# Properties

 Label 1183.2.e.i Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

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: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 11x^{14} + 85x^{12} + 334x^{10} + 952x^{8} + 1050x^{6} + 853x^{4} + 93x^{2} + 9$$ x^16 + 11*x^14 + 85*x^12 + 334*x^10 + 952*x^8 + 1050*x^6 + 853*x^4 + 93*x^2 + 9 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{4} - 1) q^{3} + ( - \beta_{10} - \beta_{6} - 1) q^{4} + \beta_{13} q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{11} + \beta_{7} + 2 \beta_{6} + \cdots + 1) q^{9}+O(q^{10})$$ q + b9 * q^2 + (-b6 - b4 - 1) * q^3 + (-b10 - b6 - 1) * q^4 + b13 * q^5 + (-b13 - b12 - b9 + b8 - b1) * q^6 + (b14 + b13 - b12 + b9) * q^7 - b12 * q^8 + (-b11 + b7 + 2*b6 + b5 + b4 + 1) * q^9 $$q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{4} - 1) q^{3} + ( - \beta_{10} - \beta_{6} - 1) q^{4} + \beta_{13} q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{6}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots - 2 \beta_1) q^{99}+O(q^{100})$$ q + b9 * q^2 + (-b6 - b4 - 1) * q^3 + (-b10 - b6 - 1) * q^4 + b13 * q^5 + (-b13 - b12 - b9 + b8 - b1) * q^6 + (b14 + b13 - b12 + b9) * q^7 - b12 * q^8 + (-b11 + b7 + 2*b6 + b5 + b4 + 1) * q^9 + (b11 + b10 - 2*b4) * q^10 + (b15 - b14 - b8 - b3 - b1) * q^11 + (-b11 + b10 + b7 + 3*b6 + b5 + b4 - b2 + 1) * q^12 + (b11 + b10 - b4 - b2 - 2) * q^14 + (-b13 - 2*b9 + b8 - 2*b1) * q^15 + (-b10 + b5 + b4 + b2 + 1) * q^16 + (-b11 - b6 - 1) * q^17 + (b15 - b14 - 4*b8 - b3 + 3*b1) * q^18 + (-b12 + 2*b9 - b3) * q^19 + (-2*b15 - b14 - 3*b13 + b12 - 2*b9 + 3*b8 + b3 - b1) * q^20 + (-b14 - b13 - 2*b12 - b9 - b3) * q^21 + (-b7 - b5 - b2 - 3) * q^22 + (-b6 + b5 + b4 + 1) * q^23 + (b15 - b14 - 2*b8 + 3*b1) * q^24 + (-b11 - 2*b10 + b4) * q^25 + (-b7 - b5 + 2*b2 + 2) * q^27 + (b14 - 2*b13 - b12 - b9 + 2*b8 + b1) * q^28 + (-3*b5 - 2*b2 - 3) * q^29 + (-b11 + b10 + b7 + 6*b6 + 2*b5 + 2*b4 - b2 + 2) * q^30 + (b15 - b14 - b1) * q^31 + (-b8 + 2*b3 - b1) * q^32 + (b15 + 2*b14 + b9 + b3 + b1) * q^33 + (2*b15 + b14 + 3*b13 - 2*b12 + 2*b9 - 3*b8 - b3 + b1) * q^34 + (-b11 - 2*b10 + b7 - 3*b6 + b5 + b4 - 1) * q^35 + (-2*b7 - 5*b5 + 1) * q^36 + (-b15 - 2*b14 - 3*b9 - b3 - b1) * q^37 + (-b10 - 4*b6 + b4 - 4) * q^38 + (-b11 + b7 + b6 + b5 + b4 + 1) * q^40 + (-2*b15 - b14 + b13 + 2*b12 - b9 - b8 + b3) * q^41 + (-b11 + 2*b10 + 6*b6 + 2*b5 + 4*b4 - b2 + 6) * q^42 + (b7 - 2*b5 - 2) * q^43 + (-b15 - 2*b14 + 2*b13 - 2*b9 - b3 - b1) * q^44 + (-2*b3 + 4*b1) * q^45 + (-b8 - b3 - b1) * q^46 + (-b15 - 2*b14 + b13 + 3*b12 - b9 + 2*b3 - b1) * q^47 + (-b2 + 2) * q^48 + (-2*b10 + b7 + 3*b5 + 3*b4 + 2*b2) * q^49 + (2*b15 + b14 + 4*b13 - 3*b12 + b9 - 4*b8 - b3) * q^50 + (-2*b10 + b6 - b5 - b4 + 2*b2 - 1) * q^51 + (-2*b10 - 3*b6 - 3) * q^53 + (b15 + 2*b14 + 4*b13 + b12 + 8*b9 + 2*b3 + b1) * q^54 + (b7 + 3*b5 + 4*b2 + 4) * q^55 + (2*b6 + 2*b5 + 3*b4 + 5) * q^56 + (2*b15 + b14 - 4*b12 + 3*b9 - b3 + 2*b1) * q^57 + (3*b13 + b12 - 2*b9 + b3) * q^58 + (-b15 + b14 + 3*b8 - b3 - 3*b1) * q^59 + (b15 - b14 - 3*b8 - 3*b3 + 4*b1) * q^60 + (-2*b10 + b6 + 2*b2) * q^61 + (2*b5 - b2 - 2) * q^62 + (b15 + 3*b13 + 3*b9 - 3*b8 + 7*b1) * q^63 + (-b7 + 2*b5 - 2*b2 - 5) * q^64 - b10 * q^66 + (-2*b8 - 3*b3) * q^67 + (b11 + 3*b10 - b7 - b6 - 4*b5 - 4*b4 - 3*b2 - 4) * q^68 + (-b7 + b5 + 4) * q^69 + (b15 - b14 - 2*b12 - 4*b9 - 4*b8 - b3 - 4*b1) * q^70 + (b13 + 3*b12 - 4*b9 - b8 - 4*b1) * q^71 + (3*b13 + 3*b12 + 2*b9 + 3*b3) * q^72 + (2*b15 - 2*b14 - 3*b8 - 2*b3 + 2*b1) * q^73 + (3*b10 + 6*b6 + 6) * q^74 + (-b11 + b7 - 2*b5 - 2*b4 - 2) * q^75 + (b13 - 2*b12 - b9 - b8 - b1) * q^76 + (4*b10 + b7 - 3*b6 - b5 - 3*b4 - 2*b2 + 2) * q^77 + (b11 - b7 - 3*b6 - b5 - b4 - 1) * q^79 + (-b15 + b14 + 2*b8 + b3 + 2*b1) * q^80 + (-4*b10 - 5*b6 - 2*b4 - 5) * q^81 + (b11 - b7 - 4*b6 - 4*b5 - 4*b4 - 4) * q^82 + (2*b15 + b14 + b13 - 2*b12 + 7*b9 - b8 - b3 + 6*b1) * q^83 + (-b14 + 3*b13 + b12 + 4*b9 - 5*b8 + 8*b1) * q^84 + (-3*b13 + 2*b12 + 2*b9 + 3*b8 + 2*b1) * q^85 + (-b15 - 2*b14 - b13 + 4*b12 - 3*b9 + 3*b3 - b1) * q^86 + (b11 + 2*b10 - 8*b6 - 8) * q^87 + (2*b10 - b6 - 2*b4 - 1) * q^88 + (-b15 - 2*b14 - 2*b13 - 2*b12 + 3*b9 - 3*b3 - b1) * q^89 + (-2*b5 + 6*b2 + 14) * q^90 + (-b7 - b5 - b2) * q^92 + (-3*b13 + 2*b12 - 6*b9 + 2*b3) * q^93 + (b11 - b10 - 6*b6 - 5*b4 - 6) * q^94 + (b11 + 2*b10 + b6 - 3*b4 + 1) * q^95 + (-2*b15 - 4*b14 - 4*b13 + 3*b12 - 9*b9 + b3 - 2*b1) * q^96 + (2*b15 + b14 + b13 + 3*b9 - b8 - b3 + 2*b1) * q^97 + (-b15 - 2*b14 - 3*b13 + 2*b12 - 6*b9 - 3*b8 - 3*b1) * q^98 + (2*b15 + b14 + 2*b13 - b12 - b9 - 2*b8 - b3 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{3} - 6 q^{4} - 12 q^{9}+O(q^{10})$$ 16 * q - 4 * q^3 - 6 * q^4 - 12 * q^9 $$16 q - 4 q^{3} - 6 q^{4} - 12 q^{9} + 6 q^{10} - 18 q^{12} - 26 q^{14} + 2 q^{16} - 8 q^{17} - 36 q^{22} + 12 q^{23} + 32 q^{27} - 16 q^{29} - 38 q^{30} + 56 q^{36} - 34 q^{38} - 4 q^{40} + 16 q^{42} - 16 q^{43} + 36 q^{48} - 40 q^{49} - 16 q^{51} - 20 q^{53} + 24 q^{55} + 36 q^{56} - 12 q^{61} - 44 q^{62} - 88 q^{64} + 2 q^{66} - 2 q^{68} + 56 q^{69} + 42 q^{74} - 8 q^{75} + 76 q^{77} + 20 q^{79} - 24 q^{81} + 16 q^{82} - 68 q^{87} - 4 q^{88} + 216 q^{90} + 12 q^{92} - 26 q^{94} + 16 q^{95}+O(q^{100})$$ 16 * q - 4 * q^3 - 6 * q^4 - 12 * q^9 + 6 * q^10 - 18 * q^12 - 26 * q^14 + 2 * q^16 - 8 * q^17 - 36 * q^22 + 12 * q^23 + 32 * q^27 - 16 * q^29 - 38 * q^30 + 56 * q^36 - 34 * q^38 - 4 * q^40 + 16 * q^42 - 16 * q^43 + 36 * q^48 - 40 * q^49 - 16 * q^51 - 20 * q^53 + 24 * q^55 + 36 * q^56 - 12 * q^61 - 44 * q^62 - 88 * q^64 + 2 * q^66 - 2 * q^68 + 56 * q^69 + 42 * q^74 - 8 * q^75 + 76 * q^77 + 20 * q^79 - 24 * q^81 + 16 * q^82 - 68 * q^87 - 4 * q^88 + 216 * q^90 + 12 * q^92 - 26 * q^94 + 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 11x^{14} + 85x^{12} + 334x^{10} + 952x^{8} + 1050x^{6} + 853x^{4} + 93x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 24498 \nu^{14} + 246060 \nu^{12} + 1852321 \nu^{10} + 6411671 \nu^{8} + 17193085 \nu^{6} + \cdots - 40872159 ) / 14163622$$ (24498*v^14 + 246060*v^12 + 1852321*v^10 + 6411671*v^8 + 17193085*v^6 + 6321845*v^4 + 690027*v^2 - 40872159) / 14163622 $$\beta_{3}$$ $$=$$ $$( - 24498 \nu^{15} - 246060 \nu^{13} - 1852321 \nu^{11} - 6411671 \nu^{9} - 17193085 \nu^{7} + \cdots + 55035781 \nu ) / 14163622$$ (-24498*v^15 - 246060*v^13 - 1852321*v^11 - 6411671*v^9 - 17193085*v^7 - 6321845*v^5 - 690027*v^3 + 55035781*v) / 14163622 $$\beta_{4}$$ $$=$$ $$( 172099 \nu^{14} + 2170865 \nu^{12} + 17340370 \nu^{10} + 78484018 \nu^{8} + 236538400 \nu^{6} + \cdots + 23829231 ) / 42490866$$ (172099*v^14 + 2170865*v^12 + 17340370*v^10 + 78484018*v^8 + 236538400*v^6 + 377649654*v^4 + 218482087*v^2 + 23829231) / 42490866 $$\beta_{5}$$ $$=$$ $$( 99072 \nu^{14} + 1000291 \nu^{12} + 7490944 \nu^{10} + 25929344 \nu^{8} + 66564370 \nu^{6} + \cdots - 20454191 ) / 14163622$$ (99072*v^14 + 1000291*v^12 + 7490944*v^10 + 25929344*v^8 + 66564370*v^6 + 25566080*v^4 + 2790528*v^2 - 20454191) / 14163622 $$\beta_{6}$$ $$=$$ $$( 539569 \nu^{14} + 5861765 \nu^{12} + 45125185 \nu^{10} + 174659083 \nu^{8} + 494434675 \nu^{6} + \cdots + 5618970 ) / 42490866$$ (539569*v^14 + 5861765*v^12 + 45125185*v^10 + 174659083*v^8 + 494434675*v^6 + 514968195*v^4 + 441286822*v^2 + 5618970) / 42490866 $$\beta_{7}$$ $$=$$ $$( - 102312 \nu^{14} - 1048444 \nu^{12} - 7735924 \nu^{10} - 26777324 \nu^{8} - 67022271 \nu^{6} + \cdots + 23341327 ) / 7081811$$ (-102312*v^14 - 1048444*v^12 - 7735924*v^10 - 26777324*v^8 - 67022271*v^6 - 26402180*v^4 - 2881788*v^2 + 23341327) / 7081811 $$\beta_{8}$$ $$=$$ $$( 123570 \nu^{15} + 1246351 \nu^{13} + 9343265 \nu^{11} + 32341015 \nu^{9} + 83757455 \nu^{7} + \cdots - 61326350 \nu ) / 14163622$$ (123570*v^15 + 1246351*v^13 + 9343265*v^11 + 32341015*v^9 + 83757455*v^7 + 31887925*v^5 + 3480555*v^3 - 61326350*v) / 14163622 $$\beta_{9}$$ $$=$$ $$( - 539569 \nu^{15} - 5861765 \nu^{13} - 45125185 \nu^{11} - 174659083 \nu^{9} + \cdots - 48109836 \nu ) / 42490866$$ (-539569*v^15 - 5861765*v^13 - 45125185*v^11 - 174659083*v^9 - 494434675*v^7 - 514968195*v^5 - 441286822*v^3 - 48109836*v) / 42490866 $$\beta_{10}$$ $$=$$ $$( - 515071 \nu^{14} - 5615705 \nu^{12} - 43272864 \nu^{10} - 168247412 \nu^{8} - 477241590 \nu^{6} + \cdots - 46491129 ) / 14163622$$ (-515071*v^14 - 5615705*v^12 - 43272864*v^10 - 168247412*v^8 - 477241590*v^6 - 508646350*v^4 - 426433173*v^2 - 46491129) / 14163622 $$\beta_{11}$$ $$=$$ $$( 3598 \nu^{14} + 40712 \nu^{12} + 317920 \nu^{10} + 1287475 \nu^{8} + 3722089 \nu^{6} + 4460568 \nu^{4} + \cdots + 366555 ) / 95271$$ (3598*v^14 + 40712*v^12 + 317920*v^10 + 1287475*v^8 + 3722089*v^6 + 4460568*v^4 + 3361729*v^2 + 366555) / 95271 $$\beta_{12}$$ $$=$$ $$( 1079138 \nu^{15} + 11723530 \nu^{13} + 90250370 \nu^{11} + 349318166 \nu^{9} + \cdots + 11237940 \nu ) / 21245433$$ (1079138*v^15 + 11723530*v^13 + 90250370*v^11 + 349318166*v^9 + 988869350*v^7 + 1029936390*v^5 + 861328211*v^3 + 11237940*v) / 21245433 $$\beta_{13}$$ $$=$$ $$( - 205171 \nu^{15} - 2261795 \nu^{13} - 17480377 \nu^{11} - 68898667 \nu^{9} - 196608895 \nu^{7} + \cdots - 19219314 \nu ) / 3862806$$ (-205171*v^15 - 2261795*v^13 - 17480377*v^11 - 68898667*v^9 - 196608895*v^7 - 219868809*v^5 - 176278948*v^3 - 19219314*v) / 3862806 $$\beta_{14}$$ $$=$$ $$( 1137374 \nu^{15} + 12639109 \nu^{13} + 98087264 \nu^{11} + 390741062 \nu^{9} + \cdots + 167644965 \nu ) / 11588418$$ (1137374*v^15 + 12639109*v^13 + 98087264*v^11 + 390741062*v^9 + 1125399698*v^7 + 1313214930*v^5 + 1094093084*v^3 + 167644965*v) / 11588418 $$\beta_{15}$$ $$=$$ $$( 17689120 \nu^{15} + 191553668 \nu^{13} + 1470474691 \nu^{11} + 5653350919 \nu^{9} + \cdots - 359333319 \nu ) / 127472598$$ (17689120*v^15 + 191553668*v^13 + 1470474691*v^11 + 5653350919*v^9 + 15847248841*v^7 + 15781577445*v^5 + 12180871093*v^3 - 359333319*v) / 127472598
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3\beta_{6} - \beta_{2}$$ b10 + 3*b6 - b2 $$\nu^{3}$$ $$=$$ $$-\beta_{12} - 4\beta_{9} - 4\beta_1$$ -b12 - 4*b9 - 4*b1 $$\nu^{4}$$ $$=$$ $$-5\beta_{10} - 14\beta_{6} - \beta_{4} - 14$$ -5*b10 - 14*b6 - b4 - 14 $$\nu^{5}$$ $$=$$ $$\beta_{13} + 6\beta_{12} + 19\beta_{9} + 6\beta_{3}$$ b13 + 6*b12 + 19*b9 + 6*b3 $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 8\beta_{5} + 24\beta_{2} + 61$$ -b7 - 8*b5 + 24*b2 + 61 $$\nu^{7}$$ $$=$$ $$\beta_{15} - \beta_{14} - 11\beta_{8} - 32\beta_{3} + 94\beta_1$$ b15 - b14 - 11*b8 - 32*b3 + 94*b1 $$\nu^{8}$$ $$=$$ $$-11\beta_{11} + 115\beta_{10} + 11\beta_{7} + 345\beta_{6} + 52\beta_{5} + 52\beta_{4} - 115\beta_{2} + 52$$ -11*b11 + 115*b10 + 11*b7 + 345*b6 + 52*b5 + 52*b4 - 115*b2 + 52 $$\nu^{9}$$ $$=$$ $$-22\beta_{15} - 11\beta_{14} - 85\beta_{13} - 145\beta_{12} - 493\beta_{9} + 85\beta_{8} + 11\beta_{3} - 482\beta_1$$ -22*b15 - 11*b14 - 85*b13 - 145*b12 - 493*b9 + 85*b8 + 11*b3 - 482*b1 $$\nu^{10}$$ $$=$$ $$85\beta_{11} - 553\beta_{10} - 1736\beta_{6} - 315\beta_{4} - 1736$$ 85*b11 - 553*b10 - 1736*b6 - 315*b4 - 1736 $$\nu^{11}$$ $$=$$ $$85\beta_{15} + 170\beta_{14} + 570\beta_{13} + 698\beta_{12} + 2544\beta_{9} + 783\beta_{3} + 85\beta_1$$ 85*b15 + 170*b14 + 570*b13 + 698*b12 + 2544*b9 + 783*b3 + 85*b1 $$\nu^{12}$$ $$=$$ $$-570\beta_{7} - 1838\beta_{5} + 2672\beta_{2} + 6935$$ -570*b7 - 1838*b5 + 2672*b2 + 6935 $$\nu^{13}$$ $$=$$ $$570\beta_{15} - 570\beta_{14} - 3548\beta_{8} - 4510\beta_{3} + 12015\beta_1$$ 570*b15 - 570*b14 - 3548*b8 - 4510*b3 + 12015*b1 $$\nu^{14}$$ $$=$$ $$- 3548 \beta_{11} + 12977 \beta_{10} + 3548 \beta_{7} + 44495 \beta_{6} + 10466 \beta_{5} + \cdots + 10466$$ -3548*b11 + 12977*b10 + 3548*b7 + 44495*b6 + 10466*b5 + 10466*b4 - 12977*b2 + 10466 $$\nu^{15}$$ $$=$$ $$- 7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} - 68116 \beta_{9} + \cdots - 64568 \beta_1$$ -7096*b15 - 3548*b14 - 21110*b13 - 16347*b12 - 68116*b9 + 21110*b8 + 3548*b3 - 64568*b1

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1
 −1.14241 − 1.97871i −1.06275 − 1.84073i −0.536527 − 0.929293i −0.166188 − 0.287846i 0.166188 + 0.287846i 0.536527 + 0.929293i 1.06275 + 1.84073i 1.14241 + 1.97871i −1.14241 + 1.97871i −1.06275 + 1.84073i −0.536527 + 0.929293i −0.166188 + 0.287846i 0.166188 − 0.287846i 0.536527 − 0.929293i 1.06275 − 1.84073i 1.14241 − 1.97871i
−1.14241 + 1.97871i −1.57521 2.72835i −1.61019 2.78892i −1.06250 + 1.84030i 7.19813 0.331665 + 2.62488i 2.78832 −3.46258 + 5.99736i −2.42760 4.20473i
170.2 −1.06275 + 1.84073i 0.0894272 + 0.154892i −1.25885 2.18040i 1.80301 3.12291i −0.380153 2.35320 1.20931i 1.10038 1.48401 2.57037i 3.83229 + 6.63772i
170.3 −0.536527 + 0.929293i 1.21570 + 2.10566i 0.424277 + 0.734868i 0.312716 0.541640i −2.60903 −1.21561 + 2.34996i −3.05665 −1.45586 + 2.52163i 0.335561 + 0.581209i
170.4 −0.166188 + 0.287846i −0.729919 1.26426i 0.944763 + 1.63638i −0.722811 + 1.25195i 0.485214 −1.36920 2.26391i −1.29278 0.434437 0.752468i −0.240245 0.416116i
170.5 0.166188 0.287846i −0.729919 1.26426i 0.944763 + 1.63638i 0.722811 1.25195i −0.485214 1.36920 + 2.26391i 1.29278 0.434437 0.752468i −0.240245 0.416116i
170.6 0.536527 0.929293i 1.21570 + 2.10566i 0.424277 + 0.734868i −0.312716 + 0.541640i 2.60903 1.21561 2.34996i 3.05665 −1.45586 + 2.52163i 0.335561 + 0.581209i
170.7 1.06275 1.84073i 0.0894272 + 0.154892i −1.25885 2.18040i −1.80301 + 3.12291i 0.380153 −2.35320 + 1.20931i −1.10038 1.48401 2.57037i 3.83229 + 6.63772i
170.8 1.14241 1.97871i −1.57521 2.72835i −1.61019 2.78892i 1.06250 1.84030i −7.19813 −0.331665 2.62488i −2.78832 −3.46258 + 5.99736i −2.42760 4.20473i
508.1 −1.14241 1.97871i −1.57521 + 2.72835i −1.61019 + 2.78892i −1.06250 1.84030i 7.19813 0.331665 2.62488i 2.78832 −3.46258 5.99736i −2.42760 + 4.20473i
508.2 −1.06275 1.84073i 0.0894272 0.154892i −1.25885 + 2.18040i 1.80301 + 3.12291i −0.380153 2.35320 + 1.20931i 1.10038 1.48401 + 2.57037i 3.83229 6.63772i
508.3 −0.536527 0.929293i 1.21570 2.10566i 0.424277 0.734868i 0.312716 + 0.541640i −2.60903 −1.21561 2.34996i −3.05665 −1.45586 2.52163i 0.335561 0.581209i
508.4 −0.166188 0.287846i −0.729919 + 1.26426i 0.944763 1.63638i −0.722811 1.25195i 0.485214 −1.36920 + 2.26391i −1.29278 0.434437 + 0.752468i −0.240245 + 0.416116i
508.5 0.166188 + 0.287846i −0.729919 + 1.26426i 0.944763 1.63638i 0.722811 + 1.25195i −0.485214 1.36920 2.26391i 1.29278 0.434437 + 0.752468i −0.240245 + 0.416116i
508.6 0.536527 + 0.929293i 1.21570 2.10566i 0.424277 0.734868i −0.312716 0.541640i 2.60903 1.21561 + 2.34996i 3.05665 −1.45586 2.52163i 0.335561 0.581209i
508.7 1.06275 + 1.84073i 0.0894272 0.154892i −1.25885 + 2.18040i −1.80301 3.12291i 0.380153 −2.35320 1.20931i −1.10038 1.48401 + 2.57037i 3.83229 6.63772i
508.8 1.14241 + 1.97871i −1.57521 + 2.72835i −1.61019 + 2.78892i 1.06250 + 1.84030i −7.19813 −0.331665 + 2.62488i −2.78832 −3.46258 5.99736i −2.42760 + 4.20473i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 170.8 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.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.i 16
7.c even 3 1 inner 1183.2.e.i 16
7.c even 3 1 8281.2.a.ck 8
7.d odd 6 1 8281.2.a.cj 8
13.b even 2 1 inner 1183.2.e.i 16
13.d odd 4 2 91.2.r.a 16
39.f even 4 2 819.2.dl.e 16
91.i even 4 2 637.2.r.f 16
91.r even 6 1 inner 1183.2.e.i 16
91.r even 6 1 8281.2.a.ck 8
91.s odd 6 1 8281.2.a.cj 8
91.z odd 12 2 91.2.r.a 16
91.z odd 12 2 637.2.c.f 8
91.bb even 12 2 637.2.c.e 8
91.bb even 12 2 637.2.r.f 16
273.cd even 12 2 819.2.dl.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 13.d odd 4 2
91.2.r.a 16 91.z odd 12 2
637.2.c.e 8 91.bb even 12 2
637.2.c.f 8 91.z odd 12 2
637.2.r.f 16 91.i even 4 2
637.2.r.f 16 91.bb even 12 2
819.2.dl.e 16 39.f even 4 2
819.2.dl.e 16 273.cd even 12 2
1183.2.e.i 16 1.a even 1 1 trivial
1183.2.e.i 16 7.c even 3 1 inner
1183.2.e.i 16 13.b even 2 1 inner
1183.2.e.i 16 91.r even 6 1 inner
8281.2.a.cj 8 7.d odd 6 1
8281.2.a.cj 8 91.s odd 6 1
8281.2.a.ck 8 7.c even 3 1
8281.2.a.ck 8 91.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{16} + 11T_{2}^{14} + 85T_{2}^{12} + 334T_{2}^{10} + 952T_{2}^{8} + 1050T_{2}^{6} + 853T_{2}^{4} + 93T_{2}^{2} + 9$$ T2^16 + 11*T2^14 + 85*T2^12 + 334*T2^10 + 952*T2^8 + 1050*T2^6 + 853*T2^4 + 93*T2^2 + 9 $$T_{3}^{8} + 2T_{3}^{7} + 11T_{3}^{6} + 6T_{3}^{5} + 67T_{3}^{4} + 62T_{3}^{3} + 114T_{3}^{2} - 20T_{3} + 4$$ T3^8 + 2*T3^7 + 11*T3^6 + 6*T3^5 + 67*T3^4 + 62*T3^3 + 114*T3^2 - 20*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 11 T^{14} + \cdots + 9$$
$3$ $$(T^{8} + 2 T^{7} + 11 T^{6} + \cdots + 4)^{2}$$
$5$ $$T^{16} + 20 T^{14} + \cdots + 2304$$
$7$ $$T^{16} + 20 T^{14} + \cdots + 5764801$$
$11$ $$T^{16} + 52 T^{14} + \cdots + 729$$
$13$ $$T^{16}$$
$17$ $$(T^{8} + 4 T^{7} + \cdots + 15129)^{2}$$
$19$ $$T^{16} + 44 T^{14} + \cdots + 10673289$$
$23$ $$(T^{8} - 6 T^{7} + 31 T^{6} + \cdots + 36)^{2}$$
$29$ $$(T^{4} + 4 T^{3} + \cdots + 624)^{4}$$
$31$ $$T^{16} + \cdots + 1136229264$$
$37$ $$T^{16} + 120 T^{14} + \cdots + 76527504$$
$41$ $$(T^{8} - 132 T^{6} + \cdots + 292032)^{2}$$
$43$ $$(T^{4} + 4 T^{3} + \cdots - 104)^{4}$$
$47$ $$T^{16} + \cdots + 57728231289$$
$53$ $$(T^{8} + 10 T^{7} + \cdots + 7569)^{2}$$
$59$ $$T^{16} + \cdots + 12487392009$$
$61$ $$(T^{8} + 6 T^{7} + \cdots + 49729)^{2}$$
$67$ $$T^{16} + \cdots + 66330457209$$
$71$ $$(T^{8} - 292 T^{6} + \cdots + 397488)^{2}$$
$73$ $$T^{16} + \cdots + 8437677133824$$
$79$ $$(T^{8} - 10 T^{7} + \cdots + 64)^{2}$$
$83$ $$(T^{8} - 296 T^{6} + \cdots + 5483712)^{2}$$
$89$ $$T^{16} + \cdots + 58102628210064$$
$97$ $$(T^{8} - 104 T^{6} + \cdots + 192)^{2}$$