# Properties

 Label 1617.4.a.bc Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $0$ Dimension $16$ 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: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199466 x^{9} + \cdots - 738304$$ x^16 - 4*x^15 - 92*x^14 + 346*x^13 + 3385*x^12 - 11756*x^11 - 63875*x^10 + 199466*x^9 + 657606*x^8 - 1772388*x^7 - 3590705*x^6 + 7831536*x^5 + 8898560*x^4 - 14389664*x^3 - 4854640*x^2 + 6896896*x - 738304 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 7^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{5} - \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} + \cdots - 4) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 5) * q^4 + (b5 - b1) * q^5 + 3*b1 * q^6 + (-b6 + b5 - b2 - 4*b1 - 4) * q^8 + 9 * q^9 $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{5} - \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} + \cdots - 4) q^{8}+ \cdots - 99 q^{99}+O(q^{100})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 5) * q^4 + (b5 - b1) * q^5 + 3*b1 * q^6 + (-b6 + b5 - b2 - 4*b1 - 4) * q^8 + 9 * q^9 + (-b9 - b7 - b5 + 2*b2 - b1 + 13) * q^10 - 11 * q^11 + (-3*b2 - 15) * q^12 + (b13 + b11 - b6 - 2*b5 + b4 + 2*b3 + 7) * q^13 + (-3*b5 + 3*b1) * q^15 + (b15 + b13 + 3*b12 - 2*b11 + 2*b10 - 2*b9 - b8 + 3*b6 - b5 - b4 + 6*b2 + 5*b1 + 16) * q^16 + (b15 + b14 - b13 + b12 - b10 - b8 - b7 - b5 - 2*b3 + 3*b2 - 5*b1 + 17) * q^17 - 9*b1 * q^18 + (-3*b15 + b10 - b9 - b8 + b4 - 2*b3 + 4*b2 - 2*b1 + 13) * q^19 + (2*b15 + 2*b12 - 2*b11 + 3*b10 - b8 + b7 + 3*b5 - 2*b4 + b3 + b2 - 18*b1 + 15) * q^20 + 11*b1 * q^22 + (2*b14 + b13 - 3*b12 + 3*b11 - b10 - 2*b8 - 4*b7 - b6 + 4*b4 + b3 + b1 - 10) * q^23 + (3*b6 - 3*b5 + 3*b2 + 12*b1 + 12) * q^24 + (3*b14 - b13 + b11 - 3*b10 - 3*b9 + b8 - 2*b7 - 2*b6 + b5 + 2*b4 + 2*b3 - 2*b2 + 5*b1 + 16) * q^25 + (4*b15 - b14 + 2*b13 + b12 - 5*b11 - 2*b10 + 3*b9 + 4*b8 + 5*b7 - b6 + b5 - b4 + 4*b2 - 12*b1) * q^26 - 27 * q^27 + (-b15 - 4*b14 + 2*b13 - b12 + 6*b11 - 4*b7 - b6 + 2*b5 + b4 - b3 - 5*b2 + 13*b1 - 42) * q^29 + (3*b9 + 3*b7 + 3*b5 - 6*b2 + 3*b1 - 39) * q^30 + (-b15 + b14 + b13 - 4*b12 + 3*b11 - 2*b10 + b9 + b8 - 2*b7 - 3*b6 - 2*b4 + 2*b3 - 3*b2 - 4*b1 + 24) * q^31 + (-2*b15 - 4*b14 - 3*b13 - 3*b12 - 2*b10 - b9 + 5*b8 - 6*b6 + 5*b5 + 4*b3 - 12*b2 - 6*b1 - 44) * q^32 + 33 * q^33 + (-2*b14 - 2*b12 + b11 + 4*b10 - 4*b9 - b7 - 3*b6 + 8*b5 + 2*b2 - 35*b1 + 49) * q^34 + (9*b2 + 45) * q^36 + (-5*b15 - b13 + 3*b12 - 2*b11 + 6*b10 - 3*b9 - 4*b8 - b7 + 3*b6 + 5*b5 - 5*b4 - b3 + 11*b2 + 15*b1 + 13) * q^37 + (2*b15 - b14 - 2*b13 + 5*b12 - 9*b11 + b10 + 3*b9 - b8 + 10*b7 - 2*b6 + 13*b5 - 7*b4 + 3*b3 + 4*b2 - 47*b1 + 23) * q^38 + (-3*b13 - 3*b11 + 3*b6 + 6*b5 - 3*b4 - 6*b3 - 21) * q^39 + (-5*b15 - 3*b13 + 3*b12 + 2*b11 - b10 + 2*b8 - 3*b7 - 4*b6 - 3*b5 - b4 - 7*b3 + 17*b2 - 10*b1 + 124) * q^40 + (-6*b15 - 4*b13 - 2*b12 + 7*b11 + b9 - 4*b8 - b7 - 3*b6 + 2*b5 + 3*b4 - 5*b3 - 6*b2 - 13*b1 + 8) * q^41 + (-4*b15 - 2*b14 + b13 + 2*b12 + 4*b11 - 2*b10 - b9 - 4*b8 + 5*b7 + 5*b5 + 4*b4 + 2*b3 + 6*b2 + 6*b1 + 19) * q^43 + (-11*b2 - 55) * q^44 + (9*b5 - 9*b1) * q^45 + (11*b15 + 3*b14 + b13 - 4*b12 - 9*b11 - 5*b10 + 5*b9 + 11*b8 + b7 - b6 + 9*b5 - 8*b4 - b3 + 4*b2 - 4*b1 - 16) * q^46 + (2*b15 - 3*b14 - 2*b13 - 6*b12 + 2*b11 - 3*b10 - b9 + 3*b8 - 8*b7 - 5*b6 + 6*b5 + 3*b4 + 2*b3 + 2*b2 - 4*b1 + 92) * q^47 + (-3*b15 - 3*b13 - 9*b12 + 6*b11 - 6*b10 + 6*b9 + 3*b8 - 9*b6 + 3*b5 + 3*b4 - 18*b2 - 15*b1 - 48) * q^48 + (9*b15 + 2*b13 + 4*b12 - 9*b11 + 8*b10 - 4*b9 - 6*b8 + 6*b7 + 6*b6 + 25*b5 - 7*b4 + 6*b3 + 3*b2 - 11*b1 - 55) * q^50 + (-3*b15 - 3*b14 + 3*b13 - 3*b12 + 3*b10 + 3*b8 + 3*b7 + 3*b5 + 6*b3 - 9*b2 + 15*b1 - 51) * q^51 + (-10*b15 - 7*b14 + 5*b13 + 2*b12 + 8*b11 - b10 - 4*b8 - b7 - b6 - 10*b5 + 11*b4 + 15*b3 + 8*b2 - 19*b1 + 34) * q^52 + (-3*b15 + 12*b14 - 8*b12 + 4*b11 - 11*b10 + 9*b9 + 7*b8 + 2*b7 - 10*b6 + 7*b5 + 5*b4 - 9*b3 + 8*b2 - 35*b1 - 85) * q^53 + 27*b1 * q^54 + (-11*b5 + 11*b1) * q^55 + (9*b15 - 3*b10 + 3*b9 + 3*b8 - 3*b4 + 6*b3 - 12*b2 + 6*b1 - 39) * q^57 + (15*b15 - 4*b14 + 5*b13 + b12 - 7*b11 + 5*b10 - 4*b9 + 6*b8 - 4*b7 + 14*b6 - 4*b5 - b4 - 9*b3 + 6*b2 + 58*b1 - 99) * q^58 + (b15 + 3*b13 + 17*b11 - 5*b10 - b9 + 5*b8 - 14*b7 + b6 - 3*b5 + 8*b4 + 4*b3 + 10*b2 - 35*b1 + 100) * q^59 + (-6*b15 - 6*b12 + 6*b11 - 9*b10 + 3*b8 - 3*b7 - 9*b5 + 6*b4 - 3*b3 - 3*b2 + 54*b1 - 45) * q^60 + (17*b15 + 11*b14 + 9*b13 - 11*b12 - 5*b11 - 7*b10 - 6*b9 + 2*b8 - 2*b7 + 4*b6 - 6*b5 + 12*b4 + 4*b3 + 20*b2 - 3*b1 + 189) * q^61 + (11*b15 + 20*b14 - 10*b12 - 2*b11 - 2*b10 + 7*b9 - 3*b8 - b7 + 8*b6 + 16*b5 + b4 - 16*b3 + 5*b2 - 31*b1 + 49) * q^62 + (-8*b15 - 6*b14 + 2*b13 + 12*b12 + 8*b11 + 13*b10 + 3*b9 - 15*b8 - 2*b7 + 21*b6 - 5*b5 + 4*b4 + 11*b3 + 21*b2 + 60*b1 + 45) * q^64 + (-8*b15 - 9*b14 + 2*b13 + 8*b12 - 5*b11 + 9*b10 + 8*b9 - b8 + 3*b7 + 8*b6 + 8*b5 - 12*b3 + 10*b2 - 11*b1 - 252) * q^65 - 33*b1 * q^66 + (-9*b15 - 4*b14 - 6*b13 + 2*b12 - 3*b11 + b10 - 6*b9 + b8 + 7*b7 + 3*b6 + 10*b5 + 10*b4 - 6*b3 + 10*b2 + 5*b1 + 79) * q^67 + (-6*b15 - 2*b14 + 3*b13 + 17*b12 - 3*b11 + 11*b10 - 10*b9 - 4*b8 - 5*b7 + 9*b6 + 11*b5 - 14*b4 - 13*b3 + 55*b2 - 45*b1 + 351) * q^68 + (-6*b14 - 3*b13 + 9*b12 - 9*b11 + 3*b10 + 6*b8 + 12*b7 + 3*b6 - 12*b4 - 3*b3 - 3*b1 + 30) * q^69 + (10*b14 - 5*b13 + 2*b12 + 2*b11 - 10*b10 + 12*b9 - 17*b8 + 7*b7 - 11*b6 + 6*b5 - b4 + 4*b3 + b2 - 55*b1 - 113) * q^71 + (-9*b6 + 9*b5 - 9*b2 - 36*b1 - 36) * q^72 + (8*b15 + 5*b14 - b13 - 9*b12 - 15*b11 - 4*b10 + 9*b9 + b8 + 12*b7 - 6*b6 - 20*b5 + b4 + 36*b3 + 6*b2 - 65*b1 + 116) * q^73 + (-b15 + 9*b14 - 16*b13 - b12 - 16*b11 + 4*b10 - 6*b9 + b8 - b7 - 10*b6 + 37*b5 - 14*b4 - 16*b3 - 17*b2 - 90*b1 - 192) * q^74 + (-9*b14 + 3*b13 - 3*b11 + 9*b10 + 9*b9 - 3*b8 + 6*b7 + 6*b6 - 3*b5 - 6*b4 - 6*b3 + 6*b2 - 15*b1 - 48) * q^75 + (-6*b15 + b14 - b13 + 12*b12 + 8*b11 - 4*b10 - 14*b9 + b8 - 8*b7 - 8*b6 - 36*b5 + 9*b4 + 8*b3 + 42*b2 - 44*b1 + 439) * q^76 + (-12*b15 + 3*b14 - 6*b13 - 3*b12 + 15*b11 + 6*b10 - 9*b9 - 12*b8 - 15*b7 + 3*b6 - 3*b5 + 3*b4 - 12*b2 + 36*b1) * q^78 + (12*b15 + 7*b14 - 2*b13 + b12 - 3*b11 - 12*b10 + 20*b9 + 5*b8 + 7*b7 - 12*b6 + 2*b5 + b4 - 10*b3 + 18*b2 - 32*b1 - 23) * q^79 + (-12*b14 + 7*b13 - 5*b12 - 4*b11 + 9*b10 - 11*b9 - 8*b8 + 5*b7 + 8*b6 + 20*b5 + 4*b4 + 23*b3 + 19*b2 - 117*b1 - 15) * q^80 + 81 * q^81 + (4*b15 + 2*b14 - 9*b13 + 7*b12 - 13*b11 + b10 - 4*b9 - 4*b8 + 7*b7 + 17*b6 - 9*b5 - 16*b4 - 39*b3 + 37*b2 - b1 + 213) * q^82 + (2*b15 + 17*b14 - 4*b13 + 7*b12 - 6*b11 - 2*b10 + 2*b9 - 4*b8 - 4*b7 + 16*b6 + b5 - 13*b4 - 18*b3 + 13*b2 + 6*b1 + 239) * q^83 + (-3*b15 - 11*b14 + 2*b13 - 3*b12 + 5*b11 + 15*b10 - 5*b9 - 8*b8 + b7 - 6*b6 + 29*b5 - 14*b4 + 19*b3 + 18*b2 - 107*b1 + 76) * q^85 + (-4*b15 - 7*b14 - 3*b13 + 4*b12 - 13*b11 - 15*b10 + 11*b8 + 25*b7 - 18*b6 - 5*b5 + 3*b4 - 33*b3 - 19*b2 - 84*b1 - 99) * q^86 + (3*b15 + 12*b14 - 6*b13 + 3*b12 - 18*b11 + 12*b7 + 3*b6 - 6*b5 - 3*b4 + 3*b3 + 15*b2 - 39*b1 + 126) * q^87 + (11*b6 - 11*b5 + 11*b2 + 44*b1 + 44) * q^88 + (-21*b15 - 5*b14 - 12*b13 - 10*b12 + 18*b11 - 10*b10 + 15*b9 - 7*b8 + 2*b7 - 6*b6 + b5 + 19*b4 - 3*b3 - 31*b2 + 9*b1 + 35) * q^89 + (-9*b9 - 9*b7 - 9*b5 + 18*b2 - 9*b1 + 117) * q^90 + (-9*b15 + 3*b14 + 8*b13 + 3*b12 + 27*b11 + 14*b10 - 7*b9 - 9*b8 - 15*b7 + 5*b6 - 20*b5 + 42*b3 - 19*b2 + 10) * q^92 + (3*b15 - 3*b14 - 3*b13 + 12*b12 - 9*b11 + 6*b10 - 3*b9 - 3*b8 + 6*b7 + 9*b6 + 6*b4 - 6*b3 + 9*b2 + 12*b1 - 72) * q^93 + (11*b15 - 7*b14 + 14*b13 + 29*b12 - 6*b11 + 32*b10 - 16*b9 - 10*b8 - 20*b7 + 33*b6 + 9*b5 - 8*b4 + 20*b3 + 59*b2 - 131*b1 + 104) * q^94 + (20*b15 + 9*b14 + 3*b13 + 21*b12 - 12*b11 + 24*b10 - 7*b9 + 2*b8 + 9*b7 + 4*b6 + 11*b5 - b4 + 25*b3 - 13*b2 - 135*b1 - 40) * q^95 + (6*b15 + 12*b14 + 9*b13 + 9*b12 + 6*b10 + 3*b9 - 15*b8 + 18*b6 - 15*b5 - 12*b3 + 36*b2 + 18*b1 + 132) * q^96 + (-8*b15 - 15*b14 + 13*b13 + 18*b12 + 4*b11 + 13*b10 + 14*b9 - 11*b8 - 5*b7 + 33*b6 - 44*b5 - 9*b4 - 42*b3 + 52*b2 - 53*b1 + 121) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{2} - 48 q^{3} + 72 q^{4} + 12 q^{6} - 66 q^{8} + 144 q^{9}+O(q^{10})$$ 16 * q - 4 * q^2 - 48 * q^3 + 72 * q^4 + 12 * q^6 - 66 * q^8 + 144 * q^9 $$16 q - 4 q^{2} - 48 q^{3} + 72 q^{4} + 12 q^{6} - 66 q^{8} + 144 q^{9} + 178 q^{10} - 176 q^{11} - 216 q^{12} + 104 q^{13} + 220 q^{16} + 220 q^{17} - 36 q^{18} + 152 q^{19} + 182 q^{20} + 44 q^{22} - 180 q^{23} + 198 q^{24} + 284 q^{25} + 10 q^{26} - 432 q^{27} - 604 q^{29} - 534 q^{30} + 380 q^{31} - 592 q^{32} + 528 q^{33} + 632 q^{34} + 648 q^{36} + 148 q^{37} + 266 q^{38} - 312 q^{39} + 1792 q^{40} + 60 q^{41} + 252 q^{43} - 792 q^{44} - 116 q^{46} + 1468 q^{47} - 660 q^{48} - 850 q^{50} - 660 q^{51} + 310 q^{52} - 1456 q^{53} + 108 q^{54} - 456 q^{57} - 1350 q^{58} + 1312 q^{59} - 546 q^{60} + 2880 q^{61} + 708 q^{62} + 630 q^{64} - 4064 q^{65} - 132 q^{66} + 1220 q^{67} + 4956 q^{68} + 540 q^{69} - 2040 q^{71} - 594 q^{72} + 1628 q^{73} - 3126 q^{74} - 852 q^{75} + 6286 q^{76} - 30 q^{78} - 416 q^{79} - 874 q^{80} + 1296 q^{81} + 3040 q^{82} + 3724 q^{83} + 628 q^{85} - 1608 q^{86} + 1812 q^{87} + 726 q^{88} + 752 q^{89} + 1602 q^{90} - 32 q^{92} - 1140 q^{93} + 610 q^{94} - 912 q^{95} + 1776 q^{96} + 1088 q^{97} - 1584 q^{99}+O(q^{100})$$ 16 * q - 4 * q^2 - 48 * q^3 + 72 * q^4 + 12 * q^6 - 66 * q^8 + 144 * q^9 + 178 * q^10 - 176 * q^11 - 216 * q^12 + 104 * q^13 + 220 * q^16 + 220 * q^17 - 36 * q^18 + 152 * q^19 + 182 * q^20 + 44 * q^22 - 180 * q^23 + 198 * q^24 + 284 * q^25 + 10 * q^26 - 432 * q^27 - 604 * q^29 - 534 * q^30 + 380 * q^31 - 592 * q^32 + 528 * q^33 + 632 * q^34 + 648 * q^36 + 148 * q^37 + 266 * q^38 - 312 * q^39 + 1792 * q^40 + 60 * q^41 + 252 * q^43 - 792 * q^44 - 116 * q^46 + 1468 * q^47 - 660 * q^48 - 850 * q^50 - 660 * q^51 + 310 * q^52 - 1456 * q^53 + 108 * q^54 - 456 * q^57 - 1350 * q^58 + 1312 * q^59 - 546 * q^60 + 2880 * q^61 + 708 * q^62 + 630 * q^64 - 4064 * q^65 - 132 * q^66 + 1220 * q^67 + 4956 * q^68 + 540 * q^69 - 2040 * q^71 - 594 * q^72 + 1628 * q^73 - 3126 * q^74 - 852 * q^75 + 6286 * q^76 - 30 * q^78 - 416 * q^79 - 874 * q^80 + 1296 * q^81 + 3040 * q^82 + 3724 * q^83 + 628 * q^85 - 1608 * q^86 + 1812 * q^87 + 726 * q^88 + 752 * q^89 + 1602 * q^90 - 32 * q^92 - 1140 * q^93 + 610 * q^94 - 912 * q^95 + 1776 * q^96 + 1088 * q^97 - 1584 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199466 x^{9} + \cdots - 738304$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 13$$ v^2 - 13 $$\beta_{3}$$ $$=$$ $$( - 24386725 \nu^{15} + 511703837 \nu^{14} + 407411307 \nu^{13} - 43182271817 \nu^{12} + \cdots + 242365506733056 ) / 13638029623296$$ (-24386725*v^15 + 511703837*v^14 + 407411307*v^13 - 43182271817*v^12 + 67314870208*v^11 + 1397967245020*v^10 - 3112536460957*v^9 - 21574672317489*v^8 + 53205221499111*v^7 + 157432044759041*v^6 - 403931289180160*v^5 - 426498880265264*v^4 + 1214562688624112*v^3 - 45430035674256*v^2 - 901101588367616*v + 242365506733056) / 13638029623296 $$\beta_{4}$$ $$=$$ $$( 8675592017 \nu^{15} - 368142774057 \nu^{14} - 191259357567 \nu^{13} + 33079986356389 \nu^{12} + \cdots - 46\!\cdots\!36 ) / 44\!\cdots\!84$$ (8675592017*v^15 - 368142774057*v^14 - 191259357567*v^13 + 33079986356389*v^12 - 13783878777280*v^11 - 1156907649087724*v^10 + 526291554762473*v^9 + 19742357820205293*v^8 - 4959747448933643*v^7 - 166814457864514557*v^6 - 9525383602399872*v^5 + 601655026268069744*v^4 + 250191101918873040*v^3 - 465407014149676464*v^2 - 304043784610198272*v - 4676886891099136) / 4486911746064384 $$\beta_{5}$$ $$=$$ $$( - 11116152549 \nu^{15} + 55793030093 \nu^{14} + 1117784822795 \nu^{13} + \cdots - 91\!\cdots\!72 ) / 44\!\cdots\!84$$ (-11116152549*v^15 + 55793030093*v^14 + 1117784822795*v^13 - 5130113718905*v^12 - 45747028666464*v^11 + 185523493180028*v^10 + 973401009255107*v^9 - 3342164481015969*v^8 - 11316526077503161*v^7 + 31278104506945361*v^6 + 67858979637404416*v^5 - 143350736084409648*v^4 - 166029245271465488*v^3 + 262228332855600752*v^2 + 39055790817482496*v - 91356723212188672) / 4486911746064384 $$\beta_{6}$$ $$=$$ $$( - 11116152549 \nu^{15} + 55793030093 \nu^{14} + 1117784822795 \nu^{13} + \cdots - 50\!\cdots\!16 ) / 44\!\cdots\!84$$ (-11116152549*v^15 + 55793030093*v^14 + 1117784822795*v^13 - 5130113718905*v^12 - 45747028666464*v^11 + 185523493180028*v^10 + 973401009255107*v^9 - 3342164481015969*v^8 - 11316526077503161*v^7 + 31278104506945361*v^6 + 67858979637404416*v^5 - 143350736084409648*v^4 - 161542333525401104*v^3 + 257741421109536368*v^2 - 50682444103805184*v - 50974517497609216) / 4486911746064384 $$\beta_{7}$$ $$=$$ $$( - 2978675587 \nu^{15} + 689934803 \nu^{14} + 270159417005 \nu^{13} + \cdots + 19\!\cdots\!96 ) / 560863968258048$$ (-2978675587*v^15 + 689934803*v^14 + 270159417005*v^13 + 142500974105*v^12 - 9773715999008*v^11 - 12696252316204*v^10 + 180335106144133*v^9 + 358630039076897*v^8 - 1783730938938223*v^7 - 4407740878916129*v^6 + 8811338000792848*v^5 + 23722932537584880*v^4 - 15858308631305712*v^3 - 46959875794172528*v^2 - 680810619956736*v + 19415640266064896) / 560863968258048 $$\beta_{8}$$ $$=$$ $$( - 26097117323 \nu^{15} - 227967094893 \nu^{14} + 3429090741733 \nu^{13} + \cdots + 13\!\cdots\!76 ) / 44\!\cdots\!84$$ (-26097117323*v^15 - 227967094893*v^14 + 3429090741733*v^13 + 20367469930073*v^12 - 168541114854464*v^11 - 711852606141340*v^10 + 4012951960503789*v^9 + 12274806963406785*v^8 - 48863345541989719*v^7 - 107429993938577041*v^6 + 288045734203306368*v^5 + 429840747836347184*v^4 - 661545470141613296*v^3 - 542735210180524656*v^2 + 271383252434542848*v + 136461067105893376) / 4486911746064384 $$\beta_{9}$$ $$=$$ $$( 23136988571 \nu^{15} + 16893139885 \nu^{14} - 2281492547893 \nu^{13} + \cdots - 65\!\cdots\!92 ) / 22\!\cdots\!92$$ (23136988571*v^15 + 16893139885*v^14 - 2281492547893*v^13 - 2064373181017*v^12 + 89389380236256*v^11 + 89701645268668*v^10 - 1770475927542653*v^9 - 1766676647982337*v^8 + 18581171355968647*v^7 + 15963988811665521*v^6 - 97321935429585792*v^5 - 56772109530652720*v^4 + 197583173512329200*v^3 + 51514228625424496*v^2 - 26392996382746880*v - 65252675753352192) / 2243455873032192 $$\beta_{10}$$ $$=$$ $$( - 20452910127 \nu^{15} - 16432133993 \nu^{14} + 1982131425633 \nu^{13} + \cdots + 30\!\cdots\!00 ) / 11\!\cdots\!96$$ (-20452910127*v^15 - 16432133993*v^14 + 1982131425633*v^13 + 2169014124837*v^12 - 75334767740704*v^11 - 103025583874348*v^10 + 1415246268576297*v^9 + 2292740351829613*v^8 - 13556102808218987*v^7 - 24988416622422525*v^6 + 60922720207914400*v^5 + 121623927844604400*v^4 - 102752828818061616*v^3 - 181320869441535920*v^2 + 90356442243841792*v + 30276992909184000) / 1121727936516096 $$\beta_{11}$$ $$=$$ $$( 55080257475 \nu^{15} - 416033083595 \nu^{14} - 4000919116205 \nu^{13} + \cdots + 67\!\cdots\!80 ) / 22\!\cdots\!92$$ (55080257475*v^15 - 416033083595*v^14 - 4000919116205*v^13 + 34836352162815*v^12 + 98534493596608*v^11 - 1124180539665284*v^10 - 742376566559221*v^9 + 17527725342224727*v^8 - 5644241932392465*v^7 - 134566224611264903*v^6 + 105280918475971456*v^5 + 450413544205672016*v^4 - 395971786053532560*v^3 - 439073769149318160*v^2 + 203228452229230336*v + 67512197092910080) / 2243455873032192 $$\beta_{12}$$ $$=$$ $$( 16429085063 \nu^{15} - 61887969455 \nu^{14} - 1425014908457 \nu^{13} + \cdots + 27\!\cdots\!52 ) / 640987392294912$$ (16429085063*v^15 - 61887969455*v^14 - 1425014908457*v^13 + 5069037776595*v^12 + 48005794589120*v^11 - 159909796260276*v^10 - 792184730281809*v^9 + 2445675915973707*v^8 + 6644854661703491*v^7 - 18660594842400827*v^6 - 26820458842631680*v^5 + 64484715819608336*v^4 + 46799050268116400*v^3 - 76705880314714832*v^2 - 31117490231966976*v + 27546322883761152) / 640987392294912 $$\beta_{13}$$ $$=$$ $$( - 124480546525 \nu^{15} + 423884464197 \nu^{14} + 11651829308051 \nu^{13} + \cdots - 26\!\cdots\!48 ) / 44\!\cdots\!84$$ (-124480546525*v^15 + 423884464197*v^14 + 11651829308051*v^13 - 36442139620369*v^12 - 437458640983136*v^11 + 1228453930615196*v^10 + 8449200566667787*v^9 - 20574051190613753*v^8 - 89312992241402001*v^7 + 177782230339390377*v^6 + 503158240925972096*v^5 - 730614857431534000*v^4 - 1312073483295782800*v^3 + 1076496281477084400*v^2 + 908868648216930048*v - 261665860041858048) / 4486911746064384 $$\beta_{14}$$ $$=$$ $$( - 131806954357 \nu^{15} + 549377494605 \nu^{14} + 10915371529179 \nu^{13} + \cdots + 25\!\cdots\!60 ) / 44\!\cdots\!84$$ (-131806954357*v^15 + 549377494605*v^14 + 10915371529179*v^13 - 43769703338809*v^12 - 341494467380224*v^11 + 1315054098824412*v^10 + 4922759967764307*v^9 - 18339731245082401*v^8 - 30213427224226153*v^7 + 114920418025327025*v^6 + 27911033860517504*v^5 - 222972078285061936*v^4 + 284112608491247856*v^3 - 223889905872433040*v^2 - 242098087057641728*v + 257558983134888960) / 4486911746064384 $$\beta_{15}$$ $$=$$ $$( 130386402597 \nu^{15} - 648517868837 \nu^{14} - 10855476276667 \nu^{13} + \cdots + 10\!\cdots\!04 ) / 22\!\cdots\!92$$ (130386402597*v^15 - 648517868837*v^14 - 10855476276667*v^13 + 53717916482321*v^12 + 346439827784944*v^11 - 1711933178665100*v^10 - 5297129929505811*v^9 + 26309311331571289*v^8 + 39388771834884137*v^7 - 198606004842486329*v^6 - 126336021691190848*v^5 + 650347149591150768*v^4 + 132502497563507728*v^3 - 609544754308250608*v^2 - 67451517843751680*v + 104358555533589504) / 2243455873032192
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 13$$ b2 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{2} + 20\beta _1 + 4$$ b6 - b5 + b2 + 20*b1 + 4 $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{13} + 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 264$$ b15 + b13 + 3*b12 - 2*b11 + 2*b10 - 2*b9 - b8 + 3*b6 - b5 - b4 + 30*b2 + 5*b1 + 264 $$\nu^{5}$$ $$=$$ $$2 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 2 \beta_{10} + \beta_{9} - 5 \beta_{8} + \cdots + 172$$ 2*b15 + 4*b14 + 3*b13 + 3*b12 + 2*b10 + b9 - 5*b8 + 38*b6 - 37*b5 - 4*b3 + 44*b2 + 454*b1 + 172 $$\nu^{6}$$ $$=$$ $$32 \beta_{15} - 6 \beta_{14} + 42 \beta_{13} + 132 \beta_{12} - 72 \beta_{11} + 93 \beta_{10} + \cdots + 6125$$ 32*b15 - 6*b14 + 42*b13 + 132*b12 - 72*b11 + 93*b10 - 77*b9 - 55*b8 - 2*b7 + 141*b6 - 45*b5 - 36*b4 + 11*b3 + 837*b2 + 260*b1 + 6125 $$\nu^{7}$$ $$=$$ $$86 \beta_{15} + 200 \beta_{14} + 148 \beta_{13} + 170 \beta_{12} + 34 \beta_{11} + 120 \beta_{10} + \cdots + 6080$$ 86*b15 + 200*b14 + 148*b13 + 170*b12 + 34*b11 + 120*b10 + 34*b9 - 290*b8 - 4*b7 + 1202*b6 - 1064*b5 + 8*b4 - 138*b3 + 1546*b2 + 11103*b1 + 6080 $$\nu^{8}$$ $$=$$ $$870 \beta_{15} - 300 \beta_{14} + 1482 \beta_{13} + 4502 \beta_{12} - 2040 \beta_{11} + 3312 \beta_{10} + \cdots + 152607$$ 870*b15 - 300*b14 + 1482*b13 + 4502*b12 - 2040*b11 + 3312*b10 - 2332*b9 - 2190*b8 - 56*b7 + 5048*b6 - 1560*b5 - 1018*b4 + 868*b3 + 23161*b2 + 10268*b1 + 152607 $$\nu^{9}$$ $$=$$ $$2870 \beta_{15} + 7144 \beta_{14} + 5520 \beta_{13} + 7188 \beta_{12} + 1972 \beta_{11} + 5580 \beta_{10} + \cdots + 203330$$ 2870*b15 + 7144*b14 + 5520*b13 + 7188*b12 + 1972*b11 + 5580*b10 + 658*b9 - 12136*b8 - 232*b7 + 36529*b6 - 28603*b5 + 294*b4 - 3220*b3 + 50573*b2 + 284702*b1 + 203330 $$\nu^{10}$$ $$=$$ $$23727 \beta_{15} - 10580 \beta_{14} + 49469 \beta_{13} + 141151 \beta_{12} - 54070 \beta_{11} + \cdots + 3969318$$ 23727*b15 - 10580*b14 + 49469*b13 + 141151*b12 - 54070*b11 + 107844*b10 - 65838*b9 - 77651*b8 - 100*b7 + 165473*b6 - 49313*b5 - 26747*b4 + 41942*b3 + 644442*b2 + 364575*b1 + 3969318 $$\nu^{11}$$ $$=$$ $$91212 \beta_{15} + 225584 \beta_{14} + 188183 \beta_{13} + 269451 \beta_{12} + 76760 \beta_{11} + \cdots + 6611554$$ 91212*b15 + 225584*b14 + 188183*b13 + 269451*b12 + 76760*b11 + 227204*b10 + 2985*b9 - 444483*b8 - 7772*b7 + 1100274*b6 - 752799*b5 + 5630*b4 - 50842*b3 + 1607342*b2 + 7534202*b1 + 6611554 $$\nu^{12}$$ $$=$$ $$672264 \beta_{15} - 324594 \beta_{14} + 1600284 \beta_{13} + 4274966 \beta_{12} - 1409044 \beta_{11} + \cdots + 106209093$$ 672264*b15 - 324594*b14 + 1600284*b13 + 4274966*b12 - 1409044*b11 + 3382283*b10 - 1817137*b9 - 2603587*b8 + 67106*b7 + 5232335*b6 - 1491377*b5 - 689260*b4 + 1656325*b3 + 18092177*b2 + 12266162*b1 + 106209093 $$\nu^{13}$$ $$=$$ $$2900502 \beta_{15} + 6727236 \beta_{14} + 6182820 \beta_{13} + 9467666 \beta_{12} + 2514238 \beta_{11} + \cdots + 211230852$$ 2900502*b15 + 6727236*b14 + 6182820*b13 + 9467666*b12 + 2514238*b11 + 8507482*b10 - 444596*b9 - 15180728*b8 - 160832*b7 + 33109246*b6 - 19703736*b5 - 4140*b4 + 23432*b3 + 50387236*b2 + 203887461*b1 + 211230852 $$\nu^{14}$$ $$=$$ $$19794032 \beta_{15} - 9225444 \beta_{14} + 50676348 \beta_{13} + 127515168 \beta_{12} + \cdots + 2900153889$$ 19794032*b15 - 9225444*b14 + 50676348*b13 + 127515168*b12 - 36794400*b11 + 104237964*b10 - 49909192*b9 - 84594940*b8 + 4342560*b7 + 162699644*b6 - 43962272*b5 - 17849080*b4 + 58952536*b3 + 512646921*b2 + 400160860*b1 + 2900153889 $$\nu^{15}$$ $$=$$ $$92925716 \beta_{15} + 194720512 \beta_{14} + 199422020 \beta_{13} + 319828308 \beta_{12} + \cdots + 6669175096$$ 92925716*b15 + 194720512*b14 + 199422020*b13 + 319828308*b12 + 74732976*b11 + 301336576*b10 - 28307360*b9 - 497699900*b8 - 64480*b7 + 997227709*b6 - 516170229*b5 - 6201988*b4 + 47160880*b3 + 1567370693*b2 + 5611477904*b1 + 6669175096

## 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
 5.50208 5.02984 4.51833 3.65350 3.30535 3.04558 1.20555 0.626276 0.120661 −0.903074 −2.39847 −2.47130 −3.48862 −3.90605 −4.70685 −5.13282
−5.50208 −3.00000 22.2728 1.21430 16.5062 0 −78.5303 9.00000 −6.68115
1.2 −5.02984 −3.00000 17.2993 −20.5790 15.0895 0 −46.7737 9.00000 103.509
1.3 −4.51833 −3.00000 12.4153 3.55084 13.5550 0 −19.9499 9.00000 −16.0439
1.4 −3.65350 −3.00000 5.34810 −11.9317 10.9605 0 9.68874 9.00000 43.5923
1.5 −3.30535 −3.00000 2.92536 9.14781 9.91606 0 16.7735 9.00000 −30.2367
1.6 −3.04558 −3.00000 1.27555 15.0528 9.13674 0 20.4798 9.00000 −45.8446
1.7 −1.20555 −3.00000 −6.54664 −9.84913 3.61666 0 17.5368 9.00000 11.8737
1.8 −0.626276 −3.00000 −7.60778 −4.84903 1.87883 0 9.77478 9.00000 3.03683
1.9 −0.120661 −3.00000 −7.98544 −18.6516 0.361983 0 1.92882 9.00000 2.25052
1.10 0.903074 −3.00000 −7.18446 21.8819 −2.70922 0 −13.7127 9.00000 19.7610
1.11 2.39847 −3.00000 −2.24736 −1.37648 −7.19540 0 −24.5779 9.00000 −3.30143
1.12 2.47130 −3.00000 −1.89269 −4.05950 −7.41389 0 −24.4478 9.00000 −10.0322
1.13 3.48862 −3.00000 4.17050 0.768490 −10.4659 0 −13.3597 9.00000 2.68097
1.14 3.90605 −3.00000 7.25725 −7.35832 −11.7182 0 −2.90120 9.00000 −28.7420
1.15 4.70685 −3.00000 14.1544 15.5088 −14.1205 0 28.9678 9.00000 72.9976
1.16 5.13282 −3.00000 18.3458 11.5297 −15.3984 0 53.1030 9.00000 59.1800
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.16 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.bc 16
7.b odd 2 1 1617.4.a.bd yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.4.a.bc 16 1.a even 1 1 trivial
1617.4.a.bd yes 16 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{16} + 4 T_{2}^{15} - 92 T_{2}^{14} - 346 T_{2}^{13} + 3385 T_{2}^{12} + 11756 T_{2}^{11} + \cdots - 738304$$ T2^16 + 4*T2^15 - 92*T2^14 - 346*T2^13 + 3385*T2^12 + 11756*T2^11 - 63875*T2^10 - 199466*T2^9 + 657606*T2^8 + 1772388*T2^7 - 3590705*T2^6 - 7831536*T2^5 + 8898560*T2^4 + 14389664*T2^3 - 4854640*T2^2 - 6896896*T2 - 738304 $$T_{5}^{16} - 1142 T_{5}^{14} - 508 T_{5}^{13} + 477806 T_{5}^{12} + 450684 T_{5}^{11} + \cdots + 16055671813888$$ T5^16 - 1142*T5^14 - 508*T5^13 + 477806*T5^12 + 450684*T5^11 - 92455556*T5^10 - 156678844*T5^9 + 8594367185*T5^8 + 21847618212*T5^7 - 346913866966*T5^6 - 1124056558600*T5^5 + 4147579649424*T5^4 + 12709673472416*T5^3 - 14198466606352*T5^2 - 18890333580160*T5 + 16055671813888

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 4 T^{15} + \cdots - 738304$$
$3$ $$(T + 3)^{16}$$
$5$ $$T^{16} + \cdots + 16055671813888$$
$7$ $$T^{16}$$
$11$ $$(T + 11)^{16}$$
$13$ $$T^{16} + \cdots - 51\!\cdots\!68$$
$17$ $$T^{16} + \cdots + 29\!\cdots\!64$$
$19$ $$T^{16} + \cdots + 22\!\cdots\!96$$
$23$ $$T^{16} + \cdots - 18\!\cdots\!36$$
$29$ $$T^{16} + \cdots + 60\!\cdots\!56$$
$31$ $$T^{16} + \cdots - 28\!\cdots\!68$$
$37$ $$T^{16} + \cdots + 13\!\cdots\!52$$
$41$ $$T^{16} + \cdots - 24\!\cdots\!12$$
$43$ $$T^{16} + \cdots - 57\!\cdots\!96$$
$47$ $$T^{16} + \cdots - 58\!\cdots\!44$$
$53$ $$T^{16} + \cdots - 18\!\cdots\!52$$
$59$ $$T^{16} + \cdots - 24\!\cdots\!76$$
$61$ $$T^{16} + \cdots + 27\!\cdots\!84$$
$67$ $$T^{16} + \cdots + 45\!\cdots\!28$$
$71$ $$T^{16} + \cdots - 58\!\cdots\!72$$
$73$ $$T^{16} + \cdots + 13\!\cdots\!24$$
$79$ $$T^{16} + \cdots + 42\!\cdots\!32$$
$83$ $$T^{16} + \cdots - 59\!\cdots\!04$$
$89$ $$T^{16} + \cdots + 87\!\cdots\!44$$
$97$ $$T^{16} + \cdots - 26\!\cdots\!52$$