# Properties

 Label 289.4.a.d Level $289$ Weight $4$ Character orbit 289.a Self dual yes Analytic conductor $17.052$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2555057.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 21x^{2} - 4x + 36$$ x^4 - x^3 - 21*x^2 - 4*x + 36 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{4} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 11) q^{6} + (5 \beta_{2} - \beta_1 - 10) q^{7} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 + 15) q^{8} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 5) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - b1 + 1) * q^3 + (b3 - b2 + b1 + 3) * q^4 + (-2*b3 - b2 + b1 - 4) * q^5 + (-b3 + 3*b2 + 2*b1 - 11) * q^6 + (5*b2 - b1 - 10) * q^7 + (3*b3 + b2 + 2*b1 + 15) * q^8 + (2*b3 - 3*b2 - 9*b1 + 5) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{4} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 11) q^{6} + (5 \beta_{2} - \beta_1 - 10) q^{7} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 + 15) q^{8} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 5) q^{9} + ( - 3 \beta_{3} + \beta_{2} - 9 \beta_1 + 3) q^{10} + (4 \beta_{3} - 2 \beta_{2} - 1) q^{11} + ( - 11 \beta_1 + 10) q^{12} + ( - 2 \beta_{3} + 5 \beta_{2} - 9 \beta_1 - 5) q^{13} + ( - \beta_{3} - 9 \beta_{2} - 21 \beta_1 - 11) q^{14} + (10 \beta_{2} + 8 \beta_1 + 9) q^{15} + (4 \beta_{2} + 19 \beta_1 + 10) q^{16} + ( - 5 \beta_{3} + 15 \beta_{2} + 10 \beta_1 - 91) q^{18} + (4 \beta_{3} + 11 \beta_{2} - 13 \beta_1 + 7) q^{19} + (\beta_{3} + 15 \beta_{2} - 28 \beta_1 - 79) q^{20} + ( - 4 \beta_{3} + 2 \beta_{2} + 38 \beta_1 - 99) q^{21} + (8 \beta_{3} + 4 \beta_{2} + 19 \beta_1 + 16) q^{22} + (4 \beta_{3} - \beta_{2} + 25 \beta_1 - 100) q^{23} + ( - 3 \beta_{3} - 13 \beta_{2} - 17 \beta_1 - 33) q^{24} + ( - 18 \beta_{3} + 7 \beta_{2} + 25 \beta_1 + 82) q^{25} + ( - 13 \beta_{3} - \beta_{2} - 32 \beta_1 - 107) q^{26} + (12 \beta_{3} - 4 \beta_{2} - 22 \beta_1 + 133) q^{27} + ( - 23 \beta_{3} - \beta_{2} - 10 \beta_1 - 155) q^{28} + ( - 6 \beta_{3} + \beta_{2} + 11 \beta_1 + 15) q^{29} + (8 \beta_{3} - 28 \beta_{2} - 3 \beta_1 + 88) q^{30} + (8 \beta_{3} + 16 \beta_{2} + 2 \beta_1 - 93) q^{31} + ( - 5 \beta_{3} - 35 \beta_{2} + 5 \beta_1 + 89) q^{32} + (2 \beta_{3} - \beta_{2} - 27 \beta_1 + 31) q^{33} + (28 \beta_{3} - 51 \beta_{2} + 9 \beta_1 - 43) q^{35} + ( - 16 \beta_{3} - 16 \beta_{2} - 59 \beta_1 + 50) q^{36} + ( - 4 \beta_{3} - 46 \beta_{2} - 38 \beta_1 - 108) q^{37} + ( - 5 \beta_{3} - 9 \beta_{2} - 12 \beta_1 - 127) q^{38} + (4 \beta_{3} - 25 \beta_{2} + 25 \beta_1 - 2) q^{39} + ( - 2 \beta_{3} - 10 \beta_{2} - 61 \beta_1 - 328) q^{40} + (12 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 210) q^{41} + (30 \beta_{3} - 42 \beta_{2} - 81 \beta_1 + 402) q^{42} + (24 \beta_{3} + 32 \beta_{2} + 50 \beta_1 - 43) q^{43} + (3 \beta_{3} - 11 \beta_{2} + 59 \beta_1 + 249) q^{44} + (36 \beta_{3} + 32 \beta_{2} + 40 \beta_1 - 171) q^{45} + (33 \beta_{3} - 23 \beta_{2} - 57 \beta_1 + 291) q^{46} + ( - 4 \beta_{3} + 42 \beta_{2} + 32 \beta_1 - 17) q^{47} + ( - 23 \beta_{3} + 43 \beta_{2} + 52 \beta_1 - 279) q^{48} + (26 \beta_{3} + 19 \beta_{2} - 59 \beta_1 + 268) q^{49} + ( - 11 \beta_{3} - 39 \beta_{2} + 21 \beta_1 + 203) q^{50} + ( - 42 \beta_{3} - 6 \beta_{2} - 117 \beta_1 - 364) q^{52} + ( - 2 \beta_{3} - 65 \beta_{2} + 93 \beta_1 + 5) q^{53} + (2 \beta_{3} + 30 \beta_{2} + 167 \beta_1 - 194) q^{54} + (44 \beta_{3} + 45 \beta_{2} - 85 \beta_1 - 308) q^{55} + ( - 48 \beta_{3} + 84 \beta_{2} - 87 \beta_1 - 114) q^{56} + (2 \beta_{3} - 61 \beta_{2} + 17 \beta_1 - 78) q^{57} + ( - \beta_{3} - 13 \beta_{2} + 97) q^{58} + (12 \beta_{3} + 64 \beta_{2} + 56 \beta_1 + 222) q^{59} + (13 \beta_{3} - 21 \beta_{2} + 109 \beta_1 - 73) q^{60} + ( - 60 \beta_{3} + 46 \beta_{2} + 2 \beta_1 - 139) q^{61} + (18 \beta_{3} - 34 \beta_{2} - 91 \beta_1 + 54) q^{62} + ( - 40 \beta_{3} + 80 \beta_{2} + 230 \beta_1 - 279) q^{63} + ( - 5 \beta_{3} + 33 \beta_{2} - 8 \beta_1 - 45) q^{64} + (20 \beta_{3} - 74 \beta_{2} + 126 \beta_1 + 85) q^{65} + ( - 23 \beta_{3} + 29 \beta_{2} + 14 \beta_1 - 289) q^{66} + (16 \beta_{3} + 10 \beta_{2} - 158 \beta_1 - 176) q^{67} + ( - 24 \beta_{3} + 172 \beta_{2} + 128 \beta_1 - 363) q^{69} + (65 \beta_{3} + 93 \beta_{2} + 180 \beta_1 + 211) q^{70} + (4 \beta_{3} + 44 \beta_{2} + 36 \beta_1 - 298) q^{71} + ( - 51 \beta_{3} - 29 \beta_{2} - 121 \beta_1 + 15) q^{72} + (16 \beta_{3} - 62 \beta_{2} - 130 \beta_1 - 83) q^{73} + ( - 46 \beta_{3} + 130 \beta_{2} - 70 \beta_1 - 434) q^{74} + ( - 32 \beta_{3} + 4 \beta_{2} + 82 \beta_1 - 297) q^{75} + ( - 54 \beta_{3} - 58 \beta_{2} - 37 \beta_1 - 208) q^{76} + ( - 78 \beta_{3} - 9 \beta_{2} + 21 \beta_1 - 246) q^{77} + (33 \beta_{3} + 25 \beta_{2} + 89 \beta_1 + 291) q^{78} + ( - 76 \beta_{3} - 52 \beta_{2} + 212 \beta_1 - 294) q^{79} + ( - 73 \beta_{3} - 39 \beta_{2} - 153 \beta_1 - 47) q^{80} + ( - 28 \beta_{3} - 126 \beta_{2} - 6 \beta_1 + 296) q^{81} + (32 \beta_{3} - 16 \beta_{2} - 162 \beta_1 + 136) q^{82} + ( - 84 \beta_{3} - 3 \beta_{2} + 37 \beta_1 + 183) q^{83} + (11 \beta_{3} + 149 \beta_{2} + 221 \beta_1 + 21) q^{84} + (98 \beta_{3} - 114 \beta_{2} + 39 \beta_1 + 646) q^{86} + ( - 12 \beta_{3} + 23 \beta_{2} + 37 \beta_1 - 114) q^{87} + (\beta_{3} - 69 \beta_{2} + 190 \beta_1 + 533) q^{88} + ( - 44 \beta_{3} - 112 \beta_{2} - 20 \beta_1 + 218) q^{89} + (112 \beta_{3} - 104 \beta_{2} - 51 \beta_1 + 584) q^{90} + (68 \beta_{3} + 106 \beta_{2} + 112 \beta_1 + 677) q^{91} + ( - 23 \beta_{3} + 111 \beta_{2} + 212 \beta_1 + 305) q^{92} + ( - 18 \beta_{3} + 75 \beta_{2} + 161 \beta_1 - 451) q^{93} + (24 \beta_{3} - 116 \beta_{2} - 85 \beta_1 + 336) q^{94} + (80 \beta_{3} - 132 \beta_{2} + 66 \beta_1 - 587) q^{95} + (30 \beta_{3} - 34 \beta_{2} - 269 \beta_1 + 744) q^{96} + (162 \beta_{3} - 149 \beta_{2} - 67 \beta_1 + 224) q^{97} + ( - 7 \beta_{3} + 21 \beta_{2} + 275 \beta_1 - 545) q^{98} + ( - 80 \beta_{3} - 59 \beta_{2} - 99 \beta_1 + 371) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - b1 + 1) * q^3 + (b3 - b2 + b1 + 3) * q^4 + (-2*b3 - b2 + b1 - 4) * q^5 + (-b3 + 3*b2 + 2*b1 - 11) * q^6 + (5*b2 - b1 - 10) * q^7 + (3*b3 + b2 + 2*b1 + 15) * q^8 + (2*b3 - 3*b2 - 9*b1 + 5) * q^9 + (-3*b3 + b2 - 9*b1 + 3) * q^10 + (4*b3 - 2*b2 - 1) * q^11 + (-11*b1 + 10) * q^12 + (-2*b3 + 5*b2 - 9*b1 - 5) * q^13 + (-b3 - 9*b2 - 21*b1 - 11) * q^14 + (10*b2 + 8*b1 + 9) * q^15 + (4*b2 + 19*b1 + 10) * q^16 + (-5*b3 + 15*b2 + 10*b1 - 91) * q^18 + (4*b3 + 11*b2 - 13*b1 + 7) * q^19 + (b3 + 15*b2 - 28*b1 - 79) * q^20 + (-4*b3 + 2*b2 + 38*b1 - 99) * q^21 + (8*b3 + 4*b2 + 19*b1 + 16) * q^22 + (4*b3 - b2 + 25*b1 - 100) * q^23 + (-3*b3 - 13*b2 - 17*b1 - 33) * q^24 + (-18*b3 + 7*b2 + 25*b1 + 82) * q^25 + (-13*b3 - b2 - 32*b1 - 107) * q^26 + (12*b3 - 4*b2 - 22*b1 + 133) * q^27 + (-23*b3 - b2 - 10*b1 - 155) * q^28 + (-6*b3 + b2 + 11*b1 + 15) * q^29 + (8*b3 - 28*b2 - 3*b1 + 88) * q^30 + (8*b3 + 16*b2 + 2*b1 - 93) * q^31 + (-5*b3 - 35*b2 + 5*b1 + 89) * q^32 + (2*b3 - b2 - 27*b1 + 31) * q^33 + (28*b3 - 51*b2 + 9*b1 - 43) * q^35 + (-16*b3 - 16*b2 - 59*b1 + 50) * q^36 + (-4*b3 - 46*b2 - 38*b1 - 108) * q^37 + (-5*b3 - 9*b2 - 12*b1 - 127) * q^38 + (4*b3 - 25*b2 + 25*b1 - 2) * q^39 + (-2*b3 - 10*b2 - 61*b1 - 328) * q^40 + (12*b3 + 4*b2 + 8*b1 - 210) * q^41 + (30*b3 - 42*b2 - 81*b1 + 402) * q^42 + (24*b3 + 32*b2 + 50*b1 - 43) * q^43 + (3*b3 - 11*b2 + 59*b1 + 249) * q^44 + (36*b3 + 32*b2 + 40*b1 - 171) * q^45 + (33*b3 - 23*b2 - 57*b1 + 291) * q^46 + (-4*b3 + 42*b2 + 32*b1 - 17) * q^47 + (-23*b3 + 43*b2 + 52*b1 - 279) * q^48 + (26*b3 + 19*b2 - 59*b1 + 268) * q^49 + (-11*b3 - 39*b2 + 21*b1 + 203) * q^50 + (-42*b3 - 6*b2 - 117*b1 - 364) * q^52 + (-2*b3 - 65*b2 + 93*b1 + 5) * q^53 + (2*b3 + 30*b2 + 167*b1 - 194) * q^54 + (44*b3 + 45*b2 - 85*b1 - 308) * q^55 + (-48*b3 + 84*b2 - 87*b1 - 114) * q^56 + (2*b3 - 61*b2 + 17*b1 - 78) * q^57 + (-b3 - 13*b2 + 97) * q^58 + (12*b3 + 64*b2 + 56*b1 + 222) * q^59 + (13*b3 - 21*b2 + 109*b1 - 73) * q^60 + (-60*b3 + 46*b2 + 2*b1 - 139) * q^61 + (18*b3 - 34*b2 - 91*b1 + 54) * q^62 + (-40*b3 + 80*b2 + 230*b1 - 279) * q^63 + (-5*b3 + 33*b2 - 8*b1 - 45) * q^64 + (20*b3 - 74*b2 + 126*b1 + 85) * q^65 + (-23*b3 + 29*b2 + 14*b1 - 289) * q^66 + (16*b3 + 10*b2 - 158*b1 - 176) * q^67 + (-24*b3 + 172*b2 + 128*b1 - 363) * q^69 + (65*b3 + 93*b2 + 180*b1 + 211) * q^70 + (4*b3 + 44*b2 + 36*b1 - 298) * q^71 + (-51*b3 - 29*b2 - 121*b1 + 15) * q^72 + (16*b3 - 62*b2 - 130*b1 - 83) * q^73 + (-46*b3 + 130*b2 - 70*b1 - 434) * q^74 + (-32*b3 + 4*b2 + 82*b1 - 297) * q^75 + (-54*b3 - 58*b2 - 37*b1 - 208) * q^76 + (-78*b3 - 9*b2 + 21*b1 - 246) * q^77 + (33*b3 + 25*b2 + 89*b1 + 291) * q^78 + (-76*b3 - 52*b2 + 212*b1 - 294) * q^79 + (-73*b3 - 39*b2 - 153*b1 - 47) * q^80 + (-28*b3 - 126*b2 - 6*b1 + 296) * q^81 + (32*b3 - 16*b2 - 162*b1 + 136) * q^82 + (-84*b3 - 3*b2 + 37*b1 + 183) * q^83 + (11*b3 + 149*b2 + 221*b1 + 21) * q^84 + (98*b3 - 114*b2 + 39*b1 + 646) * q^86 + (-12*b3 + 23*b2 + 37*b1 - 114) * q^87 + (b3 - 69*b2 + 190*b1 + 533) * q^88 + (-44*b3 - 112*b2 - 20*b1 + 218) * q^89 + (112*b3 - 104*b2 - 51*b1 + 584) * q^90 + (68*b3 + 106*b2 + 112*b1 + 677) * q^91 + (-23*b3 + 111*b2 + 212*b1 + 305) * q^92 + (-18*b3 + 75*b2 + 161*b1 - 451) * q^93 + (24*b3 - 116*b2 - 85*b1 + 336) * q^94 + (80*b3 - 132*b2 + 66*b1 - 587) * q^95 + (30*b3 - 34*b2 - 269*b1 + 744) * q^96 + (162*b3 - 149*b2 - 67*b1 + 224) * q^97 + (-7*b3 + 21*b2 + 275*b1 - 545) * q^98 + (-80*b3 - 59*b2 - 99*b1 + 371) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 2 q^{3} + 11 q^{4} - 14 q^{5} - 38 q^{6} - 36 q^{7} + 60 q^{8} + 6 q^{9}+O(q^{10})$$ 4 * q + q^2 + 2 * q^3 + 11 * q^4 - 14 * q^5 - 38 * q^6 - 36 * q^7 + 60 * q^8 + 6 * q^9 $$4 q + q^{2} + 2 q^{3} + 11 q^{4} - 14 q^{5} - 38 q^{6} - 36 q^{7} + 60 q^{8} + 6 q^{9} + 7 q^{10} - 10 q^{11} + 29 q^{12} - 22 q^{13} - 73 q^{14} + 54 q^{15} + 63 q^{16} - 334 q^{18} + 22 q^{19} - 330 q^{20} - 352 q^{21} + 79 q^{22} - 380 q^{23} - 159 q^{24} + 378 q^{25} - 448 q^{26} + 494 q^{27} - 608 q^{28} + 78 q^{29} + 313 q^{30} - 362 q^{31} + 331 q^{32} + 94 q^{33} - 242 q^{35} + 141 q^{36} - 512 q^{37} - 524 q^{38} - 12 q^{39} - 1381 q^{40} - 840 q^{41} + 1455 q^{42} - 114 q^{43} + 1041 q^{44} - 648 q^{45} + 1051 q^{46} + 10 q^{47} - 998 q^{48} + 1006 q^{49} + 805 q^{50} - 1537 q^{52} + 50 q^{53} - 581 q^{54} - 1316 q^{55} - 411 q^{56} - 358 q^{57} + 376 q^{58} + 996 q^{59} - 217 q^{60} - 448 q^{61} + 73 q^{62} - 766 q^{63} - 150 q^{64} + 372 q^{65} - 1090 q^{66} - 868 q^{67} - 1128 q^{69} + 1052 q^{70} - 1116 q^{71} - 39 q^{72} - 540 q^{73} - 1630 q^{74} - 1070 q^{75} - 873 q^{76} - 894 q^{77} + 1245 q^{78} - 940 q^{79} - 307 q^{80} + 1080 q^{81} + 334 q^{82} + 850 q^{83} + 443 q^{84} + 2411 q^{86} - 384 q^{87} + 2252 q^{88} + 784 q^{89} + 2069 q^{90} + 2858 q^{91} + 1566 q^{92} - 1550 q^{93} + 1119 q^{94} - 2494 q^{95} + 2643 q^{96} + 518 q^{97} - 1877 q^{98} + 1406 q^{99}+O(q^{100})$$ 4 * q + q^2 + 2 * q^3 + 11 * q^4 - 14 * q^5 - 38 * q^6 - 36 * q^7 + 60 * q^8 + 6 * q^9 + 7 * q^10 - 10 * q^11 + 29 * q^12 - 22 * q^13 - 73 * q^14 + 54 * q^15 + 63 * q^16 - 334 * q^18 + 22 * q^19 - 330 * q^20 - 352 * q^21 + 79 * q^22 - 380 * q^23 - 159 * q^24 + 378 * q^25 - 448 * q^26 + 494 * q^27 - 608 * q^28 + 78 * q^29 + 313 * q^30 - 362 * q^31 + 331 * q^32 + 94 * q^33 - 242 * q^35 + 141 * q^36 - 512 * q^37 - 524 * q^38 - 12 * q^39 - 1381 * q^40 - 840 * q^41 + 1455 * q^42 - 114 * q^43 + 1041 * q^44 - 648 * q^45 + 1051 * q^46 + 10 * q^47 - 998 * q^48 + 1006 * q^49 + 805 * q^50 - 1537 * q^52 + 50 * q^53 - 581 * q^54 - 1316 * q^55 - 411 * q^56 - 358 * q^57 + 376 * q^58 + 996 * q^59 - 217 * q^60 - 448 * q^61 + 73 * q^62 - 766 * q^63 - 150 * q^64 + 372 * q^65 - 1090 * q^66 - 868 * q^67 - 1128 * q^69 + 1052 * q^70 - 1116 * q^71 - 39 * q^72 - 540 * q^73 - 1630 * q^74 - 1070 * q^75 - 873 * q^76 - 894 * q^77 + 1245 * q^78 - 940 * q^79 - 307 * q^80 + 1080 * q^81 + 334 * q^82 + 850 * q^83 + 443 * q^84 + 2411 * q^86 - 384 * q^87 + 2252 * q^88 + 784 * q^89 + 2069 * q^90 + 2858 * q^91 + 1566 * q^92 - 1550 * q^93 + 1119 * q^94 - 2494 * q^95 + 2643 * q^96 + 518 * q^97 - 1877 * q^98 + 1406 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 21x^{2} - 4x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 3\nu^{2} - 15\nu + 18 ) / 4$$ (v^3 - 3*v^2 - 15*v + 18) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 19\nu - 26 ) / 4$$ (v^3 + v^2 - 19*v - 26) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} + \beta _1 + 11$$ b3 - b2 + b1 + 11 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + \beta_{2} + 18\beta _1 + 15$$ 3*b3 + b2 + 18*b1 + 15

## 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
 −3.68488 −1.58184 1.22501 5.04171
−3.68488 9.05894 5.57832 −7.08909 −33.3811 −28.1854 8.92359 55.0643 26.1224
1.2 −1.58184 −4.98387 −5.49778 −14.4471 7.88368 29.4104 21.3513 −2.16104 22.8530
1.3 1.22501 0.534684 −6.49935 20.9528 0.654993 −15.0235 −17.7618 −26.7141 25.6674
1.4 5.04171 −2.60975 17.4188 −13.4166 −13.1576 −22.2015 47.4869 −20.1892 −67.6428
 $$n$$: e.g. 2-40 or 990-1000 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.d yes 4
17.b even 2 1 289.4.a.c 4
17.c even 4 2 289.4.b.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.c 4 17.b even 2 1
289.4.a.d yes 4 1.a even 1 1 trivial
289.4.b.d 8 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2}^{4} - T_{2}^{3} - 21T_{2}^{2} - 4T_{2} + 36$$ T2^4 - T2^3 - 21*T2^2 - 4*T2 + 36 $$T_{3}^{4} - 2T_{3}^{3} - 55T_{3}^{2} - 88T_{3} + 63$$ T3^4 - 2*T3^3 - 55*T3^2 - 88*T3 + 63

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 21 T^{2} - 4 T + 36$$
$3$ $$T^{4} - 2 T^{3} - 55 T^{2} - 88 T + 63$$
$5$ $$T^{4} + 14 T^{3} - 341 T^{2} + \cdots - 28791$$
$7$ $$T^{4} + 36 T^{3} - 541 T^{2} + \cdots - 276489$$
$11$ $$T^{4} + 10 T^{3} - 1756 T^{2} + \cdots + 316467$$
$13$ $$T^{4} + 22 T^{3} - 3509 T^{2} + \cdots - 26579$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 22 T^{3} - 9191 T^{2} + \cdots + 4371507$$
$23$ $$T^{4} + 380 T^{3} + \cdots - 176752917$$
$29$ $$T^{4} - 78 T^{3} - 2705 T^{2} + \cdots - 107811$$
$31$ $$T^{4} + 362 T^{3} + \cdots - 38411429$$
$37$ $$T^{4} + 512 T^{3} + \cdots - 1758989168$$
$41$ $$T^{4} + 840 T^{3} + \cdots + 1180838736$$
$43$ $$T^{4} + 114 T^{3} + \cdots + 4351627359$$
$47$ $$T^{4} - 10 T^{3} + \cdots - 153320769$$
$53$ $$T^{4} - 50 T^{3} + \cdots + 3826783197$$
$59$ $$T^{4} - 996 T^{3} + \cdots - 7997430672$$
$61$ $$T^{4} + 448 T^{3} + \cdots + 26730644633$$
$67$ $$T^{4} + 868 T^{3} + \cdots + 30297331184$$
$71$ $$T^{4} + 1116 T^{3} + \cdots + 589600944$$
$73$ $$T^{4} + 540 T^{3} + \cdots - 46400632803$$
$79$ $$T^{4} + 940 T^{3} + \cdots + 229066578288$$
$83$ $$T^{4} - 850 T^{3} + \cdots + 1637812071$$
$89$ $$T^{4} - 784 T^{3} + \cdots + 24704450064$$
$97$ $$T^{4} - 518 T^{3} + \cdots + 2200880766749$$