# Properties

 Label 1183.2.a.q Level $1183$ Weight $2$ Character orbit 1183.a Self dual yes Analytic conductor $9.446$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(1,1183)]

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, 0]))

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

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

chi := DirichletCharacter("1183.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## 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: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29$$ x^12 - 3*x^11 - 15*x^10 + 46*x^9 + 80*x^8 - 246*x^7 - 199*x^6 + 562*x^5 + 262*x^4 - 542*x^3 - 157*x^2 + 183*x + 29 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{5} - \beta_{4} + 2) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{4} + 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{6} - q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 3) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b6 + 1) * q^3 + (b5 - b4 + 2) * q^4 + (-b11 - b7 - b4 + 1) * q^5 + (-b11 - b10 - b9 - b8 + b7 - b4 + b2 - b1) * q^6 - q^7 + (-b3 + b2 - b1 - 1) * q^8 + (-b8 + b6 + b5 + b4 - b1 + 3) * q^9 $$q - \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{5} - \beta_{4} + 2) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{4} + 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{6} - q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 3) q^{9} + (\beta_{10} - \beta_{9} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{10} + (\beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{11} + (\beta_{11} - \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{12} + \beta_1 q^{14} + ( - 2 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 - 2) q^{15} + ( - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3) q^{17} + ( - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \cdots + 2) q^{18}+ \cdots + ( - 2 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 8) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b6 + 1) * q^3 + (b5 - b4 + 2) * q^4 + (-b11 - b7 - b4 + 1) * q^5 + (-b11 - b10 - b9 - b8 + b7 - b4 + b2 - b1) * q^6 - q^7 + (-b3 + b2 - b1 - 1) * q^8 + (-b8 + b6 + b5 + b4 - b1 + 3) * q^9 + (b10 - b9 - b5 - b3 - b2 - b1 - 1) * q^10 + (b10 - b9 + b6 - b5 - b2 - 1) * q^11 + (b11 - b10 + 3*b9 + 2*b8 + b5 - b4 - b3 + 1) * q^12 + b1 * q^14 + (-2*b10 + 4*b9 + 2*b8 - b7 - b6 - b2 + b1 - 2) * q^15 + (-b9 + b7 + b5 - b4 + b3 + b2 + 2) * q^16 + (-b10 - b9 - b8 - b7 + b6 + 3) * q^17 + (-b11 - 3*b10 - 3*b9 - 2*b8 + b7 + b5 + b4 - b3 + b2 - 4*b1 + 2) * q^18 + (3*b10 - b9 - b7 + b6 - b5 - b4 - b1 + 2) * q^19 + (2*b10 - b8 + b5 + b4 + 2*b3 + b1 + 3) * q^20 + (-b6 - 1) * q^21 + (-2*b11 + b10 - b8 - b7 - b5 + b3 + 2*b1) * q^22 + (-b11 + b10 - 2*b9 + b7 + b3 - b1 + 2) * q^23 + (b11 + b7 - b6 + 2*b5 - 4*b4 + b3 + 2*b2 - b1 + 3) * q^24 + (b11 + 2*b10 + b8 - b7 + b6 - b5 + 2*b3 + b1 + 4) * q^25 + (-b11 - 3*b10 + b9 - 2*b8 + b7 + b6 + b5 + 2*b4 + b2 + 2) * q^27 + (-b5 + b4 - 2) * q^28 + (2*b10 + b9 + 2*b8 - b7 - b5 - 2*b4 + b1 + 2) * q^29 + (-2*b10 - 2*b9 - 2*b8 + 2*b7 - b6 - 2*b5 - 2*b4 + 2*b1 - 3) * q^30 + (b11 + 2*b9 + 3*b8 + b5 + b4 + b3 - b2 - b1 + 2) * q^31 + (2*b11 + b10 + 3*b9 + 2*b8 + b6 - b4 - b2 + 1) * q^32 + (-2*b9 - 2*b8 + b7 - b6 - 2*b5 + 2*b4 - b2 + b1) * q^33 + (-b11 - b6 + 3*b4 - b2 - 3*b1 - 1) * q^34 + (b11 + b7 + b4 - 1) * q^35 + (b10 + 3*b9 + b8 - 2*b7 + 3*b5 - 4*b4 - b3 + b2 - 2*b1 + 8) * q^36 + (-b11 - b9 - 2*b8 - 2*b7 + 2*b6 + b5 - 2*b4 + b1 + 2) * q^37 + (-b11 + 2*b10 - 2*b9 + b8 - b6 + b5 - b4 + b3 - b2 - b1 + 5) * q^38 + (2*b11 - 4*b10 + b8 - b5 + 6*b4 - b3 - 6) * q^40 + (2*b11 - b10 - b9 + b5 + 2*b4 + b3 + b2 + b1 + 1) * q^41 + (b11 + b10 + b9 + b8 - b7 + b4 - b2 + b1) * q^42 + (-b11 - 4*b10 - b9 - b8 - b7 + b6 + 2*b4 - 2*b3 + b2 + b1 - 4) * q^43 + (b11 - 3*b10 + b8 - b6 - 3*b5 + 5*b4 - 2*b3 - 2*b2 + 2*b1 - 9) * q^44 + (-b11 - b10 + 3*b9 - 2*b7 - b6 + 3*b5 + b4 - b3 - 2*b2) * q^45 + (b11 + 3*b10 + 2*b9 - 2*b7 + 2*b6 - b5 - 2*b4 - 2*b3 - b1 + 2) * q^46 + (2*b11 + 2*b10 + 2*b9 + 3*b8 + b5 + b4 + 2*b3 - 2*b2 + b1 + 1) * q^47 + (2*b11 + 2*b10 + 3*b8 + b7 + b5 + b4 + b2 - 4*b1 + 2) * q^48 + q^49 + (b11 + 2*b10 + 2*b8 + b7 - 2*b6 - 2*b5 + b4 - 2*b2 - b1 - 3) * q^50 + (-b11 - 2*b10 - 2*b8 + b7 + 2*b6 - 2*b5 + b4 - 2*b3 + 2*b1 - 1) * q^51 + (b11 + 3*b10 + 2*b9 - 2*b6 - 2*b4 - b3 - b2 + 5) * q^53 + (-8*b10 - 2*b9 - 2*b8 + 3*b7 + 2*b6 + 2*b4 - 2*b3 + b2 - 3*b1 - 3) * q^54 + (b11 + 4*b9 + 3*b8 + b7 - 3*b6 - b5 - 2*b4 - 2*b2 - 3*b1 - 3) * q^55 + (b3 - b2 + b1 + 1) * q^56 + (2*b11 + 3*b10 + b9 + b8 + 2*b7 - 2*b5 - 2*b4 + 2*b3 + b2 + 3*b1 + 1) * q^57 + (3*b10 - b9 + b8 + b7 - b6 - b5 - 4*b4 + b3 - b1 - 1) * q^58 + (-b11 + b10 - b9 - 2*b8 + b7 + b6 + b5 - b4 - b3 + b2 + b1 + 3) * q^59 + (b11 + 3*b10 + 3*b9 + 3*b8 - b7 + 4*b6 - 2*b5 + 7*b4 + 2*b3 - b2 + 3*b1 - 4) * q^60 + (-2*b8 - b6 - 2*b5 + 2*b4 - 2*b3 + 2*b1 - 1) * q^61 + (2*b10 - 2*b9 - 4*b8 + b7 - b6 - 7*b4 - b3 + 3*b2 - 2*b1 + 5) * q^62 + (b8 - b6 - b5 - b4 + b1 - 3) * q^63 + (-2*b11 - 3*b10 - b8 + b7 - 2*b6 - b5 - 2*b4 + b2 - b1 - 1) * q^64 + (4*b9 + 2*b8 - 4*b7 + b6 - 2*b5 + 8*b4 + 2*b3 - 4*b2 + 2*b1 - 5) * q^66 + (-2*b11 - b10 - 4*b9 - 3*b8 + 2*b6 - b5 + 2*b4 - b3 - 2) * q^67 + (3*b10 - b9 - 2*b8 - b6 + b5 - 2*b4 - b3 - b2 + b1 + 4) * q^68 + (b11 - b10 - b9 + 2*b8 + 2*b7 + b6 - b5 + 4*b4 + 2*b3 - 3*b1 - 1) * q^69 + (-b10 + b9 + b5 + b3 + b2 + b1 + 1) * q^70 + (2*b11 - b10 - b8 + b7 - 2*b6 + b5 - b3 + b2 + b1 - 3) * q^71 + (2*b11 + 4*b10 + 4*b8 + 2*b7 - 2*b6 + 2*b5 - 7*b4 + b2 - 4*b1 + 4) * q^72 + (-b10 - 3*b9 - b8 + b7 - b6 + 2*b5 - 2*b3 + 2*b2 - 2*b1 + 1) * q^73 + (-2*b11 - 2*b10 - 3*b9 + b7 - b6 - 2*b5 + 6*b4 - 2*b3 - b2 - 3*b1 - 8) * q^74 + (b11 + 9*b10 - b9 + 2*b8 + 3*b6 - 3*b5 + 2*b4 + 2*b3 + 3*b1 + 3) * q^75 + (b11 + b10 + 3*b9 - b8 - 2*b7 - b6 - 3*b3 - 3*b1 - 3) * q^76 + (-b10 + b9 - b6 + b5 + b2 + 1) * q^77 + (-2*b11 - 4*b9 - b8 - b5 + b4 + b3 + 2*b2) * q^79 + (-b11 - 2*b10 - 3*b8 - 2*b6 + b5 - 4*b4 + b2 + 4*b1 - 1) * q^80 + (-4*b11 - 6*b10 - 2*b9 - 3*b8 - 2*b7 + 4*b6 + b5 + 4*b4 - b3 - 4*b1 + 3) * q^81 + (2*b11 + 2*b10 + 2*b9 + b8 - 2*b6 + b5 + 3*b4 - b1 - 3) * q^82 + (2*b11 - b7 + 2*b6 + b4 + b3 + b2 + 2) * q^83 + (-b11 + b10 - 3*b9 - 2*b8 - b5 + b4 + b3 - 1) * q^84 + (-4*b11 - 3*b10 - 3*b9 - b8 - 3*b7 + 2*b2 - 2*b1 + 4) * q^85 + (-2*b11 - 2*b9 - 4*b8 + b7 + b5 + b3 + b2 + 2*b1 - 3) * q^86 + (3*b11 + 6*b10 + 4*b9 + 5*b8 - 2*b7 - b6 + b5 - 4*b4 + 2*b3 - b2 + 2*b1 + 6) * q^87 + (b11 + b10 + b9 - 3*b8 - b7 - b6 + b5 + b4 + 4*b3 - b2 + 6*b1 - 3) * q^88 + (-2*b11 - b10 - 3*b9 - 2*b8 + 2*b7 + b6 + b5 + b4 - 2*b3 + 3*b2 + 1) * q^89 + (-2*b11 - 3*b10 - 9*b9 - 7*b8 - b6 - 2*b5 + 3*b4 - 3*b3 + b2 - 4*b1 - 5) * q^90 + (-2*b11 - 6*b10 - 3*b9 + 2*b7 - 3*b6 + 4*b5 - 2*b4 + 2*b3 + b2 - b1 + 4) * q^92 + (-b11 + 2*b10 + 2*b9 + 3*b8 + b6 + 3*b5 - 2*b4 + 2*b3 + b2 + 8) * q^93 + (2*b10 - 2*b9 - 3*b8 - 2*b6 - 3*b5 - 2*b4 - b3 + 2*b2 - 4) * q^94 + (-2*b10 + 4*b9 + 5*b8 - 3*b7 - 4*b6 - b5 - 2*b4 - b3 - 3*b2 - 3*b1 + 1) * q^95 + (-b11 + 5*b10 - b9 - 2*b8 - b7 + b6 + 3*b5 - 5*b4 - b3 + b2 - b1 + 15) * q^96 + (-2*b11 + b10 - b9 + 2*b8 + b7 - 2*b6 - 3*b5 - b4 + b1 - 3) * q^97 - b1 * q^98 + (-2*b11 - 4*b10 - 6*b9 - 6*b8 + b7 + 2*b6 - 4*b5 + 6*b4 - b3 + b2 + 2*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9}+O(q^{10})$$ 12 * q - 3 * q^2 + 8 * q^3 + 15 * q^4 + 4 * q^5 - 2 * q^6 - 12 * q^7 - 12 * q^8 + 26 * q^9 $$12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9} - 6 q^{10} - 12 q^{11} + 13 q^{12} + 3 q^{14} - 11 q^{15} + 13 q^{16} + 31 q^{17} + 29 q^{18} + 3 q^{19} + 18 q^{20} - 8 q^{21} - 4 q^{22} + 18 q^{23} + 6 q^{24} + 32 q^{25} + 32 q^{27} - 15 q^{28} + 15 q^{29} - 10 q^{30} + 21 q^{31} + 3 q^{32} + 29 q^{33} - 3 q^{34} - 4 q^{35} + 49 q^{36} - 5 q^{37} + 45 q^{38} - 20 q^{40} + 16 q^{41} + 2 q^{42} - 22 q^{43} - 35 q^{44} - 5 q^{45} - 2 q^{46} + 4 q^{47} + 11 q^{48} + 12 q^{49} - 13 q^{50} + 18 q^{51} + 53 q^{53} + 5 q^{54} - 26 q^{55} + 12 q^{56} + 8 q^{57} - 32 q^{58} + 26 q^{59} - 38 q^{60} + 22 q^{61} + 19 q^{62} - 26 q^{63} + 2 q^{64} - 34 q^{66} - 12 q^{67} + 34 q^{68} + 3 q^{69} + 6 q^{70} - 21 q^{71} + 4 q^{72} + 15 q^{73} - 40 q^{74} + 15 q^{75} - 43 q^{76} + 12 q^{77} + 2 q^{79} - 13 q^{80} + 36 q^{81} - 32 q^{82} + 9 q^{83} - 13 q^{84} + 39 q^{85} - 44 q^{86} + 27 q^{87} - 48 q^{88} + 22 q^{89} - 26 q^{90} + 52 q^{92} + 53 q^{93} - 44 q^{94} + 29 q^{95} + 114 q^{96} - 9 q^{97} - 3 q^{98} - 37 q^{99}+O(q^{100})$$ 12 * q - 3 * q^2 + 8 * q^3 + 15 * q^4 + 4 * q^5 - 2 * q^6 - 12 * q^7 - 12 * q^8 + 26 * q^9 - 6 * q^10 - 12 * q^11 + 13 * q^12 + 3 * q^14 - 11 * q^15 + 13 * q^16 + 31 * q^17 + 29 * q^18 + 3 * q^19 + 18 * q^20 - 8 * q^21 - 4 * q^22 + 18 * q^23 + 6 * q^24 + 32 * q^25 + 32 * q^27 - 15 * q^28 + 15 * q^29 - 10 * q^30 + 21 * q^31 + 3 * q^32 + 29 * q^33 - 3 * q^34 - 4 * q^35 + 49 * q^36 - 5 * q^37 + 45 * q^38 - 20 * q^40 + 16 * q^41 + 2 * q^42 - 22 * q^43 - 35 * q^44 - 5 * q^45 - 2 * q^46 + 4 * q^47 + 11 * q^48 + 12 * q^49 - 13 * q^50 + 18 * q^51 + 53 * q^53 + 5 * q^54 - 26 * q^55 + 12 * q^56 + 8 * q^57 - 32 * q^58 + 26 * q^59 - 38 * q^60 + 22 * q^61 + 19 * q^62 - 26 * q^63 + 2 * q^64 - 34 * q^66 - 12 * q^67 + 34 * q^68 + 3 * q^69 + 6 * q^70 - 21 * q^71 + 4 * q^72 + 15 * q^73 - 40 * q^74 + 15 * q^75 - 43 * q^76 + 12 * q^77 + 2 * q^79 - 13 * q^80 + 36 * q^81 - 32 * q^82 + 9 * q^83 - 13 * q^84 + 39 * q^85 - 44 * q^86 + 27 * q^87 - 48 * q^88 + 22 * q^89 - 26 * q^90 + 52 * q^92 + 53 * q^93 - 44 * q^94 + 29 * q^95 + 114 * q^96 - 9 * q^97 - 3 * q^98 - 37 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 77279 \nu^{3} + 193411 \nu^{2} + \cdots + 25085 ) / 14629$$ (202*v^11 + 1816*v^10 - 5010*v^9 - 29197*v^8 + 39197*v^7 + 161452*v^6 - 110863*v^5 - 346540*v^4 + 77279*v^3 + 193411*v^2 + 16187*v + 25085) / 14629 $$\beta_{3}$$ $$=$$ $$( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 91908 \nu^{3} + 193411 \nu^{2} + \cdots + 10456 ) / 14629$$ (202*v^11 + 1816*v^10 - 5010*v^9 - 29197*v^8 + 39197*v^7 + 161452*v^6 - 110863*v^5 - 346540*v^4 + 91908*v^3 + 193411*v^2 - 56958*v + 10456) / 14629 $$\beta_{4}$$ $$=$$ $$( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 165757 \nu^{2} + \cdots - 14765 ) / 14629$$ (266*v^11 + 1812*v^10 - 9639*v^9 - 26136*v^8 + 98110*v^7 + 125845*v^6 - 384542*v^5 - 239507*v^4 + 573947*v^3 + 165757*v^2 - 263443*v - 14765) / 14629 $$\beta_{5}$$ $$=$$ $$( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 180386 \nu^{2} + \cdots - 73281 ) / 14629$$ (266*v^11 + 1812*v^10 - 9639*v^9 - 26136*v^8 + 98110*v^7 + 125845*v^6 - 384542*v^5 - 239507*v^4 + 573947*v^3 + 180386*v^2 - 263443*v - 73281) / 14629 $$\beta_{6}$$ $$=$$ $$( 721 \nu^{11} - 2788 \nu^{10} - 8033 \nu^{9} + 41570 \nu^{8} + 13387 \nu^{7} - 212101 \nu^{6} + 95671 \nu^{5} + 447600 \nu^{4} - 310269 \nu^{3} - 374170 \nu^{2} + \cdots + 63537 ) / 14629$$ (721*v^11 - 2788*v^10 - 8033*v^9 + 41570*v^8 + 13387*v^7 - 212101*v^6 + 95671*v^5 + 447600*v^4 - 310269*v^3 - 374170*v^2 + 204863*v + 63537) / 14629 $$\beta_{7}$$ $$=$$ $$( 1025 \nu^{11} - 2807 \nu^{10} - 19049 \nu^{9} + 45138 \nu^{8} + 135962 \nu^{7} - 245916 \nu^{6} - 469197 \nu^{5} + 506165 \nu^{4} + 761552 \nu^{3} - 264148 \nu^{2} + \cdots - 16033 ) / 14629$$ (1025*v^11 - 2807*v^10 - 19049*v^9 + 45138*v^8 + 135962*v^7 - 245916*v^6 - 469197*v^5 + 506165*v^4 + 761552*v^3 - 264148*v^2 - 384615*v - 16033) / 14629 $$\beta_{8}$$ $$=$$ $$( 1181 \nu^{11} - 6474 \nu^{10} - 9303 \nu^{9} + 90086 \nu^{8} - 23075 \nu^{7} - 408596 \nu^{6} + 301924 \nu^{5} + 705799 \nu^{4} - 620810 \nu^{3} - 446758 \nu^{2} + \cdots + 60552 ) / 14629$$ (1181*v^11 - 6474*v^10 - 9303*v^9 + 90086*v^8 - 23075*v^7 - 408596*v^6 + 301924*v^5 + 705799*v^4 - 620810*v^3 - 446758*v^2 + 361943*v + 60552) / 14629 $$\beta_{9}$$ $$=$$ $$( 1429 \nu^{11} + 825 \nu^{10} - 29069 \nu^{9} - 13256 \nu^{8} + 214356 \nu^{7} + 76988 \nu^{6} - 690923 \nu^{5} - 201544 \nu^{4} + 930739 \nu^{3} + 225077 \nu^{2} + \cdots - 68266 ) / 14629$$ (1429*v^11 + 825*v^10 - 29069*v^9 - 13256*v^8 + 214356*v^7 + 76988*v^6 - 690923*v^5 - 201544*v^4 + 930739*v^3 + 225077*v^2 - 425386*v - 68266) / 14629 $$\beta_{10}$$ $$=$$ $$( 2354 \nu^{11} - 5633 \nu^{10} - 34485 \nu^{9} + 79215 \nu^{8} + 175064 \nu^{7} - 364728 \nu^{6} - 391458 \nu^{5} + 632025 \nu^{4} + 415204 \nu^{3} - 345129 \nu^{2} + \cdots + 5396 ) / 14629$$ (2354*v^11 - 5633*v^10 - 34485*v^9 + 79215*v^8 + 175064*v^7 - 364728*v^6 - 391458*v^5 + 632025*v^4 + 415204*v^3 - 345129*v^2 - 144501*v + 5396) / 14629 $$\beta_{11}$$ $$=$$ $$( - 4628 \nu^{11} + 11261 \nu^{10} + 66841 \nu^{9} - 158261 \nu^{8} - 324031 \nu^{7} + 725177 \nu^{6} + 627337 \nu^{5} - 1236319 \nu^{4} - 443637 \nu^{3} + \cdots + 5226 ) / 14629$$ (-4628*v^11 + 11261*v^10 + 66841*v^9 - 158261*v^8 - 324031*v^7 + 725177*v^6 + 627337*v^5 - 1236319*v^4 - 443637*v^3 + 648376*v^2 + 34553*v + 5226) / 14629
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 4$$ b5 - b4 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{2} + 5\beta _1 + 1$$ b3 - b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{9} + \beta_{7} + 7\beta_{5} - 7\beta_{4} + \beta_{3} + \beta_{2} + 22$$ -b9 + b7 + 7*b5 - 7*b4 + b3 + b2 + 22 $$\nu^{5}$$ $$=$$ $$-2\beta_{11} - \beta_{10} - 3\beta_{9} - 2\beta_{8} - \beta_{6} + \beta_{4} + 8\beta_{3} - 7\beta_{2} + 28\beta _1 + 7$$ -2*b11 - b10 - 3*b9 - 2*b8 - b6 + b4 + 8*b3 - 7*b2 + 28*b1 + 7 $$\nu^{6}$$ $$=$$ $$- 2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - \beta_{8} + 11 \beta_{7} - 2 \beta_{6} + 45 \beta_{5} - 48 \beta_{4} + 10 \beta_{3} + 11 \beta_{2} - \beta _1 + 131$$ -2*b11 - 3*b10 - 10*b9 - b8 + 11*b7 - 2*b6 + 45*b5 - 48*b4 + 10*b3 + 11*b2 - b1 + 131 $$\nu^{7}$$ $$=$$ $$- 23 \beta_{11} - 9 \beta_{10} - 37 \beta_{9} - 25 \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{4} + 57 \beta_{3} - 43 \beta_{2} + 166 \beta _1 + 43$$ -23*b11 - 9*b10 - 37*b9 - 25*b8 + b7 - 13*b6 + 10*b4 + 57*b3 - 43*b2 + 166*b1 + 43 $$\nu^{8}$$ $$=$$ $$- 27 \beta_{11} - 36 \beta_{10} - 84 \beta_{9} - 19 \beta_{8} + 93 \beta_{7} - 24 \beta_{6} + 289 \beta_{5} - 330 \beta_{4} + 80 \beta_{3} + 94 \beta_{2} - 14 \beta _1 + 812$$ -27*b11 - 36*b10 - 84*b9 - 19*b8 + 93*b7 - 24*b6 + 289*b5 - 330*b4 + 80*b3 + 94*b2 - 14*b1 + 812 $$\nu^{9}$$ $$=$$ $$- 198 \beta_{11} - 61 \beta_{10} - 331 \beta_{9} - 231 \beta_{8} + 14 \beta_{7} - 120 \beta_{6} - \beta_{5} + 67 \beta_{4} + 396 \beta_{3} - 259 \beta_{2} + 1021 \beta _1 + 265$$ -198*b11 - 61*b10 - 331*b9 - 231*b8 + 14*b7 - 120*b6 - b5 + 67*b4 + 396*b3 - 259*b2 + 1021*b1 + 265 $$\nu^{10}$$ $$=$$ $$- 257 \beta_{11} - 306 \beta_{10} - 658 \beta_{9} - 224 \beta_{8} + 710 \beta_{7} - 212 \beta_{6} + 1874 \beta_{5} - 2277 \beta_{4} + 593 \beta_{3} + 734 \beta_{2} - 132 \beta _1 + 5167$$ -257*b11 - 306*b10 - 658*b9 - 224*b8 + 710*b7 - 212*b6 + 1874*b5 - 2277*b4 + 593*b3 + 734*b2 - 132*b1 + 5167 $$\nu^{11}$$ $$=$$ $$- 1539 \beta_{11} - 370 \beta_{10} - 2625 \beta_{9} - 1909 \beta_{8} + 136 \beta_{7} - 967 \beta_{6} - 16 \beta_{5} + 356 \beta_{4} + 2724 \beta_{3} - 1555 \beta_{2} + 6448 \beta _1 + 1686$$ -1539*b11 - 370*b10 - 2625*b9 - 1909*b8 + 136*b7 - 967*b6 - 16*b5 + 356*b4 + 2724*b3 - 1555*b2 + 6448*b1 + 1686

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.62803 2.47725 2.23724 2.06743 0.983820 0.842530 −0.149660 −0.961590 −1.10989 −1.35819 −2.07140 −2.58557
−2.62803 −1.76762 4.90656 2.94291 4.64535 −1.00000 −7.63852 0.124466 −7.73406
1.2 −2.47725 0.982981 4.13677 −1.35413 −2.43509 −1.00000 −5.29330 −2.03375 3.35452
1.3 −2.23724 3.02592 3.00523 3.28547 −6.76971 −1.00000 −2.24893 6.15622 −7.35037
1.4 −2.06743 2.11889 2.27425 −2.43928 −4.38065 −1.00000 −0.566992 1.48970 5.04303
1.5 −0.983820 −1.57171 −1.03210 −0.398447 1.54628 −1.00000 2.98304 −0.529731 0.392000
1.6 −0.842530 0.161973 −1.29014 3.72786 −0.136467 −1.00000 2.77204 −2.97376 −3.14083
1.7 0.149660 2.76031 −1.97760 −4.13443 0.413107 −1.00000 −0.595288 4.61930 −0.618759
1.8 0.961590 −1.98737 −1.07534 −3.39320 −1.91103 −1.00000 −2.95722 0.949635 −3.26287
1.9 1.10989 0.955760 −0.768150 3.55862 1.06079 −1.00000 −3.07233 −2.08652 3.94966
1.10 1.35819 3.39737 −0.155322 0.772491 4.61428 −1.00000 −2.92733 8.54215 1.04919
1.11 2.07140 −3.01646 2.29068 2.69245 −6.24828 −1.00000 0.602118 6.09903 5.57713
1.12 2.58557 2.93994 4.68518 −1.26031 7.60143 −1.00000 6.94272 5.64327 −3.25863
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.q 12
7.b odd 2 1 8281.2.a.cn 12
13.b even 2 1 1183.2.a.r yes 12
13.d odd 4 2 1183.2.c.j 24
91.b odd 2 1 8281.2.a.cq 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.q 12 1.a even 1 1 trivial
1183.2.a.r yes 12 13.b even 2 1
1183.2.c.j 24 13.d odd 4 2
8281.2.a.cn 12 7.b odd 2 1
8281.2.a.cq 12 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1183))$$:

 $$T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 46 T_{2}^{9} + 80 T_{2}^{8} + 246 T_{2}^{7} - 199 T_{2}^{6} - 562 T_{2}^{5} + 262 T_{2}^{4} + 542 T_{2}^{3} - 157 T_{2}^{2} - 183 T_{2} + 29$$ T2^12 + 3*T2^11 - 15*T2^10 - 46*T2^9 + 80*T2^8 + 246*T2^7 - 199*T2^6 - 562*T2^5 + 262*T2^4 + 542*T2^3 - 157*T2^2 - 183*T2 + 29 $$T_{11}^{12} + 12 T_{11}^{11} - 13 T_{11}^{10} - 613 T_{11}^{9} - 994 T_{11}^{8} + 12005 T_{11}^{7} + 30684 T_{11}^{6} - 112394 T_{11}^{5} - 339321 T_{11}^{4} + 505685 T_{11}^{3} + 1617506 T_{11}^{2} - 900676 T_{11} - 2701133$$ T11^12 + 12*T11^11 - 13*T11^10 - 613*T11^9 - 994*T11^8 + 12005*T11^7 + 30684*T11^6 - 112394*T11^5 - 339321*T11^4 + 505685*T11^3 + 1617506*T11^2 - 900676*T11 - 2701133

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 3 T^{11} - 15 T^{10} - 46 T^{9} + \cdots + 29$$
$3$ $$T^{12} - 8 T^{11} + T^{10} + 136 T^{9} + \cdots + 448$$
$5$ $$T^{12} - 4 T^{11} - 38 T^{10} + \cdots + 6208$$
$7$ $$(T + 1)^{12}$$
$11$ $$T^{12} + 12 T^{11} - 13 T^{10} + \cdots - 2701133$$
$13$ $$T^{12}$$
$17$ $$T^{12} - 31 T^{11} + 364 T^{10} + \cdots + 110272$$
$19$ $$T^{12} - 3 T^{11} - 131 T^{10} + \cdots + 1395008$$
$23$ $$T^{12} - 18 T^{11} + 12 T^{10} + \cdots + 7017331$$
$29$ $$T^{12} - 15 T^{11} - 24 T^{10} + \cdots + 177673$$
$31$ $$T^{12} - 21 T^{11} + 7 T^{10} + \cdots - 15261184$$
$37$ $$T^{12} + 5 T^{11} - 252 T^{10} + \cdots + 545930729$$
$41$ $$T^{12} - 16 T^{11} + \cdots - 101827648$$
$43$ $$T^{12} + 22 T^{11} - 28 T^{10} + \cdots - 9002449$$
$47$ $$T^{12} - 4 T^{11} - 262 T^{10} + \cdots + 17156608$$
$53$ $$T^{12} - 53 T^{11} + 1054 T^{10} + \cdots + 62267981$$
$59$ $$T^{12} - 26 T^{11} + \cdots + 913870784$$
$61$ $$T^{12} - 22 T^{11} - 129 T^{10} + \cdots - 69982144$$
$67$ $$T^{12} + 12 T^{11} - 199 T^{10} + \cdots + 1207037$$
$71$ $$T^{12} + 21 T^{11} - 125 T^{10} + \cdots - 22571863$$
$73$ $$T^{12} - 15 T^{11} + \cdots - 4111861312$$
$79$ $$T^{12} - 2 T^{11} - 364 T^{10} + \cdots - 33746539$$
$83$ $$T^{12} - 9 T^{11} - 248 T^{10} + \cdots - 478100288$$
$89$ $$T^{12} - 22 T^{11} + \cdots - 639815168$$
$97$ $$T^{12} + 9 T^{11} + \cdots - 6873362944$$