# Properties

 Label 1617.4.a.w Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1617.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: $$95.4060884793$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 4 x^{9} - 45 x^{8} + 160 x^{7} + 661 x^{6} - 1934 x^{5} - 3519 x^{4} + 6710 x^{3} + 6802 x^{2} + \cdots - 2880$$ x^10 - 4*x^9 - 45*x^8 + 160*x^7 + 661*x^6 - 1934*x^5 - 3519*x^4 + 6710*x^3 + 6802*x^2 - 3264*x - 2880 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{3} + 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q - b1 * q^2 - 3 * q^3 + (b2 + b1 + 2) * q^4 + (-b3 + 2) * q^5 + 3*b1 * q^6 + (b5 - b4 - b3 - b2 - 3*b1 - 5) * q^8 + 9 * q^9 $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{3} + 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{8}+ \cdots - 99 q^{99}+O(q^{100})$$ q - b1 * q^2 - 3 * q^3 + (b2 + b1 + 2) * q^4 + (-b3 + 2) * q^5 + 3*b1 * q^6 + (b5 - b4 - b3 - b2 - 3*b1 - 5) * q^8 + 9 * q^9 + (b9 + b8 + b7 + 3*b5 - b4 - b3 + b2 - 4*b1 - 4) * q^10 - 11 * q^11 + (-3*b2 - 3*b1 - 6) * q^12 + (-b9 - b8 + b7 + b5 + b2 - 6*b1 - 3) * q^13 + (3*b3 - 6) * q^15 + (b9 + b8 - b6 - b5 + 2*b4 - b3 + b2 + 7*b1 + 5) * q^16 + (-3*b9 - 3*b8 - b7 - b6 - 2*b5 - b4 + 3*b3 - 4*b2 + 7*b1 + 7) * q^17 - 9*b1 * q^18 + (2*b9 - b8 + b6 - 4*b5 + 2*b3 - 2*b2 - 2*b1 + 12) * q^19 + (2*b8 - 4*b7 - 4*b5 + 4*b4 + 3*b3 + 8*b2 + 8*b1 + 16) * q^20 + 11*b1 * q^22 + (-b8 + 2*b7 - b5 + 2*b4 - 5*b2 + 3*b1 - 18) * q^23 + (-3*b5 + 3*b4 + 3*b3 + 3*b2 + 9*b1 + 15) * q^24 + (-6*b9 - 3*b8 - 6*b7 + 4*b6 - 7*b5 + 3*b4 + 3*b3 - 5*b2 + 21*b1 + 33) * q^25 + (b9 + 5*b8 - 5*b7 + 2*b6 - 6*b5 + 2*b4 + b3 + 6*b2 + 2*b1 + 69) * q^26 - 27 * q^27 + (7*b9 + 2*b8 + 8*b7 + 3*b5 - 5*b4 - 5*b3 + 2*b2 - 4*b1 - 36) * q^29 + (-3*b9 - 3*b8 - 3*b7 - 9*b5 + 3*b4 + 3*b3 - 3*b2 + 12*b1 + 12) * q^30 + (6*b9 + 8*b8 - 2*b7 - 2*b6 - 2*b5 - b4 + 4*b3 + 4*b2 - 3) * q^31 + (3*b7 + 3*b6 + 3*b5 + 2*b4 + 6*b3 - 10*b2 + 7*b1 - 40) * q^32 + 33 * q^33 + (-12*b5 + 2*b4 + 12*b3 - 3*b2 + 9*b1 - 60) * q^34 + (9*b2 + 9*b1 + 18) * q^36 + (-11*b9 - 3*b6 + 3*b4 - 2*b3 - 9*b2 - 21*b1 - 8) * q^37 + (-b9 - 10*b8 + 7*b7 - 4*b6 - 3*b5 - 5*b4 + 6*b3 - 10*b2 + 23) * q^38 + (3*b9 + 3*b8 - 3*b7 - 3*b5 - 3*b2 + 18*b1 + 9) * q^39 + (-13*b9 - 15*b8 - 3*b7 + 2*b6 + 3*b5 - 3*b4 + 3*b3 - 23*b2 - 52*b1 + 2) * q^40 + (-6*b9 + 5*b8 - 5*b7 + 7*b6 - 3*b5 + 14*b4 + 14*b3 + 20*b2 + 37*b1 + 103) * q^41 + (12*b9 + b8 - 5*b7 - 8*b6 + 5*b4 + 3*b3 + 9*b2 + 14*b1 - 71) * q^43 + (-11*b2 - 11*b1 - 22) * q^44 + (-9*b3 + 18) * q^45 + (b9 + 4*b8 - 3*b7 + 4*b6 - 9*b5 + 3*b4 + 9*b3 - 19*b2 + 56*b1 - 55) * q^46 + (7*b9 - 4*b8 - 4*b7 - 7*b6 + 8*b5 + 2*b4 + 7*b3 + 9*b2 - 69*b1 + 63) * q^47 + (-3*b9 - 3*b8 + 3*b6 + 3*b5 - 6*b4 + 3*b3 - 3*b2 - 21*b1 - 15) * q^48 + (b8 + 2*b7 - 2*b6 + 11*b5 + 7*b4 + 17*b3 - 3*b2 - 40*b1 - 170) * q^50 + (9*b9 + 9*b8 + 3*b7 + 3*b6 + 6*b5 + 3*b4 - 9*b3 + 12*b2 - 21*b1 - 21) * q^51 + (2*b9 - 12*b8 + 13*b7 - 9*b6 + 32*b5 - 17*b4 - 13*b3 - 7*b2 - 86*b1 + 37) * q^52 + (3*b9 - 9*b8 - 4*b7 - 9*b6 + 3*b5 + 4*b4 + 3*b3 + 48*b1 + 27) * q^53 + 27*b1 * q^54 + (11*b3 - 22) * q^55 + (-6*b9 + 3*b8 - 3*b6 + 12*b5 - 6*b3 + 6*b2 + 6*b1 - 36) * q^57 + (3*b9 - 10*b8 + 10*b7 - 9*b6 - 14*b5 - 13*b4 - 16*b3 - 16*b2 + 22*b1 - 3) * q^58 + (-5*b9 + 18*b8 - 8*b7 + 7*b6 - 10*b5 + 12*b4 + 34*b3 - 27*b2 + 27*b1 + 3) * q^59 + (-6*b8 + 12*b7 + 12*b5 - 12*b4 - 9*b3 - 24*b2 - 24*b1 - 48) * q^60 + (-3*b9 + 21*b8 - b7 + 12*b6 + 5*b5 + 7*b4 + 11*b3 + 3*b2 - 76*b1 + 18) * q^61 + (-12*b9 - 32*b8 + 22*b7 - 10*b6 + 19*b5 - 17*b4 - 17*b3 - 13*b2 - 18*b1 - 9) * q^62 + (-14*b9 - 4*b8 - 22*b7 + 12*b6 - 36*b5 + 14*b4 + 26*b3 - 27*b2 + 81*b1 - 132) * q^64 + (6*b9 + 16*b8 + 2*b7 + 4*b6 + 24*b5 - 12*b4 - 6*b3 - 14*b2 - 96*b1 - 120) * q^65 - 33*b1 * q^66 + (-8*b9 + 13*b8 - 8*b7 - 4*b6 + 9*b5 - 2*b4 + 5*b3 - 15*b2 + 43*b1 - 79) * q^67 + (12*b9 - 8*b8 + 28*b7 + 3*b5 - 11*b4 - 7*b3 - 19*b2 + b1 - 103) * q^68 + (3*b8 - 6*b7 + 3*b5 - 6*b4 + 15*b2 - 9*b1 + 54) * q^69 + (-2*b9 + 7*b8 + 8*b7 + 10*b6 - 19*b5 - 8*b4 - 9*b3 + 15*b2 + 41*b1 - 78) * q^71 + (9*b5 - 9*b4 - 9*b3 - 9*b2 - 27*b1 - 45) * q^72 + (15*b9 - b8 + 17*b7 - 2*b6 + 35*b5 - 18*b4 - 24*b3 - 27*b2 + 42*b1 - 17) * q^73 + (2*b9 + 19*b8 - b7 + 9*b6 + 3*b5 + 4*b4 + 10*b3 + 30*b2 + 76*b1 + 175) * q^74 + (18*b9 + 9*b8 + 18*b7 - 12*b6 + 21*b5 - 9*b4 - 9*b3 + 15*b2 - 63*b1 - 99) * q^75 + (-12*b9 + 7*b8 - 3*b7 - 7*b6 - 32*b5 - 3*b4 + 15*b3 - 15*b2 + 62*b1 - 104) * q^76 + (-3*b9 - 15*b8 + 15*b7 - 6*b6 + 18*b5 - 6*b4 - 3*b3 - 18*b2 - 6*b1 - 207) * q^78 + (16*b9 - 18*b8 + 3*b7 - 3*b6 - 2*b5 - 2*b4 - 16*b3 - 41*b2 - 18*b1 - 186) * q^79 + (12*b9 + 24*b8 - 6*b7 + 10*b6 - 32*b5 + 22*b4 + 29*b3 + 42*b2 + 156*b1 + 384) * q^80 + 81 * q^81 + (-19*b9 + 4*b8 - 30*b7 + 13*b6 + 16*b5 + 13*b4 - 2*b3 - 53*b2 - 267*b1 - 189) * q^82 + (52*b9 + 37*b8 + 10*b7 + 4*b6 + 49*b5 - 11*b4 - 37*b3 + 61*b2 + 43*b1 + 132) * q^83 + (-23*b9 - 27*b8 - 3*b7 + 4*b6 - 9*b5 - 29*b4 - 41*b3 - 63*b2 + 88*b1 - 288) * q^85 + (-4*b9 - 7*b8 - 3*b7 + 17*b6 + 30*b5 - 13*b4 - 3*b3 - 52*b2 + 54*b1 - 154) * q^86 + (-21*b9 - 6*b8 - 24*b7 - 9*b5 + 15*b4 + 15*b3 - 6*b2 + 12*b1 + 108) * q^87 + (-11*b5 + 11*b4 + 11*b3 + 11*b2 + 33*b1 + 55) * q^88 + (10*b9 + 40*b8 + 26*b7 - 10*b6 + 32*b5 - 10*b4 - 28*b3 + 52*b2 - 2*b1 + 82) * q^89 + (9*b9 + 9*b8 + 9*b7 + 27*b5 - 9*b4 - 9*b3 + 9*b2 - 36*b1 - 36) * q^90 + (-13*b9 - 22*b8 - 11*b6 - 4*b5 - 5*b4 + 23*b3 - 34*b2 + 116*b1 - 465) * q^92 + (-18*b9 - 24*b8 + 6*b7 + 6*b6 + 6*b5 + 3*b4 - 12*b3 - 12*b2 + 9) * q^93 + (-3*b9 + 14*b8 - 36*b7 + 23*b6 - 15*b5 + 18*b4 + 8*b3 + 60*b2 + 22*b1 + 720) * q^94 + (36*b9 - 2*b8 + 25*b7 - 19*b6 + 16*b5 - 17*b4 - 41*b3 + 9*b2 + 124*b1 - 161) * q^95 + (-9*b7 - 9*b6 - 9*b5 - 6*b4 - 18*b3 + 30*b2 - 21*b1 + 120) * q^96 + (14*b9 + 2*b8 - 4*b7 - 10*b6 + 54*b5 - 13*b4 - 38*b3 + 4*b2 + 94*b1 - 84) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 4 q^{2} - 30 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} - 60 q^{8} + 90 q^{9}+O(q^{10})$$ 10 * q - 4 * q^2 - 30 * q^3 + 26 * q^4 + 20 * q^5 + 12 * q^6 - 60 * q^8 + 90 * q^9 $$10 q - 4 q^{2} - 30 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} - 60 q^{8} + 90 q^{9} - 38 q^{10} - 110 q^{11} - 78 q^{12} - 44 q^{13} - 60 q^{15} + 78 q^{16} + 80 q^{17} - 36 q^{18} + 78 q^{19} + 184 q^{20} + 44 q^{22} - 178 q^{23} + 180 q^{24} + 356 q^{25} + 678 q^{26} - 270 q^{27} - 348 q^{29} + 114 q^{30} - 26 q^{31} - 374 q^{32} + 330 q^{33} - 618 q^{34} + 234 q^{36} - 132 q^{37} + 198 q^{38} + 132 q^{39} - 246 q^{40} + 1216 q^{41} - 684 q^{43} - 286 q^{44} + 180 q^{45} - 408 q^{46} + 358 q^{47} - 234 q^{48} - 1806 q^{50} - 240 q^{51} + 154 q^{52} + 428 q^{53} + 108 q^{54} - 220 q^{55} - 234 q^{57} - 24 q^{58} + 90 q^{59} - 552 q^{60} - 30 q^{61} - 84 q^{62} - 1266 q^{64} - 1476 q^{65} - 132 q^{66} - 552 q^{67} - 1020 q^{68} + 534 q^{69} - 614 q^{71} - 540 q^{72} + 92 q^{73} + 2172 q^{74} - 1068 q^{75} - 872 q^{76} - 2034 q^{78} - 2140 q^{79} + 4424 q^{80} + 810 q^{81} - 3054 q^{82} + 1782 q^{83} - 2726 q^{85} - 1366 q^{86} + 1044 q^{87} + 660 q^{88} + 1288 q^{89} - 342 q^{90} - 4284 q^{92} + 78 q^{93} + 7226 q^{94} - 1046 q^{95} + 1122 q^{96} - 284 q^{97} - 990 q^{99}+O(q^{100})$$ 10 * q - 4 * q^2 - 30 * q^3 + 26 * q^4 + 20 * q^5 + 12 * q^6 - 60 * q^8 + 90 * q^9 - 38 * q^10 - 110 * q^11 - 78 * q^12 - 44 * q^13 - 60 * q^15 + 78 * q^16 + 80 * q^17 - 36 * q^18 + 78 * q^19 + 184 * q^20 + 44 * q^22 - 178 * q^23 + 180 * q^24 + 356 * q^25 + 678 * q^26 - 270 * q^27 - 348 * q^29 + 114 * q^30 - 26 * q^31 - 374 * q^32 + 330 * q^33 - 618 * q^34 + 234 * q^36 - 132 * q^37 + 198 * q^38 + 132 * q^39 - 246 * q^40 + 1216 * q^41 - 684 * q^43 - 286 * q^44 + 180 * q^45 - 408 * q^46 + 358 * q^47 - 234 * q^48 - 1806 * q^50 - 240 * q^51 + 154 * q^52 + 428 * q^53 + 108 * q^54 - 220 * q^55 - 234 * q^57 - 24 * q^58 + 90 * q^59 - 552 * q^60 - 30 * q^61 - 84 * q^62 - 1266 * q^64 - 1476 * q^65 - 132 * q^66 - 552 * q^67 - 1020 * q^68 + 534 * q^69 - 614 * q^71 - 540 * q^72 + 92 * q^73 + 2172 * q^74 - 1068 * q^75 - 872 * q^76 - 2034 * q^78 - 2140 * q^79 + 4424 * q^80 + 810 * q^81 - 3054 * q^82 + 1782 * q^83 - 2726 * q^85 - 1366 * q^86 + 1044 * q^87 + 660 * q^88 + 1288 * q^89 - 342 * q^90 - 4284 * q^92 + 78 * q^93 + 7226 * q^94 - 1046 * q^95 + 1122 * q^96 - 284 * q^97 - 990 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 45 x^{8} + 160 x^{7} + 661 x^{6} - 1934 x^{5} - 3519 x^{4} + 6710 x^{3} + 6802 x^{2} + \cdots - 2880$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 10$$ v^2 - v - 10 $$\beta_{3}$$ $$=$$ $$( 403 \nu^{9} - 1681 \nu^{8} - 17472 \nu^{7} + 66496 \nu^{6} + 237199 \nu^{5} - 793331 \nu^{4} + \cdots - 1301952 ) / 36288$$ (403*v^9 - 1681*v^8 - 17472*v^7 + 66496*v^6 + 237199*v^5 - 793331*v^4 - 997800*v^3 + 2721338*v^2 + 763840*v - 1301952) / 36288 $$\beta_{4}$$ $$=$$ $$( 59 \nu^{9} - 293 \nu^{8} - 2436 \nu^{7} + 11960 \nu^{6} + 29879 \nu^{5} - 147175 \nu^{4} + \cdots - 217104 ) / 3024$$ (59*v^9 - 293*v^8 - 2436*v^7 + 11960*v^6 + 29879*v^5 - 147175*v^4 - 92100*v^3 + 507202*v^2 - 18088*v - 217104) / 3024 $$\beta_{5}$$ $$=$$ $$( 1111 \nu^{9} - 5197 \nu^{8} - 46704 \nu^{7} + 210016 \nu^{6} + 595747 \nu^{5} - 2559431 \nu^{4} + \cdots - 4088640 ) / 36288$$ (1111*v^9 - 5197*v^8 - 46704*v^7 + 210016*v^6 + 595747*v^5 - 2559431*v^4 - 2139288*v^3 + 8844050*v^2 + 1199968*v - 4088640) / 36288 $$\beta_{6}$$ $$=$$ $$( 1633 \nu^{9} - 7879 \nu^{8} - 67116 \nu^{7} + 317728 \nu^{6} + 821557 \nu^{5} - 3879533 \nu^{4} + \cdots - 7014144 ) / 18144$$ (1633*v^9 - 7879*v^8 - 67116*v^7 + 317728*v^6 + 821557*v^5 - 3879533*v^4 - 2598396*v^3 + 13622366*v^2 + 295288*v - 7014144) / 18144 $$\beta_{7}$$ $$=$$ $$( - 1885 \nu^{9} + 8887 \nu^{8} + 78456 \nu^{7} - 358048 \nu^{6} - 988129 \nu^{5} + 4360853 \nu^{4} + \cdots + 7243968 ) / 12096$$ (-1885*v^9 + 8887*v^8 + 78456*v^7 - 358048*v^6 - 988129*v^5 + 4360853*v^4 + 3485184*v^3 - 15156038*v^2 - 1900528*v + 7243968) / 12096 $$\beta_{8}$$ $$=$$ $$( - 1615 \nu^{9} + 7717 \nu^{8} + 66864 \nu^{7} - 310816 \nu^{6} - 834139 \nu^{5} + 3784655 \nu^{4} + \cdots + 6368736 ) / 6048$$ (-1615*v^9 + 7717*v^8 + 66864*v^7 - 310816*v^6 - 834139*v^5 + 3784655*v^4 + 2850936*v^3 - 13161314*v^2 - 1187872*v + 6368736) / 6048 $$\beta_{9}$$ $$=$$ $$( 6527 \nu^{9} - 30953 \nu^{8} - 270564 \nu^{7} + 1244912 \nu^{6} + 3381899 \nu^{5} - 15125635 \nu^{4} + \cdots - 24958464 ) / 18144$$ (6527*v^9 - 30953*v^8 - 270564*v^7 + 1244912*v^6 + 3381899*v^5 - 15125635*v^4 - 11614548*v^3 + 52348978*v^2 + 4949000*v - 24958464) / 18144
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 10$$ b2 + b1 + 10 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 19\beta _1 + 5$$ -b5 + b4 + b3 + b2 + 19*b1 + 5 $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 25\beta_{2} + 31\beta _1 + 181$$ b9 + b8 - b6 - b5 + 2*b4 - b3 + 25*b2 + 31*b1 + 181 $$\nu^{5}$$ $$=$$ $$-3\beta_{7} - 3\beta_{6} - 35\beta_{5} + 30\beta_{4} + 26\beta_{3} + 42\beta_{2} + 409\beta _1 + 200$$ -3*b7 - 3*b6 - 35*b5 + 30*b4 + 26*b3 + 42*b2 + 409*b1 + 200 $$\nu^{6}$$ $$=$$ $$26 \beta_{9} + 36 \beta_{8} - 22 \beta_{7} - 28 \beta_{6} - 76 \beta_{5} + 94 \beta_{4} - 14 \beta_{3} + \cdots + 3780$$ 26*b9 + 36*b8 - 22*b7 - 28*b6 - 76*b5 + 94*b4 - 14*b3 + 589*b2 + 937*b1 + 3780 $$\nu^{7}$$ $$=$$ $$22 \beta_{9} + 48 \beta_{8} - 154 \beta_{7} - 104 \beta_{6} - 1043 \beta_{5} + 807 \beta_{4} + \cdots + 6775$$ 22*b9 + 48*b8 - 154*b7 - 104*b6 - 1043*b5 + 807*b4 + 581*b3 + 1377*b2 + 9387*b1 + 6775 $$\nu^{8}$$ $$=$$ $$629 \beta_{9} + 1171 \beta_{8} - 1134 \beta_{7} - 627 \beta_{6} - 3189 \beta_{5} + 3220 \beta_{4} + \cdots + 84897$$ 629*b9 + 1171*b8 - 1134*b7 - 627*b6 - 3189*b5 + 3220*b4 + 85*b3 + 14111*b2 + 27491*b1 + 84897 $$\nu^{9}$$ $$=$$ $$1256 \beta_{9} + 2994 \beta_{8} - 6011 \beta_{7} - 2707 \beta_{6} - 29825 \beta_{5} + 21664 \beta_{4} + \cdots + 210820$$ 1256*b9 + 2994*b8 - 6011*b7 - 2707*b6 - 29825*b5 + 21664*b4 + 13148*b3 + 41590*b2 + 225725*b1 + 210820

## 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
 5.25538 4.48887 3.37565 2.21027 0.760491 −0.737904 −0.758345 −2.17274 −4.02407 −4.39760
−5.25538 −3.00000 19.6190 17.3326 15.7661 0 −61.0623 9.00000 −91.0895
1.2 −4.48887 −3.00000 12.1499 −8.63270 13.4666 0 −18.6285 9.00000 38.7511
1.3 −3.37565 −3.00000 3.39502 −13.9352 10.1270 0 15.5448 9.00000 47.0402
1.4 −2.21027 −3.00000 −3.11472 21.6829 6.63080 0 24.5665 9.00000 −47.9251
1.5 −0.760491 −3.00000 −7.42165 −4.03718 2.28147 0 11.7280 9.00000 3.07024
1.6 0.737904 −3.00000 −7.45550 9.09244 −2.21371 0 −11.4047 9.00000 6.70935
1.7 0.758345 −3.00000 −7.42491 7.17969 −2.27503 0 −11.6974 9.00000 5.44468
1.8 2.17274 −3.00000 −3.27921 −16.5779 −6.51821 0 −24.5068 9.00000 −36.0195
1.9 4.02407 −3.00000 8.19314 −3.47545 −12.0722 0 0.777189 9.00000 −13.9854
1.10 4.39760 −3.00000 11.3389 11.3707 −13.1928 0 14.6831 9.00000 50.0039
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## 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 1617.4.a.w 10
7.b odd 2 1 1617.4.a.x 10
7.d odd 6 2 231.4.i.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.c 20 7.d odd 6 2
1617.4.a.w 10 1.a even 1 1 trivial
1617.4.a.x 10 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{10} + 4 T_{2}^{9} - 45 T_{2}^{8} - 160 T_{2}^{7} + 661 T_{2}^{6} + 1934 T_{2}^{5} - 3519 T_{2}^{4} + \cdots - 2880$$ T2^10 + 4*T2^9 - 45*T2^8 - 160*T2^7 + 661*T2^6 + 1934*T2^5 - 3519*T2^4 - 6710*T2^3 + 6802*T2^2 + 3264*T2 - 2880 $$T_{5}^{10} - 20 T_{5}^{9} - 603 T_{5}^{8} + 11400 T_{5}^{7} + 120828 T_{5}^{6} - 2074836 T_{5}^{5} + \cdots - 7806081024$$ T5^10 - 20*T5^9 - 603*T5^8 + 11400*T5^7 + 120828*T5^6 - 2074836*T5^5 - 9550876*T5^4 + 135689984*T5^3 + 366290496*T5^2 - 2672248320*T5 - 7806081024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 4 T^{9} + \cdots - 2880$$
$3$ $$(T + 3)^{10}$$
$5$ $$T^{10} + \cdots - 7806081024$$
$7$ $$T^{10}$$
$11$ $$(T + 11)^{10}$$
$13$ $$T^{10} + \cdots - 17\!\cdots\!48$$
$17$ $$T^{10} + \cdots + 10\!\cdots\!72$$
$19$ $$T^{10} + \cdots - 47\!\cdots\!36$$
$23$ $$T^{10} + \cdots + 87\!\cdots\!44$$
$29$ $$T^{10} + \cdots - 15\!\cdots\!80$$
$31$ $$T^{10} + \cdots + 12\!\cdots\!68$$
$37$ $$T^{10} + \cdots + 24\!\cdots\!56$$
$41$ $$T^{10} + \cdots + 36\!\cdots\!48$$
$43$ $$T^{10} + \cdots - 28\!\cdots\!00$$
$47$ $$T^{10} + \cdots + 23\!\cdots\!52$$
$53$ $$T^{10} + \cdots + 21\!\cdots\!88$$
$59$ $$T^{10} + \cdots + 17\!\cdots\!00$$
$61$ $$T^{10} + \cdots - 31\!\cdots\!48$$
$67$ $$T^{10} + \cdots + 37\!\cdots\!84$$
$71$ $$T^{10} + \cdots - 56\!\cdots\!24$$
$73$ $$T^{10} + \cdots + 46\!\cdots\!56$$
$79$ $$T^{10} + \cdots - 67\!\cdots\!40$$
$83$ $$T^{10} + \cdots - 10\!\cdots\!92$$
$89$ $$T^{10} + \cdots - 53\!\cdots\!12$$
$97$ $$T^{10} + \cdots + 26\!\cdots\!72$$