# Properties

 Label 289.4.a.h Level $289$ Weight $4$ Character orbit 289.a Self dual yes Analytic conductor $17.052$ Analytic rank $1$ 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] = [289,4,Mod(1,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.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: $$17.0515519917$$ Analytic rank: $$1$$ 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: $$x^{12} - 72 x^{10} - 17 x^{9} + 1872 x^{8} + 627 x^{7} - 20922 x^{6} - 5163 x^{5} + 93255 x^{4} - 4607 x^{3} - 117822 x^{2} + 21960 x + 29352$$ x^12 - 72*x^10 - 17*x^9 + 1872*x^8 + 627*x^7 - 20922*x^6 - 5163*x^5 + 93255*x^4 - 4607*x^3 - 117822*x^2 + 21960*x + 29352 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 17^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{5} - 2) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{9} - \beta_{7} + 3 \beta_1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} - \beta_{2} + 10) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b5 - 2) * q^3 + (b2 + 4) * q^4 + (-b9 - 2) * q^5 + (b9 - b7 + 3*b1) * q^6 + (-b11 + b10 + b6 + b4 - 2*b3 + 2*b1 - 2) * q^7 + (-b10 + b7 - b6 - b5 - b4 + 2*b3 - 2*b2 - 4*b1 - 3) * q^8 + (b11 + b10 + b9 + b7 - 4*b5 - b2 + 10) * q^9 $$q - \beta_1 q^{2} + (\beta_{5} - 2) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{9} - \beta_{7} + 3 \beta_1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} - \beta_{2} + 10) q^{9} + (\beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots - 3) q^{10}+ \cdots + ( - 28 \beta_{11} - 31 \beta_{10} + 9 \beta_{9} - 24 \beta_{8} - 21 \beta_{7} + \cdots + 31) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b5 - 2) * q^3 + (b2 + 4) * q^4 + (-b9 - 2) * q^5 + (b9 - b7 + 3*b1) * q^6 + (-b11 + b10 + b6 + b4 - 2*b3 + 2*b1 - 2) * q^7 + (-b10 + b7 - b6 - b5 - b4 + 2*b3 - 2*b2 - 4*b1 - 3) * q^8 + (b11 + b10 + b9 + b7 - 4*b5 - b2 + 10) * q^9 + (b11 - 2*b10 - b9 + b7 + b6 - 3*b5 + b4 + 3*b3 + 2*b1 - 3) * q^10 + (b11 - b10 - b8 + 2*b7 + 2*b5 - b4 - 2*b2 + 4*b1 - 14) * q^11 + (b10 + b9 - 3*b7 - 4*b6 - 2*b5 - 2*b4 - b3 - 3*b2 + b1 - 18) * q^12 + (b11 - b10 + 2*b9 + b8 - 2*b7 - b6 - 5*b5 - b4 - 3*b3 - 4*b1 - 5) * q^13 + (b10 + 3*b9 + 3*b8 + 2*b7 - 2*b5 - 2*b4 - 5*b2 + 4*b1 - 22) * q^14 + (-b11 - 3*b10 + 2*b9 + b7 - 4*b6 - 5*b5 - 2*b4 + 8*b3 - b2 + 12*b1 - 8) * q^15 + (-2*b11 + b10 - 2*b9 - b8 - b7 + 4*b6 - b5 + 4*b4 - 9*b3 + 2*b2 + 11*b1 + 17) * q^16 + (5*b10 - 4*b9 - 2*b8 + 5*b7 + 9*b6 + 2*b5 + 3*b4 - 5*b3 + 3*b2 + 16) * q^18 + (-b10 - 2*b9 - 9*b7 + 4*b6 + 2*b5 + 2*b4 + b3 + 3*b2 - 8) * q^19 + (-b11 + 2*b10 - 5*b9 - 6*b8 + 5*b7 - 9*b6 + 4*b5 + 4*b4 + 10*b3 + 2*b2 - 10) * q^20 + (-2*b11 + b10 - b9 + 3*b7 - 3*b6 - 7*b5 + 4*b4 + 6*b3 - b2 - 6*b1 + 25) * q^21 + (b11 + 3*b10 + 2*b9 - b7 + 12*b6 - 2*b5 + 2*b4 - 5*b3 + b2 + 30*b1 - 40) * q^22 + (-2*b11 - 3*b10 - 3*b9 + 4*b6 - 4*b5 - b4 + 12*b3 - 2*b2 + 10*b1 - 18) * q^23 + (3*b11 - 4*b10 - 3*b9 + 5*b8 - 3*b7 + 6*b6 + 7*b5 - b4 - 12*b3 + 4*b2 + 16*b1 - 9) * q^24 + (b11 + 4*b10 + 10*b9 + 3*b8 - 3*b7 - b6 - 4*b5 - 5*b4 + 2*b3 - b2 + 2*b1 + 33) * q^25 + (-3*b11 + 2*b10 - 2*b9 - 3*b7 - 12*b6 + 7*b5 - 5*b4 - 5*b3 + 2*b2 - 2*b1 + 43) * q^26 + (-3*b11 - 6*b10 - 3*b9 - 8*b7 + b6 + 14*b5 + 2*b4 + 3*b3 + 10*b2 - 96) * q^27 + (-5*b11 + 6*b10 + 3*b9 - 6*b8 - b7 + b6 - 5*b5 + 3*b4 - 25*b3 + 6*b2 + 33*b1 - 9) * q^28 + (b10 + 7*b9 + 5*b8 + 9*b7 - 4*b6 - 2*b5 - 2*b4 - b3 + 3*b2 - 54) * q^29 + (-7*b11 - b10 - 8*b9 - 2*b8 - 3*b7 - 6*b6 + 7*b5 + 4*b4 - 18*b3 - 3*b2 - 9*b1 - 155) * q^30 + (2*b11 + b9 + b8 - 10*b7 - 20*b6 - b5 - 6*b4 + 5*b3 + 10*b2 + 14*b1 - 44) * q^31 + (5*b11 + 6*b10 + 4*b9 + 11*b8 - 4*b6 + 3*b5 - 5*b4 + 8*b3 - 10*b2 + 8*b1 - 119) * q^32 + (10*b11 - 12*b10 - 5*b9 + 3*b8 + 2*b7 + 4*b6 + 6*b5 - 6*b4 - 9*b3 + 2*b2 - 2*b1 + 80) * q^33 + (b11 - 2*b10 + 6*b9 - 9*b8 - 6*b7 + 9*b6 - 4*b5 - 4*b4 - 5*b3 - 2*b2 + 18*b1 - 7) * q^35 + (2*b11 - 4*b10 - 4*b9 + 21*b7 + 37*b6 + 5*b5 + 2*b4 + 21*b3 - 3*b2 - 15*b1 - 51) * q^36 + (4*b11 + 4*b10 + 8*b9 + 6*b8 - 5*b7 - 18*b6 - 15*b5 - b4 + 15*b3 + 7*b2 + 2*b1 + 2) * q^37 + (10*b11 - 10*b10 - 6*b9 - 5*b8 - 12*b7 - 41*b6 + 26*b5 - 10*b4 + 39*b3 - 5*b2 - 2*b1 - 18) * q^38 + (-7*b11 - 3*b10 - 5*b9 - 11*b8 + 8*b7 + 15*b6 + 8*b5 + 5*b4 + 4*b3 + 10*b2 - 24*b1 - 143) * q^39 + (11*b11 - 3*b10 - b9 + 13*b8 - 2*b7 + 27*b6 + 22*b5 - b4 + 14*b3 + 5*b2 + 3*b1 + 22) * q^40 + (7*b11 - 8*b10 + 5*b9 - 10*b8 + b7 - 14*b5 - 5*b4 - 9*b2 - 28*b1 - 22) * q^41 + (-3*b11 + b10 - 17*b9 + b8 + 11*b7 - 2*b6 + 2*b5 + 10*b4 + 4*b3 + 12*b2 - 20*b1 + 76) * q^42 + (-5*b11 - 2*b10 + b9 - 16*b8 + 16*b7 - 19*b6 + 3*b5 + 14*b4 + 17*b3 - 2*b2 - 16*b1 + 42) * q^43 + (-5*b11 + 25*b10 + 2*b9 - b8 - 2*b7 + 2*b6 - 8*b5 + 3*b4 + 6*b3 - 20*b2 + 18*b1 - 206) * q^44 + (5*b11 + 10*b10 - 9*b9 + 16*b8 + 4*b7 + 21*b6 - 6*b5 - 65*b3 + 4*b2 - 62*b1 - 37) * q^45 + (-2*b11 - 3*b10 - 16*b9 - 12*b8 + 6*b7 - 8*b6 + b5 + 12*b4 - 2*b3 + 4*b2 + 10*b1 - 127) * q^46 + (17*b11 - 2*b10 + 7*b9 + 14*b8 + 19*b7 + 16*b6 + 23*b5 - 9*b4 - 36*b3 + 7*b2 - 6*b1 - 90) * q^47 + (-10*b11 - 8*b10 - 5*b9 - 10*b8 + 20*b7 - 11*b6 - 3*b5 + 9*b4 + 25*b3 - 9*b2 - 34*b1 - 91) * q^48 + (-5*b11 + 5*b10 - 10*b9 - 11*b8 - 11*b7 - 20*b6 + 22*b5 + 16*b4 + 21*b3 + 23*b2 - 18*b1 - 59) * q^49 + (-11*b11 + 16*b10 + 13*b9 - 9*b8 - 10*b7 + 11*b6 + 19*b5 + b4 - 56*b3 + 3*b2 - 31*b1 + 19) * q^50 + (-4*b11 - 21*b10 + 12*b9 + 15*b8 - 3*b7 + 26*b6 + 13*b5 - b4 + 9*b3 - 10*b2 - 38*b1 + 13) * q^52 + (-13*b11 + 6*b10 + 11*b9 - 5*b8 + 13*b6 - 26*b5 + 15*b3 + 4*b2 + 28*b1 + 98) * q^53 + (2*b11 - 24*b10 + 4*b9 + 2*b8 - 22*b7 - 72*b6 + 14*b5 - 21*b4 + 50*b3 - 20*b2 + 14*b1 - 82) * q^54 + (-10*b11 + 16*b10 + 22*b9 + 23*b8 + 2*b7 + 13*b6 - 2*b4 - 17*b3 + 6*b2 + 10*b1 + 72) * q^55 + (17*b11 - 16*b10 + 15*b9 + 22*b8 - 3*b7 + 25*b6 + 2*b5 - 30*b4 + 14*b3 - 35*b2 - 56*b1 - 228) * q^56 + (-15*b11 + 25*b10 + 17*b9 - 24*b8 + 17*b7 + 19*b6 - 46*b5 + 8*b4 + 23*b3 - b2 - 48*b1 + 31) * q^57 + (-30*b11 + 19*b10 + 6*b9 - 5*b8 + 13*b7 + 10*b6 - 27*b5 + 9*b4 - 57*b3 - 7*b2 + 6*b1 + 5) * q^58 + (-8*b11 + 7*b10 - 3*b9 - 10*b8 - 15*b7 + 12*b5 + 4*b4 - 23*b2 + 42*b1 - 73) * q^59 + (17*b11 - 3*b10 + 8*b9 + 42*b8 - 23*b7 + 14*b6 + 13*b5 - 6*b4 - 38*b3 - 2*b2 + 78*b1 + 99) * q^60 + (17*b11 - 7*b10 + 14*b9 + 2*b8 + 19*b7 + 12*b6 + b5 - 18*b4 + 5*b3 + 3*b2 - 72*b1 - 106) * q^61 + (8*b11 - 41*b10 + 2*b9 + 9*b8 - 30*b7 - 19*b6 + 12*b5 - 12*b4 + 16*b3 - 27*b2 - 39*b1 - 240) * q^62 + (12*b11 - 16*b10 - 11*b9 + 18*b8 - 12*b7 - 27*b6 + 43*b5 - 4*b4 + 22*b3 + 4*b2 - 28*b1 - 196) * q^63 + (-10*b11 + 17*b10 + 10*b9 - 28*b8 - 7*b7 - 15*b6 + 5*b5 + 15*b4 - 4*b3 + 9*b2 + 115*b1 - 167) * q^64 + (9*b11 + 8*b10 - 15*b9 - 8*b8 - 11*b7 - 3*b6 + 8*b5 + 31*b4 - 33*b3 - 5*b2 - 44*b1 - 242) * q^65 + (-8*b11 + b10 - 13*b9 - 21*b8 - 3*b7 - 29*b6 - 24*b5 + 24*b3 - b2 - 112*b1 - 28) * q^66 + (-13*b11 + 23*b10 + 26*b9 + 12*b8 + 12*b7 + 6*b6 - 59*b5 - 7*b4 - 21*b3 - 46*b2 + 24*b1 + 74) * q^67 + (-2*b11 + 10*b10 - 18*b9 + 12*b8 - 15*b7 - 12*b6 - 43*b5 - 9*b4 - 41*b3 - 7*b2 - 10*b1 - 98) * q^69 + (26*b11 + 5*b10 + 20*b9 + 12*b8 - 9*b7 + 21*b6 + 43*b5 - 34*b4 + 6*b3 - 18*b2 + 29*b1 - 177) * q^70 + (-25*b11 + 22*b10 - 2*b9 + 20*b8 + 24*b7 + 48*b6 - 33*b5 + 12*b4 - 20*b3 - 18*b2 - 96*b1 - 206) * q^71 + (-19*b11 + 17*b10 - 46*b8 + 31*b7 - 28*b6 - 53*b5 + 23*b4 + 22*b3 + 20*b2 + 61*b1 + 155) * q^72 + (-10*b11 + 12*b10 - 23*b9 - b8 - 48*b7 - 17*b6 + 18*b5 + 28*b4 - 30*b3 + 20*b2 - 46*b1 - 75) * q^73 + (-13*b11 - 5*b10 - 30*b9 - 17*b8 - 10*b7 - 26*b6 + 36*b5 + 15*b4 - 14*b3 + 3*b2 - 55*b1 - 24) * q^74 + (5*b11 + 27*b10 - 9*b9 - 3*b8 - 16*b7 + 26*b6 + 64*b5 + b4 - 9*b3 + 28*b2 - 82*b1 - 19) * q^75 + (33*b11 - 57*b10 - 8*b9 - 8*b8 - 28*b7 - 37*b6 + 56*b5 + 12*b4 + 39*b3 + 37*b2 + 86*b1 + 20) * q^76 + (20*b11 - 49*b10 - 32*b9 - 4*b8 - 22*b7 - 32*b6 - 9*b5 - 15*b4 + 64*b3 + 20*b2 - 96*b1 - 63) * q^77 + (20*b11 - 10*b10 + 11*b9 + 17*b8 + 35*b7 + 21*b6 - 19*b5 - 25*b4 + 54*b3 + b2 + 63*b1 + 269) * q^78 + (-3*b11 - 3*b10 + 14*b9 - 17*b8 + 11*b7 + 11*b6 + 30*b5 + 12*b4 + 2*b3 + 7*b2 + 6*b1 + 17) * q^79 + (-20*b11 + 25*b10 + 11*b9 - 41*b8 - 36*b7 + 10*b6 - 24*b5 + 10*b4 - 70*b3 - 4*b2 - 32*b1 + 94) * q^80 + (-27*b11 + 22*b10 - 14*b9 - 13*b8 - b7 + b6 - 136*b5 - b4 - 40*b3 - 23*b2 - 58*b1 + 293) * q^81 + (23*b11 + 12*b10 - 11*b9 + 6*b8 - 6*b7 + 34*b6 + 48*b5 - 16*b4 + 4*b3 + 53*b2 + 88*b1 + 382) * q^82 + (4*b11 - 25*b10 - 14*b9 - 27*b8 + 50*b7 + 35*b6 + 12*b5 + 27*b4 + 10*b3 + 24*b2 + 30*b1 + 67) * q^83 + (17*b11 - 37*b10 - 25*b9 + 2*b8 + 24*b7 + 20*b6 - 15*b5 - 12*b4 + 32*b3 + 5*b2 - 114*b1 - 35) * q^84 + (24*b11 - 6*b10 - 17*b9 + 44*b8 + 5*b7 + 42*b6 + 40*b5 - 5*b4 + 52*b3 + 32*b2 + 9*b1 + 212) * q^86 + (11*b10 - 26*b9 + 14*b8 - 20*b7 - 53*b6 - 67*b5 + 5*b4 - 44*b3 - 22*b2 - 6*b1 + 98) * q^87 + (15*b11 - 3*b10 + 2*b9 + 4*b8 + 17*b7 + 28*b6 + 30*b5 + 24*b4 - 27*b3 + 15*b2 + 186*b1 + 288) * q^88 + (29*b11 - 38*b10 - 35*b9 + 40*b8 - 27*b7 + 54*b6 - 32*b5 - 11*b4 - 4*b3 - 37*b2 - 51) * q^89 + (-28*b11 + 24*b10 + 34*b9 + 2*b8 + 58*b7 + 8*b6 - 140*b5 - 9*b4 - 16*b3 - 6*b2 + 13*b1 + 724) * q^90 + (29*b11 - 15*b10 + 14*b9 + 24*b8 + 25*b7 - 5*b6 + 2*b5 - 54*b4 + 9*b3 - 13*b2 - 18*b1 - 189) * q^91 + (57*b11 - 16*b10 + 38*b8 + 20*b7 - 19*b6 + 19*b5 - 5*b4 + 29*b3 - 6*b2 + 53*b1 - 35) * q^92 + (2*b11 + 23*b10 - 8*b9 + 3*b8 + 48*b7 + 56*b6 - 93*b5 - 21*b4 + 30*b3 + 14*b2 - 62*b1 + 72) * q^93 + (-53*b11 + 64*b10 + 34*b9 - 42*b8 + 29*b7 + 30*b6 - 98*b5 + 12*b4 - 70*b3 - 27*b2 + 45*b1 + 108) * q^94 + (-53*b11 - 27*b10 - 9*b9 - 27*b8 - 17*b7 - 64*b6 - 86*b5 + 24*b4 - 47*b3 - 47*b2 - 90*b1 - 10) * q^95 + (-27*b11 + 29*b10 - 5*b9 - 14*b8 + 44*b7 - 20*b6 - 62*b5 + 29*b4 + 97*b3 + 35*b2 - 5*b1 + 464) * q^96 + (-9*b11 - 13*b10 + 18*b9 + 17*b8 - 25*b7 - 53*b6 - 86*b5 - 8*b4 + 76*b3 - 19*b2 + 82*b1 + 308) * q^97 + (54*b11 - 74*b10 - 15*b9 + 31*b8 - 26*b7 - 46*b6 + 56*b5 - 25*b4 + 175*b3 - 12*b2 - 27*b1 + 130) * q^98 + (-28*b11 - 31*b10 + 9*b9 - 24*b8 - 21*b7 - 56*b6 + 13*b5 - 8*b4 + 46*b3 + 19*b2 - 70*b1 + 31) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 18 q^{3} + 48 q^{4} - 30 q^{5} + 9 q^{6} - 24 q^{7} - 51 q^{8} + 108 q^{9}+O(q^{10})$$ 12 * q - 18 * q^3 + 48 * q^4 - 30 * q^5 + 9 * q^6 - 24 * q^7 - 51 * q^8 + 108 * q^9 $$12 q - 18 q^{3} + 48 q^{4} - 30 q^{5} + 9 q^{6} - 24 q^{7} - 51 q^{8} + 108 q^{9} - 60 q^{10} - 162 q^{11} - 216 q^{12} - 72 q^{13} - 267 q^{14} - 138 q^{15} + 192 q^{16} + 189 q^{18} - 66 q^{19} - 129 q^{20} + 246 q^{21} - 456 q^{22} - 282 q^{23} - 72 q^{24} + 444 q^{25} + 528 q^{26} - 1092 q^{27} - 120 q^{28} - 648 q^{29} - 1890 q^{30} - 504 q^{31} - 1353 q^{32} + 966 q^{33} - 66 q^{35} - 663 q^{36} + 30 q^{37} - 60 q^{38} - 1758 q^{39} + 450 q^{40} - 318 q^{41} + 804 q^{42} + 486 q^{43} - 2448 q^{44} - 486 q^{45} - 1617 q^{46} - 888 q^{47} - 1257 q^{48} - 570 q^{49} + 435 q^{50} + 225 q^{52} + 1026 q^{53} - 933 q^{54} + 972 q^{55} - 2661 q^{56} + 156 q^{57} - 201 q^{58} - 792 q^{59} + 1458 q^{60} - 1212 q^{61} - 2817 q^{62} - 2112 q^{63} - 1857 q^{64} - 2742 q^{65} - 594 q^{66} + 624 q^{67} - 1506 q^{69} - 1650 q^{70} - 2802 q^{71} + 1455 q^{72} - 726 q^{73} - 270 q^{74} + 264 q^{75} + 675 q^{76} - 1008 q^{77} + 3090 q^{78} + 444 q^{79} + 1143 q^{80} + 2520 q^{81} + 4950 q^{82} + 672 q^{83} - 777 q^{84} + 2778 q^{86} + 726 q^{87} + 3750 q^{88} - 906 q^{89} + 7755 q^{90} - 2280 q^{91} - 87 q^{92} + 132 q^{93} + 735 q^{94} - 966 q^{95} + 5046 q^{96} + 3246 q^{97} + 1911 q^{98} + 282 q^{99}+O(q^{100})$$ 12 * q - 18 * q^3 + 48 * q^4 - 30 * q^5 + 9 * q^6 - 24 * q^7 - 51 * q^8 + 108 * q^9 - 60 * q^10 - 162 * q^11 - 216 * q^12 - 72 * q^13 - 267 * q^14 - 138 * q^15 + 192 * q^16 + 189 * q^18 - 66 * q^19 - 129 * q^20 + 246 * q^21 - 456 * q^22 - 282 * q^23 - 72 * q^24 + 444 * q^25 + 528 * q^26 - 1092 * q^27 - 120 * q^28 - 648 * q^29 - 1890 * q^30 - 504 * q^31 - 1353 * q^32 + 966 * q^33 - 66 * q^35 - 663 * q^36 + 30 * q^37 - 60 * q^38 - 1758 * q^39 + 450 * q^40 - 318 * q^41 + 804 * q^42 + 486 * q^43 - 2448 * q^44 - 486 * q^45 - 1617 * q^46 - 888 * q^47 - 1257 * q^48 - 570 * q^49 + 435 * q^50 + 225 * q^52 + 1026 * q^53 - 933 * q^54 + 972 * q^55 - 2661 * q^56 + 156 * q^57 - 201 * q^58 - 792 * q^59 + 1458 * q^60 - 1212 * q^61 - 2817 * q^62 - 2112 * q^63 - 1857 * q^64 - 2742 * q^65 - 594 * q^66 + 624 * q^67 - 1506 * q^69 - 1650 * q^70 - 2802 * q^71 + 1455 * q^72 - 726 * q^73 - 270 * q^74 + 264 * q^75 + 675 * q^76 - 1008 * q^77 + 3090 * q^78 + 444 * q^79 + 1143 * q^80 + 2520 * q^81 + 4950 * q^82 + 672 * q^83 - 777 * q^84 + 2778 * q^86 + 726 * q^87 + 3750 * q^88 - 906 * q^89 + 7755 * q^90 - 2280 * q^91 - 87 * q^92 + 132 * q^93 + 735 * q^94 - 966 * q^95 + 5046 * q^96 + 3246 * q^97 + 1911 * q^98 + 282 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 72 x^{10} - 17 x^{9} + 1872 x^{8} + 627 x^{7} - 20922 x^{6} - 5163 x^{5} + 93255 x^{4} - 4607 x^{3} - 117822 x^{2} + 21960 x + 29352$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12 $$\beta_{3}$$ $$=$$ $$( 46055 \nu^{11} - 314662 \nu^{10} - 2002585 \nu^{9} + 15911043 \nu^{8} + 23350245 \nu^{7} - 270879424 \nu^{6} + 23301493 \nu^{5} + 1702105407 \nu^{4} + \cdots + 668244208 ) / 47215936$$ (46055*v^11 - 314662*v^10 - 2002585*v^9 + 15911043*v^8 + 23350245*v^7 - 270879424*v^6 + 23301493*v^5 + 1702105407*v^4 - 1204403168*v^3 - 2629068300*v^2 + 1460605688*v + 668244208) / 47215936 $$\beta_{4}$$ $$=$$ $$( 52035 \nu^{11} - 153738 \nu^{10} - 2946597 \nu^{9} + 6477251 \nu^{8} + 58746693 \nu^{7} - 77589892 \nu^{6} - 467836487 \nu^{5} + 153419951 \nu^{4} + \cdots - 475178864 ) / 47215936$$ (52035*v^11 - 153738*v^10 - 2946597*v^9 + 6477251*v^8 + 58746693*v^7 - 77589892*v^6 - 467836487*v^5 + 153419951*v^4 + 1246083148*v^3 + 1028876492*v^2 - 1347214760*v - 475178864) / 47215936 $$\beta_{5}$$ $$=$$ $$( - 66901 \nu^{11} + 325629 \nu^{10} + 3458675 \nu^{9} - 16175630 \nu^{8} - 59915226 \nu^{7} + 269286443 \nu^{6} + 366397543 \nu^{5} + \cdots - 950170632 ) / 47215936$$ (-66901*v^11 + 325629*v^10 + 3458675*v^9 - 16175630*v^8 - 59915226*v^7 + 269286443*v^6 + 366397543*v^5 - 1670888088*v^4 - 374899947*v^3 + 2859296454*v^2 - 134465856*v - 950170632) / 47215936 $$\beta_{6}$$ $$=$$ $$( 86489 \nu^{11} - 491090 \nu^{10} - 4158023 \nu^{9} + 24658509 \nu^{8} + 61818003 \nu^{7} - 415569128 \nu^{6} - 215264909 \nu^{5} + 2593563441 \nu^{4} + \cdots + 1294454224 ) / 47215936$$ (86489*v^11 - 491090*v^10 - 4158023*v^9 + 24658509*v^8 + 61818003*v^7 - 415569128*v^6 - 215264909*v^5 + 2593563441*v^4 - 1028613184*v^3 - 4205166484*v^2 + 1860851624*v + 1294454224) / 47215936 $$\beta_{7}$$ $$=$$ $$( 155568 \nu^{11} - 543997 \nu^{10} - 8625496 \nu^{9} + 25329267 \nu^{8} + 164685213 \nu^{7} - 377038701 \nu^{6} - 1193754170 \nu^{5} + \cdots - 449192696 ) / 47215936$$ (155568*v^11 - 543997*v^10 - 8625496*v^9 + 25329267*v^8 + 164685213*v^7 - 377038701*v^6 - 1193754170*v^5 + 1871083949*v^4 + 2338748637*v^3 - 1403390782*v^2 - 750496792*v - 449192696) / 47215936 $$\beta_{8}$$ $$=$$ $$( - 159467 \nu^{11} + 671066 \nu^{10} + 8518605 \nu^{9} - 32568915 \nu^{8} - 154376285 \nu^{7} + 522294276 \nu^{6} + 1015904647 \nu^{5} + \cdots - 883660304 ) / 47215936$$ (-159467*v^11 + 671066*v^10 + 8518605*v^9 - 32568915*v^8 - 154376285*v^7 + 522294276*v^6 + 1015904647*v^5 - 3023861471*v^4 - 1333374804*v^3 + 4208802084*v^2 - 508116152*v - 883660304) / 47215936 $$\beta_{9}$$ $$=$$ $$( - 170061 \nu^{11} + 814200 \nu^{10} + 8687451 \nu^{9} - 39994179 \nu^{8} - 146548157 \nu^{7} + 656266478 \nu^{6} + 822543781 \nu^{5} + \cdots - 2412870848 ) / 47215936$$ (-170061*v^11 + 814200*v^10 + 8687451*v^9 - 39994179*v^8 - 146548157*v^7 + 656266478*v^6 + 822543781*v^5 - 3992868859*v^4 - 212334910*v^3 + 6613484696*v^2 - 1316688056*v - 2412870848) / 47215936 $$\beta_{10}$$ $$=$$ $$( 176055 \nu^{11} - 854122 \nu^{10} - 8984721 \nu^{9} + 42191223 \nu^{8} + 150736233 \nu^{7} - 694924972 \nu^{6} - 830447331 \nu^{5} + \cdots + 2009725648 ) / 47215936$$ (176055*v^11 - 854122*v^10 - 8984721*v^9 + 42191223*v^8 + 150736233*v^7 - 694924972*v^6 - 830447331*v^5 + 4199199459*v^4 + 134588220*v^3 - 6438965716*v^2 + 847224856*v + 2009725648) / 47215936 $$\beta_{11}$$ $$=$$ $$( 363289 \nu^{11} - 1341287 \nu^{10} - 20053311 \nu^{9} + 63469256 \nu^{8} + 382772712 \nu^{7} - 978360605 \nu^{6} - 2803100967 \nu^{5} + \cdots + 1514531544 ) / 47215936$$ (363289*v^11 - 1341287*v^10 - 20053311*v^9 + 63469256*v^8 + 382772712*v^7 - 978360605*v^6 - 2803100967*v^5 + 5315185878*v^4 + 5819146261*v^3 - 6573286898*v^2 - 2848924336*v + 1514531544) / 47215936
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ b2 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + 20\beta _1 + 3$$ b10 - b7 + b6 + b5 + b4 - 2*b3 + 2*b2 + 20*b1 + 3 $$\nu^{4}$$ $$=$$ $$- 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{5} + 4 \beta_{4} - 9 \beta_{3} + 26 \beta_{2} + 11 \beta _1 + 241$$ -2*b11 + b10 - 2*b9 - b8 - b7 + 4*b6 - b5 + 4*b4 - 9*b3 + 26*b2 + 11*b1 + 241 $$\nu^{5}$$ $$=$$ $$- 5 \beta_{11} + 26 \beta_{10} - 4 \beta_{9} - 11 \beta_{8} - 32 \beta_{7} + 36 \beta_{6} + 29 \beta_{5} + 37 \beta_{4} - 72 \beta_{3} + 74 \beta_{2} + 440 \beta _1 + 215$$ -5*b11 + 26*b10 - 4*b9 - 11*b8 - 32*b7 + 36*b6 + 29*b5 + 37*b4 - 72*b3 + 74*b2 + 440*b1 + 215 $$\nu^{6}$$ $$=$$ $$- 90 \beta_{11} + 57 \beta_{10} - 70 \beta_{9} - 68 \beta_{8} - 47 \beta_{7} + 145 \beta_{6} - 35 \beta_{5} + 175 \beta_{4} - 364 \beta_{3} + 665 \beta_{2} + 555 \beta _1 + 5377$$ -90*b11 + 57*b10 - 70*b9 - 68*b8 - 47*b7 + 145*b6 - 35*b5 + 175*b4 - 364*b3 + 665*b2 + 555*b1 + 5377 $$\nu^{7}$$ $$=$$ $$- 288 \beta_{11} + 690 \beta_{10} - 214 \beta_{9} - 535 \beta_{8} - 878 \beta_{7} + 969 \beta_{6} + 674 \beta_{5} + 1175 \beta_{4} - 2211 \beta_{3} + 2339 \beta_{2} + 10213 \beta _1 + 8750$$ -288*b11 + 690*b10 - 214*b9 - 535*b8 - 878*b7 + 969*b6 + 674*b5 + 1175*b4 - 2211*b3 + 2339*b2 + 10213*b1 + 8750 $$\nu^{8}$$ $$=$$ $$- 3099 \beta_{11} + 2324 \beta_{10} - 2122 \beta_{9} - 2888 \beta_{8} - 1782 \beta_{7} + 4157 \beta_{6} - 771 \beta_{5} + 5882 \beta_{4} - 11827 \beta_{3} + 17390 \beta_{2} + 20397 \beta _1 + 126823$$ -3099*b11 + 2324*b10 - 2122*b9 - 2888*b8 - 1782*b7 + 4157*b6 - 771*b5 + 5882*b4 - 11827*b3 + 17390*b2 + 20397*b1 + 126823 $$\nu^{9}$$ $$=$$ $$- 11793 \beta_{11} + 19364 \beta_{10} - 8580 \beta_{9} - 19644 \beta_{8} - 23200 \beta_{7} + 24069 \beta_{6} + 15337 \beta_{5} + 35428 \beta_{4} - 65281 \beta_{3} + 70103 \beta_{2} + \cdots + 297713$$ -11793*b11 + 19364*b10 - 8580*b9 - 19644*b8 - 23200*b7 + 24069*b6 + 15337*b5 + 35428*b4 - 65281*b3 + 70103*b2 + 246934*b1 + 297713 $$\nu^{10}$$ $$=$$ $$- 98283 \beta_{11} + 82403 \beta_{10} - 63776 \beta_{9} - 104086 \beta_{8} - 60379 \beta_{7} + 109232 \beta_{6} - 11098 \beta_{5} + 181721 \beta_{4} - 358247 \beta_{3} + 464214 \beta_{2} + \cdots + 3115236$$ -98283*b11 + 82403*b10 - 63776*b9 - 104086*b8 - 60379*b7 + 109232*b6 - 11098*b5 + 181721*b4 - 358247*b3 + 464214*b2 + 663508*b1 + 3115236 $$\nu^{11}$$ $$=$$ $$- 420533 \beta_{11} + 563561 \beta_{10} - 302986 \beta_{9} - 653753 \beta_{8} - 609965 \beta_{7} + 578383 \beta_{6} + 358286 \beta_{5} + 1043116 \beta_{4} - 1902404 \beta_{3} + \cdots + 9337452$$ -420533*b11 + 563561*b10 - 302986*b9 - 653753*b8 - 609965*b7 + 578383*b6 + 358286*b5 + 1043116*b4 - 1902404*b3 + 2048466*b2 + 6172255*b1 + 9337452

## 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
 5.35197 4.98104 3.73276 2.16473 1.07125 0.820352 −0.447590 −1.26162 −3.25752 −4.28857 −4.42326 −4.44354
−5.35197 −1.66219 20.6436 5.96899 8.89600 27.8210 −67.6680 −24.2371 −31.9458
1.2 −4.98104 −6.26206 16.8108 −15.7477 31.1916 −0.789949 −43.8870 12.2134 78.4401
1.3 −3.73276 3.65511 5.93351 14.5154 −13.6437 −12.9694 7.71372 −13.6402 −54.1823
1.4 −2.16473 −9.14133 −3.31395 −16.2298 19.7885 −12.2246 24.4916 56.5639 35.1331
1.5 −1.07125 5.02012 −6.85243 −4.32398 −5.37778 −4.44673 15.9106 −1.79838 4.63204
1.6 −0.820352 0.130026 −7.32702 8.70710 −0.106667 11.4024 12.5736 −26.9831 −7.14288
1.7 0.447590 −9.51519 −7.79966 11.7425 −4.25890 −2.27797 −7.07177 63.5388 5.25584
1.8 1.26162 6.09538 −6.40830 −13.2627 7.69007 27.2593 −18.1779 10.1536 −16.7325
1.9 3.25752 6.51757 2.61146 −19.1073 21.2311 −22.0995 −17.5533 15.4787 −62.2426
1.10 4.28857 −2.85039 10.3918 6.36629 −12.2241 −29.4308 10.2575 −18.8753 27.3023
1.11 4.42326 −9.44971 11.5652 8.63042 −41.7985 −12.6513 15.7698 62.2970 38.1746
1.12 4.44354 −0.537336 11.7450 −17.2592 −2.38767 6.40763 16.6411 −26.7113 −76.6918
 $$n$$: 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
$$17$$ $$-1$$

## 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 289.4.a.h 12
17.b even 2 1 289.4.a.i yes 12
17.c even 4 2 289.4.b.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.h 12 1.a even 1 1 trivial
289.4.a.i yes 12 17.b even 2 1
289.4.b.f 24 17.c even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2}^{12} - 72 T_{2}^{10} + 17 T_{2}^{9} + 1872 T_{2}^{8} - 627 T_{2}^{7} - 20922 T_{2}^{6} + 5163 T_{2}^{5} + 93255 T_{2}^{4} + 4607 T_{2}^{3} - 117822 T_{2}^{2} - 21960 T_{2} + 29352$$ T2^12 - 72*T2^10 + 17*T2^9 + 1872*T2^8 - 627*T2^7 - 20922*T2^6 + 5163*T2^5 + 93255*T2^4 + 4607*T2^3 - 117822*T2^2 - 21960*T2 + 29352 $$T_{3}^{12} + 18 T_{3}^{11} - 54 T_{3}^{10} - 2228 T_{3}^{9} - 2097 T_{3}^{8} + 99708 T_{3}^{7} + 177392 T_{3}^{6} - 1841532 T_{3}^{5} - 3422262 T_{3}^{4} + 10471190 T_{3}^{3} + 22610910 T_{3}^{2} + \cdots - 1242003$$ T3^12 + 18*T3^11 - 54*T3^10 - 2228*T3^9 - 2097*T3^8 + 99708*T3^7 + 177392*T3^6 - 1841532*T3^5 - 3422262*T3^4 + 10471190*T3^3 + 22610910*T3^2 + 6443010*T3 - 1242003

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 72 T^{10} + 17 T^{9} + \cdots + 29352$$
$3$ $$T^{12} + 18 T^{11} - 54 T^{10} + \cdots - 1242003$$
$5$ $$T^{12} + 30 T^{11} + \cdots + 2352593298027$$
$7$ $$T^{12} + 24 T^{11} + \cdots + 578421193743$$
$11$ $$T^{12} + 162 T^{11} + \cdots + 40\!\cdots\!36$$
$13$ $$T^{12} + 72 T^{11} + \cdots - 30\!\cdots\!29$$
$17$ $$T^{12}$$
$19$ $$T^{12} + 66 T^{11} + \cdots - 60\!\cdots\!27$$
$23$ $$T^{12} + 282 T^{11} + \cdots + 93\!\cdots\!51$$
$29$ $$T^{12} + 648 T^{11} + \cdots - 47\!\cdots\!61$$
$31$ $$T^{12} + 504 T^{11} + \cdots - 12\!\cdots\!93$$
$37$ $$T^{12} - 30 T^{11} + \cdots - 25\!\cdots\!77$$
$41$ $$T^{12} + 318 T^{11} + \cdots + 18\!\cdots\!93$$
$43$ $$T^{12} - 486 T^{11} + \cdots + 31\!\cdots\!01$$
$47$ $$T^{12} + 888 T^{11} + \cdots + 56\!\cdots\!91$$
$53$ $$T^{12} - 1026 T^{11} + \cdots + 59\!\cdots\!12$$
$59$ $$T^{12} + 792 T^{11} + \cdots - 29\!\cdots\!87$$
$61$ $$T^{12} + 1212 T^{11} + \cdots + 14\!\cdots\!71$$
$67$ $$T^{12} - 624 T^{11} + \cdots - 64\!\cdots\!07$$
$71$ $$T^{12} + 2802 T^{11} + \cdots + 14\!\cdots\!47$$
$73$ $$T^{12} + 726 T^{11} + \cdots - 88\!\cdots\!43$$
$79$ $$T^{12} - 444 T^{11} + \cdots - 88\!\cdots\!44$$
$83$ $$T^{12} - 672 T^{11} + \cdots - 19\!\cdots\!11$$
$89$ $$T^{12} + 906 T^{11} + \cdots - 37\!\cdots\!51$$
$97$ $$T^{12} - 3246 T^{11} + \cdots + 16\!\cdots\!44$$