 1
8281.2.a.bx 5 7.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 $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{10} - 4T_{2}^{9} + 17T_{2}^{8} - 30T_{2}^{7} + 81T_{2}^{6} - 116T_{2}^{5} + 265T_{2}^{4} - 210T_{2}^{3} + 195T_{2}^{2} + 36T_{2} + 9$$ T2^10 - 4*T2^9 + 17*T2^8 - 30*T2^7 + 81*T2^6 - 116*T2^5 + 265*T2^4 - 210*T2^3 + 195*T2^2 + 36*T2 + 9 $$T_{3}^{10} + 9T_{3}^{8} + 65T_{3}^{6} - 4T_{3}^{5} + 144T_{3}^{4} - 72T_{3}^{3} + 256T_{3}^{2} - 64T_{3} + 16$$ T3^10 + 9*T3^8 + 65*T3^6 - 4*T3^5 + 144*T3^4 - 72*T3^3 + 256*T3^2 - 64*T3 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 4 T^{9} + \cdots + 9$$
$3$ $$T^{10} + 9 T^{8} + \cdots + 16$$
$5$ $$T^{10} - 2 T^{9} + \cdots + 2304$$
$7$ $$T^{10} + T^{9} + \cdots + 16807$$
$11$ $$T^{10} - 11 T^{9} + \cdots + 1089$$
$13$ $$T^{10}$$
$17$ $$T^{10} - 5 T^{9} + \cdots + 184041$$
$19$ $$T^{10} - 9 T^{9} + \cdots + 49729$$
$23$ $$T^{10} + 10 T^{9} + \cdots + 144$$
$29$ $$(T^{5} + 3 T^{4} + \cdots - 108)^{2}$$
$31$ $$T^{10} + 6 T^{9} + \cdots + 126736$$
$37$ $$T^{10} - 4 T^{9} + \cdots + 49505296$$
$41$ $$(T^{5} + 14 T^{4} + \cdots + 1584)^{2}$$
$43$ $$(T^{5} - 2 T^{4} - 72 T^{3} + \cdots + 64)^{2}$$
$47$ $$T^{10} - T^{9} + \cdots + 26718561$$
$53$ $$T^{10} + \cdots + 398361681$$
$59$ $$T^{10} - 11 T^{9} + \cdots + 1089$$
$61$ $$T^{10} - 11 T^{9} + \cdots + 71588521$$
$67$ $$T^{10} + \cdots + 515244601$$
$71$ $$(T^{5} + 15 T^{4} + \cdots + 6336)^{2}$$
$73$ $$T^{10} + 75 T^{8} + \cdots + 506944$$
$79$ $$T^{10} + 2 T^{9} + \cdots + 1000000$$
$83$ $$(T^{5} + 6 T^{4} + \cdots + 7488)^{2}$$
$89$ $$T^{10} + 4 T^{9} + \cdots + 59166864$$
$97$ $$(T^{5} - 12 T^{4} + \cdots + 2384)^{2}$$
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