# Properties

 Label 507.4.a.p Level $507$ Weight $4$ Character orbit 507.a Self dual yes Analytic conductor $29.914$ Analytic rank $0$ 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] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ Analytic rank: $$0$$ 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: $$x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27$$ x^9 - 56*x^7 - 27*x^6 + 945*x^5 + 763*x^4 - 4139*x^3 - 2478*x^2 + 63*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{3} + 5) q^{4} + (\beta_{8} - \beta_{3} + 3) q^{5} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 9) q^{7} + ( - 2 \beta_{8} + 2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_1 + 13) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b3 + 1) * q^2 + 3 * q^3 + (b5 + b3 + 5) * q^4 + (b8 - b3 + 3) * q^5 + (3*b3 + 3) * q^6 + (-b8 + b5 + b3 + b2 - b1 + 9) * q^7 + (-2*b8 + 2*b5 + b4 + 6*b3 - b2 + b1 + 13) * q^8 + 9 * q^9 $$q + (\beta_{3} + 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{3} + 5) q^{4} + (\beta_{8} - \beta_{3} + 3) q^{5} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 9) q^{7} + ( - 2 \beta_{8} + 2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_1 + 13) q^{8} + 9 q^{9} + (\beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} - \beta_1 - 3) q^{10} + (2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - 6 \beta_{3} - \beta_{2} + 8) q^{11} + (3 \beta_{5} + 3 \beta_{3} + 15) q^{12} + ( - \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + \cdots + 27) q^{14}+ \cdots + (18 \beta_{8} - 9 \beta_{6} - 9 \beta_{5} + 9 \beta_{4} - 54 \beta_{3} - 9 \beta_{2} + \cdots + 72) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^2 + 3 * q^3 + (b5 + b3 + 5) * q^4 + (b8 - b3 + 3) * q^5 + (3*b3 + 3) * q^6 + (-b8 + b5 + b3 + b2 - b1 + 9) * q^7 + (-2*b8 + 2*b5 + b4 + 6*b3 - b2 + b1 + 13) * q^8 + 9 * q^9 + (b8 + b7 - b6 - 3*b5 + 2*b4 + 7*b3 - b1 - 3) * q^10 + (2*b8 - b6 - b5 + b4 - 6*b3 - b2 + 8) * q^11 + (3*b5 + 3*b3 + 15) * q^12 + (-b8 - 3*b7 - b6 + 2*b5 + 2*b4 + 15*b3 - b2 + 4*b1 + 27) * q^14 + (3*b8 - 3*b3 + 9) * q^15 + (-7*b8 - 2*b7 + 5*b6 + 7*b5 - 5*b4 + 10*b3 + 2*b1 + 26) * q^16 + (5*b8 + 3*b7 - b6 + b5 + 4*b4 + 4*b3 + 2*b2 - 8*b1 + 16) * q^17 + (9*b3 + 9) * q^18 + (b8 + 5*b7 + 3*b6 + b5 - 4*b4 - 8*b3 - 2*b2 + 6*b1 + 34) * q^19 + (2*b8 - 3*b7 - 2*b6 + b5 - 5*b4 - 24*b3 + 6*b2 - 5*b1 + 32) * q^20 + (-3*b8 + 3*b5 + 3*b3 + 3*b2 - 3*b1 + 27) * q^21 + (4*b8 - 2*b6 - 10*b5 - 2*b4 + 3*b3 + 3*b2 - 9*b1 - 73) * q^22 + (2*b8 + 3*b7 - b6 - 6*b5 - 11*b4 + 5*b3 - 5*b2 + 4*b1 + 16) * q^23 + (-6*b8 + 6*b5 + 3*b4 + 18*b3 - 3*b2 + 3*b1 + 39) * q^24 + (5*b8 + 5*b7 + 2*b6 + b5 - 4*b4 - 20*b3 + 10*b2 + 6*b1 - 15) * q^25 + 27 * q^27 + (-7*b7 + 6*b6 + 20*b5 + b4 + 22*b3 - 3*b2 - 8*b1 + 106) * q^28 + (11*b8 - 12*b7 + 2*b6 - 8*b5 + 11*b4 + 11*b3 - 4*b2 + 17*b1 + 9) * q^29 + (3*b8 + 3*b7 - 3*b6 - 9*b5 + 6*b4 + 21*b3 - 3*b1 - 9) * q^30 + (-13*b8 - 2*b7 - 4*b6 + 4*b5 - 11*b4 - 5*b3 - 6*b2 - 25*b1 + 69) * q^31 + (-12*b8 + 6*b7 + 14*b6 + 19*b5 - b4 + 26*b3 - 7*b2 + 23*b1 + 91) * q^32 + (6*b8 - 3*b6 - 3*b5 + 3*b4 - 18*b3 - 3*b2 + 24) * q^33 + (17*b8 - 8*b7 - 13*b6 - 15*b5 + 9*b4 + 35*b3 + 4*b2 + 10*b1 + 95) * q^34 + (8*b8 - 9*b7 + b6 + 12*b5 - 4*b4 - 21*b3 - 5*b2 - 23*b1 - 66) * q^35 + (9*b5 + 9*b3 + 45) * q^36 + (8*b8 + 3*b7 - 2*b6 - 4*b5 + 30*b4 - b3 + 15*b2 + 33*b1 + 145) * q^37 + (-13*b8 + 30*b7 + 9*b6 - 7*b5 - 7*b4 + 63*b3 - 14*b2 + 22*b1 - 53) * q^38 + (-2*b8 - 15*b7 - 14*b6 - 18*b5 + 9*b4 + 18*b3 - 8*b2 - b1 - 140) * q^40 + (-5*b8 - 6*b7 + 12*b6 - 15*b5 - 7*b4 + b3 + 11*b2 - 8*b1 + 32) * q^41 + (-3*b8 - 9*b7 - 3*b6 + 6*b5 + 6*b4 + 45*b3 - 3*b2 + 12*b1 + 81) * q^42 + (-17*b8 + 2*b7 - 3*b6 - 7*b5 - 6*b4 - 39*b3 - 3*b2 - 41*b1 + 1) * q^43 + (18*b8 - 6*b7 - 14*b6 - 23*b5 - 5*b4 - 86*b3 + 14*b2 - 12*b1 - 67) * q^44 + (9*b8 - 9*b3 + 27) * q^45 + (-6*b8 + 38*b7 - 6*b6 - 11*b5 - 25*b4 + 13*b3 - 19*b2 - 13*b1 + 90) * q^46 + (-15*b8 - 15*b7 - 5*b6 - 27*b5 - 26*b4 - 20*b3 - 16*b2 - 32*b1 - 30) * q^47 + (-21*b8 - 6*b7 + 15*b6 + 21*b5 - 15*b4 + 30*b3 + 6*b1 + 78) * q^48 + (-15*b8 - 8*b7 - 2*b6 + 6*b5 + 7*b4 + 21*b3 + 24*b2 + 17*b1 + 16) * q^49 + (3*b8 + 21*b7 - 9*b6 - 40*b5 + 35*b4 + 43*b3 - 14*b2 + 25*b1 - 180) * q^50 + (15*b8 + 9*b7 - 3*b6 + 3*b5 + 12*b4 + 12*b3 + 6*b2 - 24*b1 + 48) * q^51 + (6*b8 + 9*b7 - 15*b6 - 21*b5 + 18*b4 + 21*b3 + 26*b2 + 38*b1 + 13) * q^53 + (27*b3 + 27) * q^54 + (3*b8 + 3*b7 - b6 + 2*b5 - 11*b4 - 24*b3 + 9*b2 + 36*b1 + 228) * q^55 + (-26*b8 + 9*b7 + 16*b6 + 35*b5 + 21*b4 + 146*b3 - 3*b2 + 2*b1 + 304) * q^56 + (3*b8 + 15*b7 + 9*b6 + 3*b5 - 12*b4 - 24*b3 - 6*b2 + 18*b1 + 102) * q^57 + (17*b8 - 12*b7 + 25*b6 + 27*b5 + 30*b4 - 81*b3 + 42*b2 - 66*b1 + 62) * q^58 + (-7*b8 - 3*b7 + 3*b6 - 25*b5 + 13*b4 + 2*b3 - 10*b2 + 19*b1 + 85) * q^59 + (6*b8 - 9*b7 - 6*b6 + 3*b5 - 15*b4 - 72*b3 + 18*b2 - 15*b1 + 96) * q^60 + (-39*b8 - 19*b7 + 15*b6 + 29*b5 - 14*b4 - 14*b3 - 30*b2 - 92*b1 - 184) * q^61 + (-13*b8 - 18*b7 - 25*b6 - 7*b5 - 44*b4 + 93*b3 - 24*b2 + 10*b1 + 82) * q^62 + (-9*b8 + 9*b5 + 9*b3 + 9*b2 - 9*b1 + 81) * q^63 + (-25*b8 + 62*b7 + 29*b6 + 50*b5 + 30*b4 + 123*b3 - 27*b2 + 73*b1 + 152) * q^64 + (12*b8 - 6*b6 - 30*b5 - 6*b4 + 9*b3 + 9*b2 - 27*b1 - 219) * q^66 + (-21*b8 - 49*b7 + 12*b6 + 9*b5 - 33*b4 + 4*b3 - 20*b2 + 69*b1 + 301) * q^67 + (10*b8 - 48*b7 - 20*b6 - 12*b5 - 11*b4 - 31*b3 + 25*b2 - 63*b1 + 288) * q^68 + (6*b8 + 9*b7 - 3*b6 - 18*b5 - 33*b4 + 15*b3 - 15*b2 + 12*b1 + 48) * q^69 + (-2*b8 - 19*b7 - 28*b6 - 37*b5 + 33*b4 + 75*b3 - 11*b2 - 89) * q^70 + (4*b8 + 42*b7 + 23*b6 - 8*b5 + 21*b4 + 18*b3 + 8*b2 - 21*b1 - 11) * q^71 + (-18*b8 + 18*b5 + 9*b4 + 54*b3 - 9*b2 + 9*b1 + 117) * q^72 + (-7*b8 - 60*b7 - 10*b6 + 2*b5 + 10*b4 - 3*b3 - 14*b2 + 14*b1 + 323) * q^73 + (28*b8 - 30*b7 + 36*b6 + 22*b5 + 47*b4 + 41*b3 + 61*b2 - 31*b1 - 138) * q^74 + (15*b8 + 15*b7 + 6*b6 + 3*b5 - 12*b4 - 60*b3 + 30*b2 + 18*b1 - 45) * q^75 + (-50*b8 + 66*b7 + 36*b6 + 82*b5 - 85*b4 - 69*b3 - 21*b2 + 57*b1 + 194) * q^76 + (17*b8 + 6*b7 + 5*b6 + 8*b5 - 2*b4 - 77*b3 - 2*b2 - 106*b1 - 323) * q^77 + (-6*b8 + 14*b7 - 9*b6 + 16*b5 + 10*b4 - 30*b3 - 24*b2 + 112*b1 - 132) * q^79 + (20*b8 - 33*b7 + 6*b6 + 13*b5 - 4*b4 - 163*b3 + 3*b2 - 98*b1 - 394) * q^80 + 81 * q^81 + (37*b8 - 10*b7 + 3*b6 - 12*b5 + 44*b4 - 102*b3 + 7*b2 + 51*b1 + 169) * q^82 + (5*b8 + 30*b7 - 33*b6 - 29*b5 - 3*b4 + 119*b3 + 27*b2 + 158*b1 + 276) * q^83 + (-21*b7 + 18*b6 + 60*b5 + 3*b4 + 66*b3 - 9*b2 - 24*b1 + 318) * q^84 + (27*b8 + 32*b7 + 2*b6 - 17*b5 - 9*b4 + 29*b3 + 53*b2 + 20*b1 + 460) * q^85 + (29*b8 - 42*b7 - 33*b6 - 61*b5 - 60*b4 - 104*b3 - 7*b2 + 39*b1 - 427) * q^86 + (33*b8 - 36*b7 + 6*b6 - 24*b5 + 33*b4 + 33*b3 - 12*b2 + 51*b1 + 27) * q^87 + (53*b8 - 24*b7 - 61*b6 - 104*b5 + 55*b4 - 184*b3 - 5*b2 - 31*b1 - 426) * q^88 + (49*b8 + 63*b7 - 26*b6 - 31*b5 + 61*b4 + 86*b3 + 14*b2 - 115*b1 - 93) * q^89 + (9*b8 + 9*b7 - 9*b6 - 27*b5 + 18*b4 + 63*b3 - 9*b1 - 27) * q^90 + (-31*b8 + 96*b7 - 17*b6 - b5 - 80*b4 + 52*b3 - 37*b2 + 41*b1 + 39) * q^92 + (-39*b8 - 6*b7 - 12*b6 + 12*b5 - 33*b4 - 15*b3 - 18*b2 - 75*b1 + 207) * q^93 + (29*b8 - 14*b7 - 37*b6 - 49*b5 - 85*b4 - 255*b3 - 10*b2 - 66*b1 - 207) * q^94 + (71*b8 + 36*b7 + 18*b6 - 7*b5 - 23*b4 - 219*b3 + 41*b2 + 28*b1 + 338) * q^95 + (-36*b8 + 18*b7 + 42*b6 + 57*b5 - 3*b4 + 78*b3 - 21*b2 + 69*b1 + 273) * q^96 + (-8*b8 - 11*b7 - 10*b6 - b5 + 13*b4 - 73*b3 - 86*b2 - 95*b1 + 130) * q^97 + (-13*b8 - 54*b7 + 11*b6 + 63*b5 + 60*b4 + 16*b2 - 6*b1 + 193) * q^98 + (18*b8 - 9*b6 - 9*b5 + 9*b4 - 54*b3 - 9*b2 + 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10})$$ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 $$9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100})$$ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( 4276 \nu^{8} + 993 \nu^{7} - 231050 \nu^{6} - 177378 \nu^{5} + 3665268 \nu^{4} + 4377940 \nu^{3} - 13739936 \nu^{2} - 15644592 \nu - 1151901 ) / 1169766$$ (4276*v^8 + 993*v^7 - 231050*v^6 - 177378*v^5 + 3665268*v^4 + 4377940*v^3 - 13739936*v^2 - 15644592*v - 1151901) / 1169766 $$\beta_{2}$$ $$=$$ $$( 64999 \nu^{8} + 190713 \nu^{7} - 3890801 \nu^{6} - 11363973 \nu^{5} + 68463819 \nu^{4} + 197174581 \nu^{3} - 296579777 \nu^{2} - 794287746 \nu - 40902156 ) / 15206958$$ (64999*v^8 + 190713*v^7 - 3890801*v^6 - 11363973*v^5 + 68463819*v^4 + 197174581*v^3 - 296579777*v^2 - 794287746*v - 40902156) / 15206958 $$\beta_{3}$$ $$=$$ $$( 24987 \nu^{8} - 7166 \nu^{7} - 1397001 \nu^{6} - 297443 \nu^{5} + 23628597 \nu^{4} + 13299657 \nu^{3} - 105271205 \nu^{2} - 45679772 \nu + 3317895 ) / 5068986$$ (24987*v^8 - 7166*v^7 - 1397001*v^6 - 297443*v^5 + 23628597*v^4 + 13299657*v^3 - 105271205*v^2 - 45679772*v + 3317895) / 5068986 $$\beta_{4}$$ $$=$$ $$( 90230 \nu^{8} + 32184 \nu^{7} - 5333239 \nu^{6} - 3836655 \nu^{5} + 94118139 \nu^{4} + 86430869 \nu^{3} - 422915233 \nu^{2} - 219614865 \nu + 6113457 ) / 15206958$$ (90230*v^8 + 32184*v^7 - 5333239*v^6 - 3836655*v^5 + 94118139*v^4 + 86430869*v^3 - 422915233*v^2 - 219614865*v + 6113457) / 15206958 $$\beta_{5}$$ $$=$$ $$( 135002 \nu^{8} - 25263 \nu^{7} - 7390255 \nu^{6} - 2116755 \nu^{5} + 120654441 \nu^{4} + 76680635 \nu^{3} - 485797741 \nu^{2} - 246575877 \nu - 154170810 ) / 15206958$$ (135002*v^8 - 25263*v^7 - 7390255*v^6 - 2116755*v^5 + 120654441*v^4 + 76680635*v^3 - 485797741*v^2 - 246575877*v - 154170810) / 15206958 $$\beta_{6}$$ $$=$$ $$( 279719 \nu^{8} - 175299 \nu^{7} - 15606016 \nu^{6} + 2115444 \nu^{5} + 265872168 \nu^{4} + 57886724 \nu^{3} - 1250692846 \nu^{2} + \cdots + 313226181 ) / 15206958$$ (279719*v^8 - 175299*v^7 - 15606016*v^6 + 2115444*v^5 + 265872168*v^4 + 57886724*v^3 - 1250692846*v^2 - 96613761*v + 313226181) / 15206958 $$\beta_{7}$$ $$=$$ $$( 450169 \nu^{8} - 76689 \nu^{7} - 24966176 \nu^{6} - 9175896 \nu^{5} + 417633228 \nu^{4} + 314702644 \nu^{3} - 1808976728 \nu^{2} - 1087356531 \nu + 15335469 ) / 15206958$$ (450169*v^8 - 76689*v^7 - 24966176*v^6 - 9175896*v^5 + 417633228*v^4 + 314702644*v^3 - 1808976728*v^2 - 1087356531*v + 15335469) / 15206958 $$\beta_{8}$$ $$=$$ $$( 735613 \nu^{8} - 134067 \nu^{7} - 41110133 \nu^{6} - 11744913 \nu^{5} + 695505123 \nu^{4} + 411567511 \nu^{3} - 3109582931 \nu^{2} + \cdots + 129805956 ) / 15206958$$ (735613*v^8 - 134067*v^7 - 41110133*v^6 - 11744913*v^5 + 695505123*v^4 + 411567511*v^3 - 3109582931*v^2 - 1105342458*v + 129805956) / 15206958
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 12\beta_{3} - \beta _1 - 5 ) / 13$$ (b7 + b6 + b5 + b4 - 12*b3 - b1 - 5) / 13 $$\nu^{2}$$ $$=$$ $$( 3\beta_{7} - 3\beta_{6} + 10\beta_{5} - 10\beta_{3} - 20\beta _1 + 147 ) / 13$$ (3*b7 - 3*b6 + 10*b5 - 10*b3 - 20*b1 + 147) / 13 $$\nu^{3}$$ $$=$$ $$( - 13 \beta_{8} + 28 \beta_{7} + 36 \beta_{6} + 49 \beta_{5} + 19 \beta_{4} - 258 \beta_{3} + 13 \beta_{2} - 53 \beta _1 - 18 ) / 13$$ (-13*b8 + 28*b7 + 36*b6 + 49*b5 + 19*b4 - 258*b3 + 13*b2 - 53*b1 - 18) / 13 $$\nu^{4}$$ $$=$$ $$( 13 \beta_{8} + 104 \beta_{7} - 53 \beta_{6} + 233 \beta_{5} - 46 \beta_{4} - 494 \beta_{3} + 26 \beta_{2} - 603 \beta _1 + 3056 ) / 13$$ (13*b8 + 104*b7 - 53*b6 + 233*b5 - 46*b4 - 494*b3 + 26*b2 - 603*b1 + 3056) / 13 $$\nu^{5}$$ $$=$$ $$( - 377 \beta_{8} + 633 \beta_{7} + 987 \beta_{6} + 1585 \beta_{5} + 407 \beta_{4} - 6283 \beta_{3} + 442 \beta_{2} - 1658 \beta _1 + 1824 ) / 13$$ (-377*b8 + 633*b7 + 987*b6 + 1585*b5 + 407*b4 - 6283*b3 + 442*b2 - 1658*b1 + 1824) / 13 $$\nu^{6}$$ $$=$$ $$( 507 \beta_{8} + 3033 \beta_{7} - 872 \beta_{6} + 6044 \beta_{5} - 2007 \beta_{4} - 17455 \beta_{3} + 1027 \beta_{2} - 15967 \beta _1 + 71121 ) / 13$$ (507*b8 + 3033*b7 - 872*b6 + 6044*b5 - 2007*b4 - 17455*b3 + 1027*b2 - 15967*b1 + 71121) / 13 $$\nu^{7}$$ $$=$$ $$( - 8788 \beta_{8} + 14072 \beta_{7} + 25574 \beta_{6} + 46062 \beta_{5} + 8136 \beta_{4} - 162988 \beta_{3} + 13598 \beta_{2} - 47536 \beta _1 + 94223 ) / 13$$ (-8788*b8 + 14072*b7 + 25574*b6 + 46062*b5 + 8136*b4 - 162988*b3 + 13598*b2 - 47536*b1 + 94223) / 13 $$\nu^{8}$$ $$=$$ $$( 15964 \beta_{8} + 82361 \beta_{7} - 9523 \beta_{6} + 167537 \beta_{5} - 69817 \beta_{4} - 554392 \beta_{3} + 35074 \beta_{2} - 413731 \beta _1 + 1753215 ) / 13$$ (15964*b8 + 82361*b7 - 9523*b6 + 167537*b5 - 69817*b4 - 554392*b3 + 35074*b2 - 413731*b1 + 1753215) / 13

## 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
 5.39246 2.37150 5.06791 −0.588238 0.107680 −0.100291 −4.83218 −4.14324 −3.27560
−4.83750 3.00000 15.4014 21.1983 −14.5125 16.2806 −35.8043 9.00000 −102.547
1.2 −3.17344 3.00000 2.07074 6.74147 −9.52033 −14.1726 18.8162 9.00000 −21.3937
1.3 −2.82093 3.00000 −0.0423641 −3.41089 −8.46278 13.3442 22.6869 9.00000 9.62187
1.4 −0.213700 3.00000 −7.95433 −15.3391 −0.641100 32.3928 3.40944 9.00000 3.27797
1.5 0.447278 3.00000 −7.79994 1.93073 1.34183 −8.14537 −7.06697 9.00000 0.863573
1.6 2.34727 3.00000 −2.49032 15.3991 7.04181 −10.1317 −24.6236 9.00000 36.1458
1.7 4.03025 3.00000 8.24289 8.08864 12.0907 5.95078 0.978887 9.00000 32.5992
1.8 4.69820 3.00000 14.0731 4.47249 14.0946 27.2096 28.5326 9.00000 21.0127
1.9 5.52257 3.00000 22.4988 −6.08065 16.5677 20.2718 80.0709 9.00000 −33.5808
 $$n$$: 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
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 507.4.a.p yes 9
3.b odd 2 1 1521.4.a.bf 9
13.b even 2 1 507.4.a.o 9
13.d odd 4 2 507.4.b.k 18
39.d odd 2 1 1521.4.a.bi 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.o 9 13.b even 2 1
507.4.a.p yes 9 1.a even 1 1 trivial
507.4.b.k 18 13.d odd 4 2
1521.4.a.bf 9 3.b odd 2 1
1521.4.a.bi 9 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}^{9} - 6T_{2}^{8} - 40T_{2}^{7} + 251T_{2}^{6} + 452T_{2}^{5} - 3075T_{2}^{4} - 1401T_{2}^{3} + 11386T_{2}^{2} - 2288T_{2} - 1016$$ T2^9 - 6*T2^8 - 40*T2^7 + 251*T2^6 + 452*T2^5 - 3075*T2^4 - 1401*T2^3 + 11386*T2^2 - 2288*T2 - 1016 $$T_{5}^{9} - 33 T_{5}^{8} - 8 T_{5}^{7} + 8831 T_{5}^{6} - 66645 T_{5}^{5} - 239411 T_{5}^{4} + 3228609 T_{5}^{3} - 2833568 T_{5}^{2} - 29476244 T_{5} + 48900601$$ T5^9 - 33*T5^8 - 8*T5^7 + 8831*T5^6 - 66645*T5^5 - 239411*T5^4 + 3228609*T5^3 - 2833568*T5^2 - 29476244*T5 + 48900601

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - 6 T^{8} - 40 T^{7} + \cdots - 1016$$
$3$ $$(T - 3)^{9}$$
$5$ $$T^{9} - 33 T^{8} - 8 T^{7} + \cdots + 48900601$$
$7$ $$T^{9} - 83 T^{8} + \cdots + 27017466139$$
$11$ $$T^{9} - 85 T^{8} + \cdots + 1288234570811$$
$13$ $$T^{9}$$
$17$ $$T^{9} - 178 T^{8} + \cdots - 25\!\cdots\!76$$
$19$ $$T^{9} - 352 T^{8} + \cdots - 17\!\cdots\!76$$
$23$ $$T^{9} - 150 T^{8} + \cdots + 60\!\cdots\!44$$
$29$ $$T^{9} + 97 T^{8} + \cdots + 34\!\cdots\!27$$
$31$ $$T^{9} - 717 T^{8} + \cdots - 40\!\cdots\!53$$
$37$ $$T^{9} - 1108 T^{8} + \cdots - 80\!\cdots\!56$$
$41$ $$T^{9} - 334 T^{8} + \cdots + 36\!\cdots\!68$$
$43$ $$T^{9} - 242 T^{8} + \cdots + 31\!\cdots\!88$$
$47$ $$T^{9} + 184 T^{8} + \cdots + 38\!\cdots\!12$$
$53$ $$T^{9} + 151 T^{8} + \cdots + 65\!\cdots\!51$$
$59$ $$T^{9} - 537 T^{8} + \cdots - 14\!\cdots\!76$$
$61$ $$T^{9} + 1340 T^{8} + \cdots - 61\!\cdots\!68$$
$67$ $$T^{9} - 2308 T^{8} + \cdots + 81\!\cdots\!92$$
$71$ $$T^{9} - 96 T^{8} + \cdots - 71\!\cdots\!12$$
$73$ $$T^{9} - 2505 T^{8} + \cdots + 19\!\cdots\!77$$
$79$ $$T^{9} + 1591 T^{8} + \cdots + 11\!\cdots\!83$$
$83$ $$T^{9} - 1539 T^{8} + \cdots - 20\!\cdots\!97$$
$89$ $$T^{9} + 592 T^{8} + \cdots + 55\!\cdots\!72$$
$97$ $$T^{9} - 1445 T^{8} + \cdots + 31\!\cdots\!39$$