# Properties

 Label 169.4.a.k Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, 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("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$9.97132279097$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096$$ x^9 - 4*x^8 - 46*x^7 + 145*x^6 + 680*x^5 - 1501*x^4 - 3203*x^3 + 4784*x^2 + 3584*x - 4096 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$13^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} + (\beta_{8} + \beta_{6} + \beta_{5} + 5) q^{4} + ( - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{5} + ( - 2 \beta_{8} - 2 \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 - 6) q^{6} + (2 \beta_{8} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 4) q^{7} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} + 4 \beta_1 - 9) q^{8} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 7) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + b2 * q^3 + (b8 + b6 + b5 + 5) * q^4 + (-b8 - b5 + b4 - b3 - b2 - 3*b1 - 2) * q^5 + (-2*b8 - 2*b6 - b3 - b2 - b1 - 6) * q^6 + (2*b8 - b5 - 2*b4 + b3 - b2 - b1 - 4) * q^7 + (-b5 + 4*b3 - b2 + 4*b1 - 9) * q^8 + (-b8 - b7 - b6 - b4 + 3*b3 - 2*b2 - b1 + 7) * q^9 $$q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} + (\beta_{8} + \beta_{6} + \beta_{5} + 5) q^{4} + ( - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{5} + ( - 2 \beta_{8} - 2 \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 - 6) q^{6} + (2 \beta_{8} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 4) q^{7} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} + 4 \beta_1 - 9) q^{8} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 7) q^{9} + (\beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots - 16) q^{10}+ \cdots + ( - 100 \beta_{8} - 14 \beta_{7} + 14 \beta_{6} + 18 \beta_{5} + \cdots - 121) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + b2 * q^3 + (b8 + b6 + b5 + 5) * q^4 + (-b8 - b5 + b4 - b3 - b2 - 3*b1 - 2) * q^5 + (-2*b8 - 2*b6 - b3 - b2 - b1 - 6) * q^6 + (2*b8 - b5 - 2*b4 + b3 - b2 - b1 - 4) * q^7 + (-b5 + 4*b3 - b2 + 4*b1 - 9) * q^8 + (-b8 - b7 - b6 - b4 + 3*b3 - 2*b2 - b1 + 7) * q^9 + (b8 + 2*b7 - 2*b6 - b5 - 3*b4 + 2*b3 + b2 - 6*b1 - 16) * q^10 + (-b8 - 2*b6 + 3*b5 - 2*b3 - 2*b2 + 2*b1 - 21) * q^11 + (-3*b6 - 2*b5 + 3*b4 - 3*b3 - 2*b2 - 14*b1 + 10) * q^12 + (-4*b8 - 4*b7 + 4*b6 - 3*b5 + 4*b4 - 2*b3 - b2 - 9*b1 - 10) * q^14 + (b8 + b7 + 9*b6 + b5 + 4*b4 - 3*b3 - b2 + 3*b1 - 20) * q^15 + (2*b8 + 7*b6 + b5 - 5*b4 - 8*b3 - 4*b2 - 7*b1 + 37) * q^16 + (-7*b8 + 5*b7 - 7*b6 + b5 + 5*b4 - 2*b3 + 2*b2 - 8*b1 - 7) * q^17 + (5*b8 - 2*b7 + 9*b6 + b5 - 5*b4 - 6*b3 + 2*b2 + 3*b1 - 5) * q^18 + (10*b8 + b7 - b6 + 3*b5 + b2 - 4*b1 - 14) * q^19 + (-9*b8 - 6*b7 - 5*b6 - 3*b5 + 10*b4 - 6*b3 + 8*b2 - 15*b1 - 36) * q^20 + (2*b8 + 3*b7 - b6 + 10*b5 - 5*b4 + 14*b3 + 2*b2 + 4*b1 - 24) * q^21 + (3*b8 - 4*b6 - 3*b5 + 8*b4 + 3*b3 + 10*b2 - 15*b1 + 43) * q^22 + (-3*b8 + 2*b6 + 14*b5 + b4 - 6*b2 + 8*b1 - 22) * q^23 + (6*b8 + 6*b7 - 7*b6 - 9*b5 + b4 + 5*b3 + 11*b2 - 3*b1 - 94) * q^24 + (3*b8 - 7*b7 + b6 + 3*b5 - 4*b4 + 3*b3 + 9*b2 + 23*b1 + 25) * q^25 + (19*b8 + 6*b6 - 6*b5 - 4*b4 - b3 + 6*b2 + 29*b1 - 83) * q^27 + (-5*b8 + 8*b7 + 12*b5 - 25*b4 + 10*b3 + 4*b2 + b1 - 46) * q^28 + (-8*b8 + 6*b6 - 5*b5 - 6*b4 + 3*b3 + 3*b2 + 29*b1 + 16) * q^29 + (17*b8 + 8*b7 + 11*b6 + 6*b5 - 16*b4 + 34*b3 - 9*b2 + 22*b1 + 50) * q^30 + (-4*b8 - 8*b7 + 12*b6 - b5 + 2*b4 - b3 + 7*b2 + 21*b1 - 82) * q^31 + (-12*b8 - 10*b7 + b6 - 18*b5 + 7*b4 + 10*b3 + 17*b2 + 18*b1 - 89) * q^32 + (5*b8 - 5*b7 + 3*b6 - 2*b5 - 4*b4 - 22*b3 - 23*b2 + 24*b1 - 59) * q^33 + (-9*b8 + 10*b7 - 29*b6 - 6*b5 + 25*b4 - 23*b3 + 10*b2 - 37*b1 - 19) * q^34 + (8*b8 + b7 - 17*b6 - 4*b5 - b4 - 12*b3 - 12*b2 + 40*b1 - 20) * q^35 + (-3*b8 - 2*b7 + 12*b6 - 12*b5 + 2*b4 + 16*b3 + 12*b2 + 31*b1 - 115) * q^36 + (-12*b8 - 11*b7 + 13*b6 + 4*b5 + 9*b4 + 14*b3 - 16*b2 - 28*b1 - 12) * q^37 + (-18*b8 - 21*b6 - 8*b5 + 20*b4 - 2*b3 - 7*b2 - 4*b1 - 90) * q^38 + (-15*b8 + 4*b7 - 35*b6 + 16*b5 - 22*b4 - 24*b3 - 9*b2 - 18*b1 - 8) * q^40 + (7*b8 + 2*b7 - 12*b6 - 21*b5 - 2*b4 + 6*b3 + 3*b2 + 14*b1 - 150) * q^41 + (-2*b8 - 10*b7 + 19*b6 - 5*b5 + 25*b4 - 26*b3 - 2*b2 - 6*b1 + 20) * q^42 + (-11*b8 - 5*b7 - 5*b6 + 21*b5 - 25*b4 - 2*b3 + 3*b2 + 14*b1 - 70) * q^43 + (-15*b8 + 16*b7 - 29*b6 - 17*b5 - 11*b4 - 15*b3 - 7*b2 - 63) * q^44 + (-11*b8 + 10*b7 - 20*b6 + 8*b5 + 13*b4 - 14*b3 - 42*b2 - 42*b1 + 26) * q^45 + (27*b8 + 2*b7 + 12*b6 - 4*b5 + 5*b4 + 26*b3 + 20*b2 + 39*b1 + 82) * q^46 + (5*b8 + 4*b7 + 10*b6 + 7*b5 - 25*b4 + b3 - 23*b2 - 25*b1 - 92) * q^47 + (-37*b8 + 2*b7 - 3*b6 + 23*b5 + 10*b4 - 33*b3 - 9*b2 - 42*b1 + 6) * q^48 + (6*b8 + 7*b7 - 19*b6 - 27*b5 + 39*b4 - 41*b3 - 5*b2 - 25*b1 + 55) * q^49 + (5*b8 - 8*b7 + 11*b6 + 22*b5 - 26*b4 - 2*b3 - 9*b2 + 53*b1 + 113) * q^50 + (30*b8 + 8*b7 + 40*b6 - 24*b5 - 8*b4 + 4*b3 - 11*b2 + 44*b1 + 49) * q^51 + (-43*b8 + 3*b7 + 13*b6 - 6*b5 + 36*b4 - 12*b3 - 12*b2 - 18*b1 - 18) * q^53 + (-5*b8 - 8*b7 + 18*b6 + 26*b5 + 10*b4 + 8*b3 - 32*b2 - 49*b1 + 325) * q^54 + (37*b8 + 7*b7 + 7*b6 + 19*b5 - 38*b4 + 83*b3 + 19*b2 + 37*b1 - 50) * q^55 + (-18*b8 - 18*b7 - b6 - 35*b5 + 55*b4 - 4*b3 + 36*b2 - 4*b1 + 30) * q^56 + (18*b8 - b7 - b6 + 21*b5 - 4*b4 + 40*b3 - 43*b2 - 34*b1 + 113) * q^57 + (28*b8 - 12*b7 + 60*b6 + 25*b5 - 16*b4 + 4*b3 - 3*b2 + 25*b1 + 346) * q^58 + (-34*b8 + 22*b7 - 22*b6 - 13*b5 + 8*b4 + b3 + 30*b2 - 49*b1 - 158) * q^59 + (20*b8 - 40*b7 + 51*b6 + 2*b5 + 7*b4 - 4*b3 - 23*b2 + 85*b1 + 354) * q^60 + (18*b8 - 13*b7 + 51*b6 - 18*b5 + 17*b4 + 52*b3 + 16*b2 + 6*b1 + 4) * q^61 + (34*b8 + 4*b7 + 32*b6 + 33*b5 - 72*b4 + 38*b3 - 17*b2 - 27*b1 + 248) * q^62 + (17*b8 + 13*b7 - 15*b6 - 33*b5 + 34*b4 - 15*b3 + 27*b2 + 27*b1 + 86) * q^63 + (15*b8 + 14*b7 - 27*b6 + 62*b5 - 66*b4 + 22*b3 - 9*b2 - 71*b1 + 49) * q^64 + (43*b8 - 8*b7 + 33*b6 + b5 + 2*b4 + 79*b3 + 39*b2 - 7*b1 + 381) * q^66 + (22*b8 - 31*b7 - 9*b6 - 3*b5 + 37*b4 - 57*b3 + 28*b2 + 45*b1 + 102) * q^67 + (-4*b8 + 10*b7 - 115*b6 - 22*b5 + 26*b4 - 51*b3 + 4*b2 - 76*b1 - 185) * q^68 + (27*b8 - 6*b7 - 10*b6 - 57*b5 + 15*b4 - 101*b3 - 39*b2 - 15*b1 - 212) * q^69 + (24*b8 - 2*b7 + 3*b6 + 29*b5 + 57*b4 - 20*b3 + 30*b2 - 26*b1 + 576) * q^70 + (-26*b8 + 23*b7 - 27*b6 + 17*b5 - 27*b4 - 11*b3 + 71*b2 + 11*b1 - 264) * q^71 + (4*b8 + 20*b7 + 4*b6 + 57*b5 - 29*b4 + 30*b3 - 67*b2 - 117*b1 + 543) * q^72 + (2*b8 + 15*b7 + 19*b6 - 39*b5 - 45*b4 + 5*b3 + 50*b2 + 85*b1 - 52) * q^73 + (72*b8 + 18*b7 + 57*b6 + 11*b5 - 121*b4 + 42*b3 - 4*b2 + 44*b1 - 232) * q^74 + (-34*b8 - 36*b7 - 64*b6 + 11*b5 - 18*b4 - 3*b3 + 36*b2 + 15*b1 + 226) * q^75 + (-35*b8 + 32*b7 - 22*b6 + 18*b5 - 22*b4 - 60*b3 + 12*b2 - 120*b1 + 434) * q^76 + (-76*b8 - 15*b7 + 65*b6 + 22*b5 + 31*b4 - 50*b3 + 24*b2 - 40*b1 - 32) * q^77 + (38*b8 + 21*b7 - 5*b6 + 9*b5 + 3*b4 + 77*b3 - 3*b2 - 5*b1 + 44) * q^79 + (-20*b8 + 4*b7 - 57*b6 - 82*b5 + 79*b4 + 12*b3 + 57*b2 - 17*b1 + 56) * q^80 + (-81*b8 + 32*b7 - 72*b6 + 74*b5 + 27*b4 - 8*b3 - 85*b2 - 124*b1 - 62) * q^81 + (-11*b8 - 4*b7 + 12*b6 + 35*b5 + 18*b4 - 74*b3 - 23*b2 - 237*b1 + 438) * q^82 + (-10*b8 + 13*b7 - 25*b6 - 16*b5 - 10*b4 + 85*b3 + 88*b2 - 21*b1 - 444) * q^83 + (37*b8 + 26*b7 - 26*b6 - 47*b5 - 79*b4 + 4*b3 - 35*b2 + 84*b1 + 100) * q^84 + (-24*b8 - 38*b7 + 88*b6 + 43*b5 - 90*b4 + 65*b3 + 21*b2 + 167*b1 + 182) * q^85 + (-33*b8 - 50*b7 + 9*b6 - 54*b5 + 47*b4 + 12*b3 + 61*b2 - 67*b1 + 14) * q^86 + (-88*b8 + 11*b7 - 71*b6 - 18*b5 - 3*b4 - 38*b3 + 36*b2 - 30*b1 - 132) * q^87 + (-77*b8 - 22*b7 + b6 - 12*b5 + 79*b4 - 107*b3 - 20*b2 - 127*b1 + 3) * q^88 + (13*b8 - 29*b7 - 17*b6 - 46*b5 - 18*b4 - 44*b3 - 9*b2 + 42*b1 - 210) * q^89 + (35*b8 + 26*b7 - 36*b6 - 48*b5 + 75*b4 + 8*b3 + 82*b2 - 5*b1 - 158) * q^90 + (42*b8 + 10*b7 + 31*b6 - 35*b5 - 35*b4 - 42*b3 - 46*b2 + 96*b1 + 418) * q^92 + (-82*b8 - 27*b7 - 67*b6 - 18*b5 + 35*b4 - 68*b3 - 96*b2 - 86*b1 + 150) * q^93 + (-27*b8 - 50*b7 + 56*b6 - 85*b5 + 61*b4 + 54*b3 + 39*b2 - 92*b1 - 226) * q^94 + (37*b8 - 55*b7 + 37*b6 + 14*b5 - 16*b4 - 4*b3 + 12*b2 + 140*b1 - 206) * q^95 + (-53*b8 - 28*b7 - 36*b6 - 8*b5 + 7*b4 + 47*b3 + 7*b2 + 38*b1 + 298) * q^96 + (8*b8 + 6*b7 - 20*b6 - 17*b5 + 87*b4 + 68*b3 + 54*b2 - 96*b1 + 157) * q^97 + (-10*b8 + 78*b7 - 153*b6 + 32*b5 + 15*b4 - 4*b3 - 7*b2 - 26*b1 - 33) * q^98 + (-100*b8 - 14*b7 + 14*b6 + 18*b5 + 20*b4 - 70*b3 - 38*b2 - 50*b1 - 121) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10})$$ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 $$9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100})$$ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 6901 \nu^{8} - 12396 \nu^{7} + 501446 \nu^{6} + 728955 \nu^{5} - 11530440 \nu^{4} - 12738943 \nu^{3} + 91159455 \nu^{2} + 52058000 \nu - 149293824 ) / 8071424$$ (-6901*v^8 - 12396*v^7 + 501446*v^6 + 728955*v^5 - 11530440*v^4 - 12738943*v^3 + 91159455*v^2 + 52058000*v - 149293824) / 8071424 $$\beta_{3}$$ $$=$$ $$( - 3843 \nu^{8} + 11372 \nu^{7} + 195178 \nu^{6} - 384275 \nu^{5} - 3297016 \nu^{4} + 4265751 \nu^{3} + 16793257 \nu^{2} - 22791952 \nu - 6050816 ) / 4035712$$ (-3843*v^8 + 11372*v^7 + 195178*v^6 - 384275*v^5 - 3297016*v^4 + 4265751*v^3 + 16793257*v^2 - 22791952*v - 6050816) / 4035712 $$\beta_{4}$$ $$=$$ $$( - 595 \nu^{8} + 2737 \nu^{7} + 19422 \nu^{6} - 72705 \nu^{5} - 171803 \nu^{4} + 485407 \nu^{3} + 252488 \nu^{2} - 1604391 \nu - 646464 ) / 504464$$ (-595*v^8 + 2737*v^7 + 19422*v^6 - 72705*v^5 - 171803*v^4 + 485407*v^3 + 252488*v^2 - 1604391*v - 646464) / 504464 $$\beta_{5}$$ $$=$$ $$( - 23843 \nu^{8} + 103372 \nu^{7} + 1059978 \nu^{6} - 3803155 \nu^{5} - 14845688 \nu^{4} + 38793527 \nu^{3} + 67400873 \nu^{2} - 97179408 \nu - 92826880 ) / 8071424$$ (-23843*v^8 + 103372*v^7 + 1059978*v^6 - 3803155*v^5 - 14845688*v^4 + 38793527*v^3 + 67400873*v^2 - 97179408*v - 92826880) / 8071424 $$\beta_{6}$$ $$=$$ $$( - 32107 \nu^{8} + 90940 \nu^{7} + 1548474 \nu^{6} - 2570427 \nu^{5} - 23878504 \nu^{4} + 12527327 \nu^{3} + 111900273 \nu^{2} + 12391680 \nu - 122506496 ) / 8071424$$ (-32107*v^8 + 90940*v^7 + 1548474*v^6 - 2570427*v^5 - 23878504*v^4 + 12527327*v^3 + 111900273*v^2 + 12391680*v - 122506496) / 8071424 $$\beta_{7}$$ $$=$$ $$( - 53515 \nu^{8} + 88524 \nu^{7} + 2837370 \nu^{6} - 2210427 \nu^{5} - 46218904 \nu^{4} + 3078399 \nu^{3} + 217315329 \nu^{2} + 51740944 \nu - 202361088 ) / 8071424$$ (-53515*v^8 + 88524*v^7 + 2837370*v^6 - 2210427*v^5 - 46218904*v^4 + 3078399*v^3 + 217315329*v^2 + 51740944*v - 202361088) / 8071424 $$\beta_{8}$$ $$=$$ $$( 27975 \nu^{8} - 97156 \nu^{7} - 1304226 \nu^{6} + 3186791 \nu^{5} + 19362096 \nu^{4} - 25660427 \nu^{3} - 85614861 \nu^{2} + 34322440 \nu + 59238144 ) / 4035712$$ (27975*v^8 - 97156*v^7 - 1304226*v^6 + 3186791*v^5 + 19362096*v^4 - 25660427*v^3 - 85614861*v^2 + 34322440*v + 59238144) / 4035712
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{6} + \beta_{5} + 2\beta _1 + 12$$ b8 + b6 + b5 + 2*b1 + 12 $$\nu^{3}$$ $$=$$ $$3\beta_{8} + 3\beta_{6} + 2\beta_{5} + 4\beta_{3} - \beta_{2} + 23\beta _1 + 12$$ 3*b8 + 3*b6 + 2*b5 + 4*b3 - b2 + 23*b1 + 12 $$\nu^{4}$$ $$=$$ $$32\beta_{8} + 37\beta_{6} + 27\beta_{5} - 5\beta_{4} + 8\beta_{3} - 8\beta_{2} + 77\beta _1 + 260$$ 32*b8 + 37*b6 + 27*b5 - 5*b4 + 8*b3 - 8*b2 + 77*b1 + 260 $$\nu^{5}$$ $$=$$ $$128 \beta_{8} - 10 \beta_{7} + 166 \beta_{6} + 75 \beta_{5} - 18 \beta_{4} + 138 \beta_{3} - 45 \beta_{2} + 636 \beta _1 + 604$$ 128*b8 - 10*b7 + 166*b6 + 75*b5 - 18*b4 + 138*b3 - 45*b2 + 636*b1 + 604 $$\nu^{6}$$ $$=$$ $$1004 \beta_{8} - 46 \beta_{7} + 1315 \beta_{6} + 748 \beta_{5} - 299 \beta_{4} + 490 \beta_{3} - 339 \beta_{2} + 2746 \beta _1 + 6752$$ 1004*b8 - 46*b7 + 1315*b6 + 748*b5 - 299*b4 + 490*b3 - 339*b2 + 2746*b1 + 6752 $$\nu^{7}$$ $$=$$ $$4677 \beta_{8} - 644 \beta_{7} + 6896 \beta_{6} + 2684 \beta_{5} - 1299 \beta_{4} + 4588 \beta_{3} - 1762 \beta_{2} + 19224 \beta _1 + 23360$$ 4677*b8 - 644*b7 + 6896*b6 + 2684*b5 - 1299*b4 + 4588*b3 - 1762*b2 + 19224*b1 + 23360 $$\nu^{8}$$ $$=$$ $$32278 \beta_{8} - 3242 \beta_{7} + 46550 \beta_{6} + 21858 \beta_{5} - 12940 \beta_{4} + 21190 \beta_{3} - 12178 \beta_{2} + 95033 \beta _1 + 192772$$ 32278*b8 - 3242*b7 + 46550*b6 + 21858*b5 - 12940*b4 + 21190*b3 - 12178*b2 + 95033*b1 + 192772

## 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
 −4.42835 −3.82555 −2.16135 −1.22799 0.850942 1.39012 2.72763 4.83438 5.84018
−5.42835 1.67510 21.4670 −7.70909 −9.09301 15.0250 −73.1038 −24.1941 41.8477
1.2 −4.82555 4.44352 15.2860 12.7712 −21.4425 −26.1871 −35.1589 −7.25513 −61.6281
1.3 −3.16135 7.08883 1.99415 −13.6039 −22.4103 14.3315 18.9866 23.2516 43.0068
1.4 −2.22799 −9.74867 −3.03607 8.20685 21.7199 −8.35495 24.5882 68.0366 −18.2848
1.5 −0.149058 −6.48858 −7.97778 10.2526 0.967177 29.6743 2.38162 15.1017 −1.52823
1.6 0.390115 3.60967 −7.84781 7.52136 1.40819 −19.5446 −6.18247 −13.9703 2.93420
1.7 1.72763 6.89591 −5.01528 −20.8281 11.9136 −7.56566 −22.4856 20.5536 −35.9833
1.8 3.83438 −0.279163 6.70249 −11.3710 −1.07042 −31.0623 −4.97517 −26.9221 −43.6008
1.9 4.84018 −6.19662 15.4273 −15.2399 −29.9927 −4.31620 35.9495 11.3981 −73.7636
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 169.4.a.k 9
3.b odd 2 1 1521.4.a.bh 9
13.b even 2 1 169.4.a.l yes 9
13.c even 3 2 169.4.c.l 18
13.d odd 4 2 169.4.b.g 18
13.e even 6 2 169.4.c.k 18
13.f odd 12 4 169.4.e.h 36
39.d odd 2 1 1521.4.a.bg 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k 9 1.a even 1 1 trivial
169.4.a.l yes 9 13.b even 2 1
169.4.b.g 18 13.d odd 4 2
169.4.c.k 18 13.e even 6 2
169.4.c.l 18 13.c even 3 2
169.4.e.h 36 13.f odd 12 4
1521.4.a.bg 9 39.d odd 2 1
1521.4.a.bh 9 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{9} + 5T_{2}^{8} - 42T_{2}^{7} - 205T_{2}^{6} + 486T_{2}^{5} + 2310T_{2}^{4} - 1257T_{2}^{3} - 5898T_{2}^{2} + 1464T_{2} + 344$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} + 5 T^{8} - 42 T^{7} - 205 T^{6} + \cdots + 344$$
$3$ $$T^{9} - T^{8} - 154 T^{7} + \cdots - 143717$$
$5$ $$T^{9} + 30 T^{8} + \cdots + 3059376152$$
$7$ $$T^{9} + 38 T^{8} + \cdots - 27715644424$$
$11$ $$T^{9} + 181 T^{8} + \cdots + 276199564381$$
$13$ $$T^{9}$$
$17$ $$T^{9} + 55 T^{8} + \cdots + 5572934105557$$
$19$ $$T^{9} + \cdots + 865058822963419$$
$23$ $$T^{9} + 204 T^{8} + \cdots + 68\!\cdots\!92$$
$29$ $$T^{9} - 280 T^{8} + \cdots + 56\!\cdots\!96$$
$31$ $$T^{9} + 706 T^{8} + \cdots - 30\!\cdots\!56$$
$37$ $$T^{9} + 298 T^{8} + \cdots - 31\!\cdots\!48$$
$41$ $$T^{9} + 1201 T^{8} + \cdots + 61\!\cdots\!53$$
$43$ $$T^{9} + 533 T^{8} + \cdots - 35\!\cdots\!77$$
$47$ $$T^{9} + 956 T^{8} + \cdots + 11\!\cdots\!84$$
$53$ $$T^{9} + 278 T^{8} + \cdots - 12\!\cdots\!76$$
$59$ $$T^{9} + 1377 T^{8} + \cdots + 27\!\cdots\!23$$
$61$ $$T^{9} + 136 T^{8} + \cdots - 27\!\cdots\!32$$
$67$ $$T^{9} - 931 T^{8} + \cdots + 32\!\cdots\!99$$
$71$ $$T^{9} + 2046 T^{8} + \cdots + 44\!\cdots\!72$$
$73$ $$T^{9} - 45 T^{8} + \cdots - 31\!\cdots\!21$$
$79$ $$T^{9} - 412 T^{8} + \cdots + 63\!\cdots\!88$$
$83$ $$T^{9} + 3709 T^{8} + \cdots + 14\!\cdots\!61$$
$89$ $$T^{9} + 1663 T^{8} + \cdots + 43\!\cdots\!23$$
$97$ $$T^{9} - 1087 T^{8} + \cdots + 20\!\cdots\!89$$