# Properties

 Label 115.4.a.e Level $115$ Weight $4$ Character orbit 115.a Self dual yes Analytic conductor $6.785$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 115.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: $$6.78521965066$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ x^5 - x^4 - 27*x^3 + 7*x^2 + 168*x + 92 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 4) q^{4} - 5 q^{5} + ( - 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 5) q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{7} + (5 \beta_{4} - 3 \beta_{3} - \beta_{2} + 5 \beta_1 + 26) q^{8} + ( - 3 \beta_{4} + 6 \beta_{3} + 16) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^2 + (b3 - b2 + b1 + 1) * q^3 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 - 5 * q^5 + (-2*b4 - 3*b2 + 2*b1 + 5) * q^6 + (b3 + 3*b2 + b1 - 2) * q^7 + (5*b4 - 3*b3 - b2 + 5*b1 + 26) * q^8 + (-3*b4 + 6*b3 + 16) * q^9 $$q + (\beta_1 + 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 4) q^{4} - 5 q^{5} + ( - 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 5) q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{7} + (5 \beta_{4} - 3 \beta_{3} - \beta_{2} + 5 \beta_1 + 26) q^{8} + ( - 3 \beta_{4} + 6 \beta_{3} + 16) q^{9} + ( - 5 \beta_1 - 5) q^{10} + (2 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 1) q^{11} + ( - 10 \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 11) q^{12} + (\beta_{4} - 4 \beta_{2} - 6 \beta_1 + 29) q^{13} + (6 \beta_{4} - 8 \beta_{3} + \beta_{2} - 5 \beta_1 + 18) q^{14} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 5) q^{15} + (13 \beta_{4} - 7 \beta_{3} + 15 \beta_{2} + 19 \beta_1 + 44) q^{16} + ( - 9 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 14 \beta_1 + 10) q^{17} + ( - 15 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 4) q^{18} + ( - 5 \beta_{4} + 5 \beta_{3} - 11 \beta_1 - 29) q^{19} + ( - 5 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 15 \beta_1 - 20) q^{20} + (9 \beta_{4} - 5 \beta_{3} - 9 \beta_{2} - 11 \beta_1 - 8) q^{21} + (18 \beta_{4} - 26 \beta_{3} + 5 \beta_{2} - 10 \beta_1 + 35) q^{22} - 23 q^{23} + ( - 14 \beta_{4} + 26 \beta_{3} - 3 \beta_{2} + 25) q^{24} + 25 q^{25} + ( - 11 \beta_{4} + 11 \beta_{3} + 4 \beta_{2} + 21 \beta_1 - 55) q^{26} + (9 \beta_{4} + 4 \beta_{3} - 7 \beta_{2} - 8 \beta_1 + 118) q^{27} + (23 \beta_{4} - 15 \beta_{3} + 2 \beta_{2} + 15 \beta_1 - 5) q^{28} + ( - 5 \beta_{4} + 9 \beta_{3} - 21 \beta_{2} - 17 \beta_1 + 93) q^{29} + (10 \beta_{4} + 15 \beta_{2} - 10 \beta_1 - 25) q^{30} + (12 \beta_{4} - 8 \beta_{3} - 17 \beta_{2} + 14 \beta_1 + 8) q^{31} + (55 \beta_{4} - 57 \beta_{3} + 37 \beta_{2} + 41 \beta_1 + 100) q^{32} + (27 \beta_{4} + 7 \beta_{3} - 14 \beta_{2} - 45 \beta_1 + 47) q^{33} + ( - 41 \beta_{4} + 49 \beta_{3} - 2 \beta_{2} - 8 \beta_1 - 122) q^{34} + ( - 5 \beta_{3} - 15 \beta_{2} - 5 \beta_1 + 10) q^{35} + ( - 44 \beta_{4} + 14 \beta_{3} - 49 \beta_{2} + 18 \beta_1 - 107) q^{36} + ( - 53 \beta_{4} + 24 \beta_{3} + 11 \beta_{2} + 28 \beta_1 - 7) q^{37} + ( - 31 \beta_{4} + 21 \beta_{3} - 4 \beta_{2} - 61 \beta_1 - 155) q^{38} + (\beta_{4} + 46 \beta_{3} + \beta_{2} + 22 \beta_1 + 62) q^{39} + ( - 25 \beta_{4} + 15 \beta_{3} + 5 \beta_{2} - 25 \beta_1 - 130) q^{40} + ( - 19 \beta_{4} - 9 \beta_{3} - 20 \beta_{2} + 67 \beta_1 - 4) q^{41} + (3 \beta_{4} + 7 \beta_{3} + 25 \beta_{2} - 11 \beta_1 - 168) q^{42} + (4 \beta_{4} - 6 \beta_{3} + 60 \beta_{2} - 10 \beta_1 - 136) q^{43} + (64 \beta_{4} - 60 \beta_{3} + 21 \beta_{2} + 46 \beta_1 - 21) q^{44} + (15 \beta_{4} - 30 \beta_{3} - 80) q^{45} + ( - 23 \beta_1 - 23) q^{46} + ( - 8 \beta_{4} + 33 \beta_{3} + 8 \beta_{2} - 29 \beta_1 + 188) q^{47} + (6 \beta_{4} + 6 \beta_{3} - 49 \beta_{2} - 56 \beta_1 - 125) q^{48} + (5 \beta_{4} - 32 \beta_{3} - 10 \beta_{2} - 38 \beta_1 - 137) q^{49} + (25 \beta_1 + 25) q^{50} + (5 \beta_{4} - 15 \beta_{3} - 30 \beta_{2} + 97 \beta_1 - 207) q^{51} + ( - 23 \beta_{4} - 7 \beta_{3} - 18 \beta_{2} + 9 \beta_1 - 51) q^{52} + (25 \beta_{4} - 12 \beta_{3} + 81 \beta_{2} - 56 \beta_1 + 205) q^{53} + (\beta_{4} - 9 \beta_{3} + 15 \beta_{2} + 101 \beta_1 - 28) q^{54} + ( - 10 \beta_{4} - 20 \beta_{3} - 35 \beta_{2} - 10 \beta_1 - 5) q^{55} + (55 \beta_{4} - 9 \beta_{3} + 40 \beta_{2} + 93 \beta_1 + 23) q^{56} + (27 \beta_{4} - 13 \beta_{3} + 21 \beta_{2} - 15 \beta_1 + 12) q^{57} + ( - 83 \beta_{4} + 65 \beta_{3} - 23 \beta_{2} + 62 \beta_1 - 195) q^{58} + ( - 9 \beta_{4} - 88 \beta_{3} - 55 \beta_{2} + 68 \beta_1 + 275) q^{59} + (50 \beta_{4} - 10 \beta_{3} + 5 \beta_{2} - 20 \beta_1 - 55) q^{60} + ( - 60 \beta_{4} + 24 \beta_{3} + 101 \beta_{2} + 10 \beta_1 - 365) q^{61} + (24 \beta_{4} - 8 \beta_{3} + \beta_{2} + 69 \beta_1 + 94) q^{62} + ( - 6 \beta_{4} + \beta_{3} + 30 \beta_{2} - 89 \beta_1 + 61) q^{63} + (233 \beta_{4} - 167 \beta_{3} + 43 \beta_{2} + 107 \beta_1 + 408) q^{64} + ( - 5 \beta_{4} + 20 \beta_{2} + 30 \beta_1 - 145) q^{65} + (\beta_{4} - 15 \beta_{3} + 78 \beta_{2} - 43 \beta_1 - 579) q^{66} + ( - 7 \beta_{4} - 4 \beta_{3} - 19 \beta_{2} + 28 \beta_1 + 115) q^{67} + ( - 112 \beta_{4} + 118 \beta_{3} - 109 \beta_{2} - 122 \beta_1 - 363) q^{68} + ( - 23 \beta_{3} + 23 \beta_{2} - 23 \beta_1 - 23) q^{69} + ( - 30 \beta_{4} + 40 \beta_{3} - 5 \beta_{2} + 25 \beta_1 - 90) q^{70} + (77 \beta_{4} + 35 \beta_{3} - 48 \beta_{2} - 69 \beta_1 - 4) q^{71} + ( - 106 \beta_{4} + 174 \beta_{3} - 73 \beta_{2} - 82 \beta_1 - 91) q^{72} + (84 \beta_{4} - 135 \beta_{3} + 64 \beta_{2} - 49 \beta_1 + 202) q^{73} + ( - 133 \beta_{4} + 85 \beta_{3} - 147 \beta_{2} - 10 \beta_1 + 379) q^{74} + (25 \beta_{3} - 25 \beta_{2} + 25 \beta_1 + 25) q^{75} + ( - 143 \beta_{4} + 101 \beta_{3} - 26 \beta_{2} - 227 \beta_1 - 611) q^{76} + (25 \beta_{4} - 73 \beta_{3} - 39 \beta_{2} - 79 \beta_1 + 548) q^{77} + ( - 19 \beta_{4} - 73 \beta_{3} - 65 \beta_{2} + 13 \beta_1 + 168) q^{78} + ( - 92 \beta_{4} + 152 \beta_{3} - 4 \beta_{2} + 136 \beta_1 + 124) q^{79} + ( - 65 \beta_{4} + 35 \beta_{3} - 75 \beta_{2} - 95 \beta_1 - 220) q^{80} + (93 \beta_{4} + 21 \beta_{3} - 27 \beta_{2} + 27 \beta_1 - 77) q^{81} + ( - 21 \beta_{4} + 39 \beta_{3} - 116 \beta_{2} + 168 \beta_1 + 718) q^{82} + ( - 9 \beta_{4} - 74 \beta_{3} - 29 \beta_{2} - 190 \beta_1 + 35) q^{83} + ( - 31 \beta_{4} - 15 \beta_{3} + 107 \beta_{2} - 141 \beta_1 - 152) q^{84} + (45 \beta_{4} + 20 \beta_{3} + 10 \beta_{2} + 70 \beta_1 - 50) q^{85} + (128 \beta_{4} - 116 \beta_{3} + 84 \beta_{2} - 204 \beta_1 + 4) q^{86} + ( - 16 \beta_{4} + 173 \beta_{3} - 26 \beta_{2} + 127 \beta_1 + 466) q^{87} + (196 \beta_{4} - 12 \beta_{3} + 123 \beta_{2} + 250 \beta_1 + 341) q^{88} + (56 \beta_{4} - 134 \beta_{3} + 120 \beta_{2} - 246 \beta_1 + 104) q^{89} + (75 \beta_{4} - 15 \beta_{3} + 60 \beta_{2} - 20 \beta_1 - 20) q^{90} + ( - 26 \beta_{4} + 106 \beta_{3} + 81 \beta_{2} + 100 \beta_1 - 365) q^{91} + ( - 23 \beta_{4} + 23 \beta_{3} + 23 \beta_{2} - 69 \beta_1 - 92) q^{92} + ( - 83 \beta_{4} + 32 \beta_{3} + 87 \beta_{2} - 20 \beta_1 + 168) q^{93} + ( - 70 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} + 56 \beta_1 - 182) q^{94} + (25 \beta_{4} - 25 \beta_{3} + 55 \beta_1 + 145) q^{95} + ( - 30 \beta_{4} - 78 \beta_{3} + 37 \beta_{2} - 200 \beta_1 - 1167) q^{96} + (26 \beta_{4} - 106 \beta_{3} - 173 \beta_{2} + 128 \beta_1 + 165) q^{97} + ( - 11 \beta_{4} + 75 \beta_{3} + 70 \beta_{2} - 139 \beta_1 - 509) q^{98} + (35 \beta_{4} + 121 \beta_{3} + 58 \beta_{2} - 295 \beta_1 + 433) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^2 + (b3 - b2 + b1 + 1) * q^3 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 - 5 * q^5 + (-2*b4 - 3*b2 + 2*b1 + 5) * q^6 + (b3 + 3*b2 + b1 - 2) * q^7 + (5*b4 - 3*b3 - b2 + 5*b1 + 26) * q^8 + (-3*b4 + 6*b3 + 16) * q^9 + (-5*b1 - 5) * q^10 + (2*b4 + 4*b3 + 7*b2 + 2*b1 + 1) * q^11 + (-10*b4 + 2*b3 - b2 + 4*b1 + 11) * q^12 + (b4 - 4*b2 - 6*b1 + 29) * q^13 + (6*b4 - 8*b3 + b2 - 5*b1 + 18) * q^14 + (-5*b3 + 5*b2 - 5*b1 - 5) * q^15 + (13*b4 - 7*b3 + 15*b2 + 19*b1 + 44) * q^16 + (-9*b4 - 4*b3 - 2*b2 - 14*b1 + 10) * q^17 + (-15*b4 + 3*b3 - 12*b2 + 4*b1 + 4) * q^18 + (-5*b4 + 5*b3 - 11*b1 - 29) * q^19 + (-5*b4 + 5*b3 + 5*b2 - 15*b1 - 20) * q^20 + (9*b4 - 5*b3 - 9*b2 - 11*b1 - 8) * q^21 + (18*b4 - 26*b3 + 5*b2 - 10*b1 + 35) * q^22 - 23 * q^23 + (-14*b4 + 26*b3 - 3*b2 + 25) * q^24 + 25 * q^25 + (-11*b4 + 11*b3 + 4*b2 + 21*b1 - 55) * q^26 + (9*b4 + 4*b3 - 7*b2 - 8*b1 + 118) * q^27 + (23*b4 - 15*b3 + 2*b2 + 15*b1 - 5) * q^28 + (-5*b4 + 9*b3 - 21*b2 - 17*b1 + 93) * q^29 + (10*b4 + 15*b2 - 10*b1 - 25) * q^30 + (12*b4 - 8*b3 - 17*b2 + 14*b1 + 8) * q^31 + (55*b4 - 57*b3 + 37*b2 + 41*b1 + 100) * q^32 + (27*b4 + 7*b3 - 14*b2 - 45*b1 + 47) * q^33 + (-41*b4 + 49*b3 - 2*b2 - 8*b1 - 122) * q^34 + (-5*b3 - 15*b2 - 5*b1 + 10) * q^35 + (-44*b4 + 14*b3 - 49*b2 + 18*b1 - 107) * q^36 + (-53*b4 + 24*b3 + 11*b2 + 28*b1 - 7) * q^37 + (-31*b4 + 21*b3 - 4*b2 - 61*b1 - 155) * q^38 + (b4 + 46*b3 + b2 + 22*b1 + 62) * q^39 + (-25*b4 + 15*b3 + 5*b2 - 25*b1 - 130) * q^40 + (-19*b4 - 9*b3 - 20*b2 + 67*b1 - 4) * q^41 + (3*b4 + 7*b3 + 25*b2 - 11*b1 - 168) * q^42 + (4*b4 - 6*b3 + 60*b2 - 10*b1 - 136) * q^43 + (64*b4 - 60*b3 + 21*b2 + 46*b1 - 21) * q^44 + (15*b4 - 30*b3 - 80) * q^45 + (-23*b1 - 23) * q^46 + (-8*b4 + 33*b3 + 8*b2 - 29*b1 + 188) * q^47 + (6*b4 + 6*b3 - 49*b2 - 56*b1 - 125) * q^48 + (5*b4 - 32*b3 - 10*b2 - 38*b1 - 137) * q^49 + (25*b1 + 25) * q^50 + (5*b4 - 15*b3 - 30*b2 + 97*b1 - 207) * q^51 + (-23*b4 - 7*b3 - 18*b2 + 9*b1 - 51) * q^52 + (25*b4 - 12*b3 + 81*b2 - 56*b1 + 205) * q^53 + (b4 - 9*b3 + 15*b2 + 101*b1 - 28) * q^54 + (-10*b4 - 20*b3 - 35*b2 - 10*b1 - 5) * q^55 + (55*b4 - 9*b3 + 40*b2 + 93*b1 + 23) * q^56 + (27*b4 - 13*b3 + 21*b2 - 15*b1 + 12) * q^57 + (-83*b4 + 65*b3 - 23*b2 + 62*b1 - 195) * q^58 + (-9*b4 - 88*b3 - 55*b2 + 68*b1 + 275) * q^59 + (50*b4 - 10*b3 + 5*b2 - 20*b1 - 55) * q^60 + (-60*b4 + 24*b3 + 101*b2 + 10*b1 - 365) * q^61 + (24*b4 - 8*b3 + b2 + 69*b1 + 94) * q^62 + (-6*b4 + b3 + 30*b2 - 89*b1 + 61) * q^63 + (233*b4 - 167*b3 + 43*b2 + 107*b1 + 408) * q^64 + (-5*b4 + 20*b2 + 30*b1 - 145) * q^65 + (b4 - 15*b3 + 78*b2 - 43*b1 - 579) * q^66 + (-7*b4 - 4*b3 - 19*b2 + 28*b1 + 115) * q^67 + (-112*b4 + 118*b3 - 109*b2 - 122*b1 - 363) * q^68 + (-23*b3 + 23*b2 - 23*b1 - 23) * q^69 + (-30*b4 + 40*b3 - 5*b2 + 25*b1 - 90) * q^70 + (77*b4 + 35*b3 - 48*b2 - 69*b1 - 4) * q^71 + (-106*b4 + 174*b3 - 73*b2 - 82*b1 - 91) * q^72 + (84*b4 - 135*b3 + 64*b2 - 49*b1 + 202) * q^73 + (-133*b4 + 85*b3 - 147*b2 - 10*b1 + 379) * q^74 + (25*b3 - 25*b2 + 25*b1 + 25) * q^75 + (-143*b4 + 101*b3 - 26*b2 - 227*b1 - 611) * q^76 + (25*b4 - 73*b3 - 39*b2 - 79*b1 + 548) * q^77 + (-19*b4 - 73*b3 - 65*b2 + 13*b1 + 168) * q^78 + (-92*b4 + 152*b3 - 4*b2 + 136*b1 + 124) * q^79 + (-65*b4 + 35*b3 - 75*b2 - 95*b1 - 220) * q^80 + (93*b4 + 21*b3 - 27*b2 + 27*b1 - 77) * q^81 + (-21*b4 + 39*b3 - 116*b2 + 168*b1 + 718) * q^82 + (-9*b4 - 74*b3 - 29*b2 - 190*b1 + 35) * q^83 + (-31*b4 - 15*b3 + 107*b2 - 141*b1 - 152) * q^84 + (45*b4 + 20*b3 + 10*b2 + 70*b1 - 50) * q^85 + (128*b4 - 116*b3 + 84*b2 - 204*b1 + 4) * q^86 + (-16*b4 + 173*b3 - 26*b2 + 127*b1 + 466) * q^87 + (196*b4 - 12*b3 + 123*b2 + 250*b1 + 341) * q^88 + (56*b4 - 134*b3 + 120*b2 - 246*b1 + 104) * q^89 + (75*b4 - 15*b3 + 60*b2 - 20*b1 - 20) * q^90 + (-26*b4 + 106*b3 + 81*b2 + 100*b1 - 365) * q^91 + (-23*b4 + 23*b3 + 23*b2 - 69*b1 - 92) * q^92 + (-83*b4 + 32*b3 + 87*b2 - 20*b1 + 168) * q^93 + (-70*b4 + 4*b3 - 12*b2 + 56*b1 - 182) * q^94 + (25*b4 - 25*b3 + 55*b1 + 145) * q^95 + (-30*b4 - 78*b3 + 37*b2 - 200*b1 - 1167) * q^96 + (26*b4 - 106*b3 - 173*b2 + 128*b1 + 165) * q^97 + (-11*b4 + 75*b3 + 70*b2 - 139*b1 - 509) * q^98 + (35*b4 + 121*b3 + 58*b2 - 295*b1 + 433) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10})$$ 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 $$5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100})$$ 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 - 30 * q^10 + 23 * q^11 + 47 * q^12 + 132 * q^13 + 93 * q^14 - 20 * q^15 + 282 * q^16 + 23 * q^17 - 15 * q^18 - 161 * q^19 - 110 * q^20 - 60 * q^21 + 193 * q^22 - 115 * q^23 + 105 * q^24 + 125 * q^25 - 257 * q^26 + 577 * q^27 + 17 * q^28 + 401 * q^29 - 95 * q^30 + 32 * q^31 + 670 * q^32 + 189 * q^33 - 663 * q^34 + 15 * q^35 - 659 * q^36 - 38 * q^37 - 875 * q^38 + 335 * q^39 - 690 * q^40 - 12 * q^41 - 798 * q^42 - 566 * q^43 + 47 * q^44 - 385 * q^45 - 138 * q^46 + 919 * q^47 - 773 * q^48 - 738 * q^49 + 150 * q^50 - 993 * q^51 - 305 * q^52 + 1156 * q^53 - 8 * q^54 - 115 * q^55 + 343 * q^56 + 114 * q^57 - 1042 * q^58 + 1324 * q^59 - 235 * q^60 - 1673 * q^61 + 565 * q^62 + 270 * q^63 + 2466 * q^64 - 660 * q^65 - 2781 * q^66 + 558 * q^67 - 2267 * q^68 - 92 * q^69 - 465 * q^70 - 108 * q^71 - 789 * q^72 + 1173 * q^73 + 1458 * q^74 + 100 * q^75 - 3477 * q^76 + 2608 * q^77 + 704 * q^78 + 656 * q^79 - 1410 * q^80 - 319 * q^81 + 3505 * q^82 - 82 * q^83 - 718 * q^84 - 115 * q^85 + 112 * q^86 + 2389 * q^87 + 2397 * q^88 + 570 * q^89 + 75 * q^90 - 1589 * q^91 - 506 * q^92 + 911 * q^93 - 948 * q^94 + 805 * q^95 - 5991 * q^96 + 633 * q^97 - 2555 * q^98 + 2021 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 25\nu^{2} - 11\nu + 98 ) / 16$$ (v^4 + v^3 - 25*v^2 - 11*v + 98) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 3\nu^{3} + 17\nu^{2} - 41\nu - 42 ) / 8$$ (-v^4 + 3*v^3 + 17*v^2 - 41*v - 42) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{4} + 7\nu^{3} + 25\nu^{2} - 109\nu - 162 ) / 16$$ (-v^4 + 7*v^3 + 25*v^2 - 109*v - 162) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 11$$ b4 - b3 - b2 + b1 + 11 $$\nu^{3}$$ $$=$$ $$2\beta_{4} + 2\beta_{2} + 15\beta _1 + 8$$ 2*b4 + 2*b2 + 15*b1 + 8 $$\nu^{4}$$ $$=$$ $$23\beta_{4} - 25\beta_{3} - 11\beta_{2} + 21\beta _1 + 169$$ 23*b4 - 25*b3 - 11*b2 + 21*b1 + 169

## 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
 −3.93900 −2.49214 −0.595043 3.41740 4.60878
−2.93900 −3.85751 0.637693 −5.00000 11.3372 −23.5932 21.6378 −12.1196 14.6950
1.2 −1.49214 9.02447 −5.77352 −5.00000 −13.4658 4.33445 20.5520 54.4411 7.46070
1.3 0.404957 −7.11323 −7.83601 −5.00000 −2.88055 13.7888 −6.41290 23.5981 −2.02479
1.4 4.41740 7.84147 11.5134 −5.00000 34.6389 −8.97260 15.5200 34.4886 −22.0870
1.5 5.60878 −1.89520 23.4584 −5.00000 −10.6297 11.4426 86.7031 −23.4082 −28.0439
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$23$$ $$1$$

## 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 115.4.a.e 5
3.b odd 2 1 1035.4.a.k 5
4.b odd 2 1 1840.4.a.n 5
5.b even 2 1 575.4.a.j 5
5.c odd 4 2 575.4.b.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 1.a even 1 1 trivial
575.4.a.j 5 5.b even 2 1
575.4.b.i 10 5.c odd 4 2
1035.4.a.k 5 3.b odd 2 1
1840.4.a.n 5 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 6T_{2}^{4} - 13T_{2}^{3} + 72T_{2}^{2} + 82T_{2} - 44$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(115))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 6 T^{4} - 13 T^{3} + 72 T^{2} + \cdots - 44$$
$3$ $$T^{5} - 4 T^{4} - 98 T^{3} + \cdots + 3680$$
$5$ $$(T + 5)^{5}$$
$7$ $$T^{5} + 3 T^{4} - 484 T^{3} + \cdots - 144774$$
$11$ $$T^{5} - 23 T^{4} - 4015 T^{3} + \cdots - 74136848$$
$13$ $$T^{5} - 132 T^{4} + 4786 T^{3} + \cdots + 1550116$$
$17$ $$T^{5} - 23 T^{4} + \cdots - 1039045340$$
$19$ $$T^{5} + 161 T^{4} + 3999 T^{3} + \cdots - 801280$$
$23$ $$(T + 23)^{5}$$
$29$ $$T^{5} - 401 T^{4} + \cdots - 6149898500$$
$31$ $$T^{5} - 32 T^{4} + \cdots - 438072447$$
$37$ $$T^{5} + 38 T^{4} + \cdots + 1590700778176$$
$41$ $$T^{5} + 12 T^{4} + \cdots + 114116030755$$
$43$ $$T^{5} + 566 T^{4} + \cdots + 504784881664$$
$47$ $$T^{5} - 919 T^{4} + \cdots + 117787714816$$
$53$ $$T^{5} - 1156 T^{4} + \cdots + 5720332226904$$
$59$ $$T^{5} - 1324 T^{4} + \cdots + 24279649927232$$
$61$ $$T^{5} + 1673 T^{4} + \cdots - 34095834816896$$
$67$ $$T^{5} - 558 T^{4} + \cdots + 5644442112$$
$71$ $$T^{5} + 108 T^{4} + \cdots + 15638892903635$$
$73$ $$T^{5} + \cdots - 100895881632176$$
$79$ $$T^{5} - 656 T^{4} + \cdots - 90481602379776$$
$83$ $$T^{5} + 82 T^{4} + \cdots + 18307318870176$$
$89$ $$T^{5} + \cdots - 115104799418880$$
$97$ $$T^{5} + \cdots - 480989167569272$$