# Properties

 Label 1183.2.c.i Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.c (of order $$2$$, degree $$1$$, not 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: $$12$$ Coefficient field: 12.0.58891012706304.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ x^12 - 5*x^10 - 2*x^9 + 15*x^8 + 2*x^7 - 30*x^6 + 4*x^5 + 60*x^4 - 16*x^3 - 80*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 48\nu ) / 32$$ (v^11 - 5*v^9 - 2*v^8 + 15*v^7 + 2*v^6 - 30*v^5 + 4*v^4 + 60*v^3 - 16*v^2 - 48*v) / 32 $$\beta_{2}$$ $$=$$ $$( \nu^{10} - 3\nu^{9} - \nu^{8} + 9\nu^{7} + \nu^{6} - 23\nu^{5} + 16\nu^{4} + 26\nu^{3} - 32\nu^{2} - 28\nu + 48 ) / 8$$ (v^10 - 3*v^9 - v^8 + 9*v^7 + v^6 - 23*v^5 + 16*v^4 + 26*v^3 - 32*v^2 - 28*v + 48) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{11} + 6 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 3 \nu^{7} + 52 \nu^{6} - 34 \nu^{5} - 88 \nu^{4} + 100 \nu^{3} + 168 \nu^{2} - 112 \nu - 160 ) / 32$$ (v^11 + 6*v^10 - 5*v^9 - 24*v^8 + 3*v^7 + 52*v^6 - 34*v^5 - 88*v^4 + 100*v^3 + 168*v^2 - 112*v - 160) / 32 $$\beta_{4}$$ $$=$$ $$( \nu^{11} + 4 \nu^{10} - 17 \nu^{9} - 6 \nu^{8} + 51 \nu^{7} + 6 \nu^{6} - 122 \nu^{5} + 68 \nu^{4} + 164 \nu^{3} - 144 \nu^{2} - 224 \nu + 192 ) / 32$$ (v^11 + 4*v^10 - 17*v^9 - 6*v^8 + 51*v^7 + 6*v^6 - 122*v^5 + 68*v^4 + 164*v^3 - 144*v^2 - 224*v + 192) / 32 $$\beta_{5}$$ $$=$$ $$( - \nu^{11} - 2 \nu^{10} + 17 \nu^{9} - 4 \nu^{8} - 55 \nu^{7} + 24 \nu^{6} + 126 \nu^{5} - 128 \nu^{4} - 156 \nu^{3} + 232 \nu^{2} + 192 \nu - 288 ) / 32$$ (-v^11 - 2*v^10 + 17*v^9 - 4*v^8 - 55*v^7 + 24*v^6 + 126*v^5 - 128*v^4 - 156*v^3 + 232*v^2 + 192*v - 288) / 32 $$\beta_{6}$$ $$=$$ $$( \nu^{11} - 8 \nu^{10} - 5 \nu^{9} + 30 \nu^{8} + 15 \nu^{7} - 78 \nu^{6} + 18 \nu^{5} + 124 \nu^{4} - 68 \nu^{3} - 192 \nu^{2} + 112 \nu + 96 ) / 32$$ (v^11 - 8*v^10 - 5*v^9 + 30*v^8 + 15*v^7 - 78*v^6 + 18*v^5 + 124*v^4 - 68*v^3 - 192*v^2 + 112*v + 96) / 32 $$\beta_{7}$$ $$=$$ $$( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32$$ (-5*v^11 + 2*v^10 + 17*v^9 - 8*v^8 - 39*v^7 + 44*v^6 + 50*v^5 - 88*v^4 - 68*v^3 + 120*v^2 + 16*v - 32) / 32 $$\beta_{8}$$ $$=$$ $$( - \nu^{11} + 5 \nu^{10} + 7 \nu^{9} - 19 \nu^{8} - 19 \nu^{7} + 49 \nu^{6} + 14 \nu^{5} - 106 \nu^{4} + 4 \nu^{3} + 172 \nu^{2} - 8 \nu - 128 ) / 16$$ (-v^11 + 5*v^10 + 7*v^9 - 19*v^8 - 19*v^7 + 49*v^6 + 14*v^5 - 106*v^4 + 4*v^3 + 172*v^2 - 8*v - 128) / 16 $$\beta_{9}$$ $$=$$ $$( - 5 \nu^{11} + 9 \nu^{9} + 18 \nu^{8} - 11 \nu^{7} - 18 \nu^{6} + 6 \nu^{5} + 68 \nu^{4} - 76 \nu^{3} - 160 \nu^{2} + 16 \nu + 288 ) / 32$$ (-5*v^11 + 9*v^9 + 18*v^8 - 11*v^7 - 18*v^6 + 6*v^5 + 68*v^4 - 76*v^3 - 160*v^2 + 16*v + 288) / 32 $$\beta_{10}$$ $$=$$ $$( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 88 ) / 8$$ (-v^11 + 3*v^10 + 5*v^9 - 13*v^8 - 13*v^7 + 35*v^6 + 12*v^5 - 70*v^4 + 8*v^3 + 108*v^2 - 16*v - 88) / 8 $$\beta_{11}$$ $$=$$ $$( - 3 \nu^{11} + \nu^{10} + 12 \nu^{9} - 3 \nu^{8} - 28 \nu^{7} + 19 \nu^{6} + 43 \nu^{5} - 52 \nu^{4} - 66 \nu^{3} + 64 \nu^{2} + 52 \nu - 16 ) / 8$$ (-3*v^11 + v^10 + 12*v^9 - 3*v^8 - 28*v^7 + 19*v^6 + 43*v^5 - 52*v^4 - 66*v^3 + 64*v^2 + 52*v - 16) / 8
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{2} + \beta_1 ) / 2$$ (-b4 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{11} + \beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta _1 + 2 ) / 2$$ (-b11 + b8 + 2*b7 + b6 - b5 - b4 - b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta _1 + 1 ) / 2$$ (b10 + b9 - b8 - b7 - b5 - 2*b4 + b3 + 2*b1 + 1) / 2 $$\nu^{4}$$ $$=$$ $$( -2\beta_{11} + \beta_{10} + 3\beta_{7} - 2\beta_{5} - 3\beta_{4} - \beta_{3} + \beta_{2} - 3\beta _1 - 1 ) / 2$$ (-2*b11 + b10 + 3*b7 - 2*b5 - 3*b4 - b3 + b2 - 3*b1 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( -\beta_{11} + 4\beta_{10} + 2\beta_{9} - 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - \beta_{2} - 2\beta _1 + 6 ) / 2$$ (-b11 + 4*b10 + 2*b9 - 2*b8 - 2*b7 + 2*b6 - b2 - 2*b1 + 6) / 2 $$\nu^{6}$$ $$=$$ $$( - 3 \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 2 \beta _1 - 3 ) / 2$$ (-3*b11 + b10 + 2*b9 - b8 + 3*b7 - 3*b6 + 3*b5 - 2*b4 - 3*b3 + 3*b2 + 2*b1 - 3) / 2 $$\nu^{7}$$ $$=$$ $$( - 5 \beta_{11} + 5 \beta_{10} + \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + 8 \beta_{6} - \beta_{5} + 2 \beta_{4} - 7 \beta_{3} - \beta_{2} - 4 \beta _1 + 7 ) / 2$$ (-5*b11 + 5*b10 + b9 + 5*b8 + 5*b7 + 8*b6 - b5 + 2*b4 - 7*b3 - b2 - 4*b1 + 7) / 2 $$\nu^{8}$$ $$=$$ $$( - 5 \beta_{11} + 12 \beta_{9} + 3 \beta_{8} - \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 13 \beta _1 - 6 ) / 2$$ (-5*b11 + 12*b9 + 3*b8 - b6 + 5*b5 - 3*b4 + 2*b3 - 2*b2 + 13*b1 - 6) / 2 $$\nu^{9}$$ $$=$$ $$( - 8 \beta_{11} + 3 \beta_{10} + \beta_{9} + 11 \beta_{8} + 9 \beta_{7} - 13 \beta_{5} - 8 \beta_{4} - 17 \beta_{3} - 6 \beta_{2} - 13 ) / 2$$ (-8*b11 + 3*b10 + b9 + 11*b8 + 9*b7 - 13*b5 - 8*b4 - 17*b3 - 6*b2 - 13) / 2 $$\nu^{10}$$ $$=$$ $$( - 4 \beta_{11} + 13 \beta_{10} + 24 \beta_{9} + 4 \beta_{8} - 25 \beta_{7} + 8 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 9 \beta_{2} - 7 \beta _1 - 9 ) / 2$$ (-4*b11 + 13*b10 + 24*b9 + 4*b8 - 25*b7 + 8*b6 - 2*b5 - 3*b4 + 7*b3 - 9*b2 - 7*b1 - 9) / 2 $$\nu^{11}$$ $$=$$ $$( - 7 \beta_{11} - 6 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} - 16 \beta_{7} - 40 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} - 26 \beta_{3} - 11 \beta_{2} + 10 \beta _1 - 20 ) / 2$$ (-7*b11 - 6*b10 + 10*b9 + 4*b8 - 16*b7 - 40*b6 + 6*b5 - 4*b4 - 26*b3 - 11*b2 + 10*b1 - 20) / 2

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 −1.12906 − 0.851598i −1.30089 − 0.554694i 0.759479 − 1.19298i 1.40744 − 0.138282i −1.08105 − 0.911778i 1.34408 + 0.439820i 1.34408 − 0.439820i −1.08105 + 0.911778i 1.40744 + 0.138282i 0.759479 + 1.19298i −1.30089 + 0.554694i −1.12906 + 0.851598i
2.70320i −0.345949 −5.30727 3.25812i 0.935168i 1.00000i 8.94020i −2.88032 −8.80735
337.2 2.10939i 2.26165 −2.44952 3.60178i 4.77070i 1.00000i 0.948212i 2.11505 −7.59755
337.3 1.38595i −2.82577 0.0791355 0.518957i 3.91639i 1.00000i 2.88158i 4.98500 −0.719250
337.4 1.27656i −1.16793 0.370384 1.81487i 1.49093i 1.00000i 3.02595i −1.63595 2.31680
337.5 0.823556i 2.66029 1.32176 3.16209i 2.19090i 1.00000i 2.73565i 4.07715 2.60416
337.6 0.120360i −0.582292 1.98551 1.68817i 0.0700846i 1.00000i 0.479696i −2.66094 0.203187
337.7 0.120360i −0.582292 1.98551 1.68817i 0.0700846i 1.00000i 0.479696i −2.66094 0.203187
337.8 0.823556i 2.66029 1.32176 3.16209i 2.19090i 1.00000i 2.73565i 4.07715 2.60416
337.9 1.27656i −1.16793 0.370384 1.81487i 1.49093i 1.00000i 3.02595i −1.63595 2.31680
337.10 1.38595i −2.82577 0.0791355 0.518957i 3.91639i 1.00000i 2.88158i 4.98500 −0.719250
337.11 2.10939i 2.26165 −2.44952 3.60178i 4.77070i 1.00000i 0.948212i 2.11505 −7.59755
337.12 2.70320i −0.345949 −5.30727 3.25812i 0.935168i 1.00000i 8.94020i −2.88032 −8.80735
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.i 12
13.b even 2 1 inner 1183.2.c.i 12
13.c even 3 1 91.2.q.a 12
13.d odd 4 1 1183.2.a.m 6
13.d odd 4 1 1183.2.a.p 6
13.e even 6 1 91.2.q.a 12
39.h odd 6 1 819.2.ct.a 12
39.i odd 6 1 819.2.ct.a 12
52.i odd 6 1 1456.2.cc.c 12
52.j odd 6 1 1456.2.cc.c 12
91.g even 3 1 637.2.u.h 12
91.h even 3 1 637.2.k.h 12
91.i even 4 1 8281.2.a.by 6
91.i even 4 1 8281.2.a.ch 6
91.k even 6 1 637.2.k.h 12
91.l odd 6 1 637.2.k.g 12
91.m odd 6 1 637.2.u.i 12
91.n odd 6 1 637.2.q.h 12
91.p odd 6 1 637.2.u.i 12
91.t odd 6 1 637.2.q.h 12
91.u even 6 1 637.2.u.h 12
91.v odd 6 1 637.2.k.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 13.c even 3 1
91.2.q.a 12 13.e even 6 1
637.2.k.g 12 91.l odd 6 1
637.2.k.g 12 91.v odd 6 1
637.2.k.h 12 91.h even 3 1
637.2.k.h 12 91.k even 6 1
637.2.q.h 12 91.n odd 6 1
637.2.q.h 12 91.t odd 6 1
637.2.u.h 12 91.g even 3 1
637.2.u.h 12 91.u even 6 1
637.2.u.i 12 91.m odd 6 1
637.2.u.i 12 91.p odd 6 1
819.2.ct.a 12 39.h odd 6 1
819.2.ct.a 12 39.i odd 6 1
1183.2.a.m 6 13.d odd 4 1
1183.2.a.p 6 13.d odd 4 1
1183.2.c.i 12 1.a even 1 1 trivial
1183.2.c.i 12 13.b even 2 1 inner
1456.2.cc.c 12 52.i odd 6 1
1456.2.cc.c 12 52.j odd 6 1
8281.2.a.by 6 91.i even 4 1
8281.2.a.ch 6 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 206T_{2}^{6} + 208T_{2}^{4} + 72T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 16 T^{10} + 88 T^{8} + 206 T^{6} + \cdots + 1$$
$3$ $$(T^{6} - 11 T^{4} - 2 T^{3} + 25 T^{2} + \cdots + 4)^{2}$$
$5$ $$T^{12} + 40 T^{10} + 600 T^{8} + \cdots + 3481$$
$7$ $$(T^{2} + 1)^{6}$$
$11$ $$T^{12} + 50 T^{10} + 587 T^{8} + \cdots + 256$$
$13$ $$T^{12}$$
$17$ $$(T^{6} - 4 T^{5} - 21 T^{4} + 60 T^{3} + \cdots - 491)^{2}$$
$19$ $$T^{12} + 58 T^{10} + 1027 T^{8} + \cdots + 55696$$
$23$ $$(T^{6} - 12 T^{5} - 20 T^{4} + 608 T^{3} + \cdots + 6208)^{2}$$
$29$ $$(T^{6} + 8 T^{5} - 44 T^{4} - 566 T^{3} + \cdots + 3169)^{2}$$
$31$ $$T^{12} + 136 T^{10} + 5854 T^{8} + \cdots + 913936$$
$37$ $$T^{12} + 318 T^{10} + \cdots + 1755945216$$
$41$ $$T^{12} + 270 T^{10} + \cdots + 884705536$$
$43$ $$(T^{6} + 2 T^{5} - 109 T^{4} - 90 T^{3} + \cdots + 1552)^{2}$$
$47$ $$T^{12} + 272 T^{10} + 21782 T^{8} + \cdots + 9461776$$
$53$ $$(T^{6} + 22 T^{5} + 91 T^{4} - 700 T^{3} + \cdots - 2339)^{2}$$
$59$ $$T^{12} + 328 T^{10} + \cdots + 4571923456$$
$61$ $$(T^{6} + 14 T^{5} - 87 T^{4} - 1416 T^{3} + \cdots + 2368)^{2}$$
$67$ $$T^{12} + 388 T^{10} + \cdots + 613651984$$
$71$ $$T^{12} + 152 T^{10} + \cdots + 46895104$$
$73$ $$T^{12} + 334 T^{10} + \cdots + 1386221824$$
$79$ $$(T^{6} + 28 T^{5} + 212 T^{4} + 192 T^{3} + \cdots - 512)^{2}$$
$83$ $$T^{12} + 304 T^{10} + \cdots + 141324544$$
$89$ $$T^{12} + 658 T^{10} + \cdots + 1834580224$$
$97$ $$T^{12} + 382 T^{10} + \cdots + 53465344$$