LMFDB - Newform orbit 755.2.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [755,2,Mod(1,755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(755, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$755 = 5 \cdot 151$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	755.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:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} - 2 x^{17} - 32 x^{16} + 64 x^{15} + 417 x^{14} - 839 x^{13} - 2829 x^{12} + 5789 x^{11} + \cdots + 576$ x^18 - 2x^17 - 32x^16 + 64x^15 + 417x^14 - 839x^13 - 2829x^12 + 5789x^11 + 10562x^10 - 22471x^9 - 20867x^8 + 48611x^7 + 18052x^6 - 54316x^5 - 788x^4 + 25424x^3 - 5248x^2 - 2240*x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{14} - \beta_{12} + \beta_{11} + \cdots + 1) q^{6} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_1) q^{7} + ( - \beta_{3} - 2 \beta_1) q^{8}+ \cdots + (3 \beta_{17} - 2 \beta_{16} + \cdots + 7) q^{99}+O(q^{100})$ q - b1 * q^2 - b10 * q^3 + (b2 + 2) * q^4 - q^5 + (b14 - b12 + b11 + b10 - 2b8 + b6 + b3 - b2 - b1 + 1) q^6 + (-b11 - b10 - b4 + b1) * q^7 + (-b3 - 2b1) q^8 + (b15 - b7 + 2) * q^9 + b1 * q^10 + (b17 + b5 + 1) * q^11 + (-b17 - b13 + b12 - b11 - 2b10 - b9 + b8 - b6 - 2b5 - b3 + b2 - 2) * q^12 + (b17 - b16 - b12 + b11 + b9 + b6 + b5 - b1 + 2) * q^13 + (-b17 + b16 - b15 + b12 - b11 + b7 - b5 + b3 - 1) * q^14 + b10 * q^15 + (b15 + b13 + b11 + b9 + b5 + b4 + 2b2 - b1 + 5) q^16 + (b16 - b14 + b8 - b6 + b1 - 1) * q^17 + (-b16 - 2b15 + b13 - b12 + 2b11 + b10 - b9 - b8 + b5 + b4 + b3 - 2b2 - 3b1) * q^18 + (-b11 - b10 + b7 + b1) * q^19 + (-b2 - 2) * q^20 + (-2b17 + b16 - b15 + b12 - b11 + 2b8 - b6 - b5 + b1) * q^21 + (-b16 - 2b14 + b12 + b8 + b2 - b1 + 2) q^22 + (b17 + b15 + b13 - b10 + 1) * q^23 + (b17 + 2b14 - b13 - 2b12 + 4b11 + 3b10 - b9 - 4b8 + b6 + b5 - b4 + 3b3 - 2b2 - b1 + 3) q^24 + q^25 + (b17 - b16 - b15 - b14 - b12 + 2b11 + b10 + b7 + 2b6 + 2b4 - 3b1 + 1) * q^26 + (-b16 - b15 - b14 + b12 - b11 - 2b10 + b8 - b5 + b2 - 1) q^27 + (-b17 + 2b16 + b15 + b14 - b13 + b12 - 4b11 - b10 + b9 + 2b8 - 2b6 - b5 - b4 - 2b3 + b2 + 3b1 - 1) * q^28 + (-b17 + b16 + b15 + b12 - 2b11 + b8 - b7 - 2b6 - b5 - b3 + b1) * q^29 + (-b14 + b12 - b11 - b10 + 2b8 - b6 - b3 + b2 + b1 - 1) q^30 + (b12 - b11 - 2b10 + b8 - b6 + b5 - b3 + b2 + 2b1) * q^31 + (-b11 + b8 - b7 - b6 + b5 - 2b3 - 2b1 - 1) * q^32 + (-b16 - b15 - b13 - b12 + 3b11 + b10 - 2b8 + b7 + 2b6 + b5 + b4 + b3 - 2b2 - 2b1 - 1) q^33 + (b13 - b11 + b10 + b9 + b8 + b7 + b6 + b4 - b3 + b2 + 1) * q^34 + (b11 + b10 + b4 - b1) * q^35 + (b17 + 4b15 + b13 + b12 - 2b11 - 2b10 + b9 + b8 - 3b7 - b6 - b5 - b4 - 2b3 + 3b2 + 2b1 + 5) q^36 + (b17 + b15 + b14 + 2b11 + b6 + b5 + b2 - b1 + 4) q^37 + (b16 - b15 - 2b13 + b12 + b10 + b7 + b6 - b5 - 2b4 + 2b3 - b2 + b1 - 2) q^38 + (b17 - 2b16 - b15 - b14 - b13 - b12 + 3b11 + b10 - b9 - b8 + 2b6 + b5 + b4 + b3 - 2b2 - 3b1 + 3) q^39 + (b3 + 2b1) q^40 + (-b16 - b14 - b13 + b12 - b11 - b10 - b9 - b7 - b6 - b5 - b4 - b3 + b1 - 1) * q^41 + (-2b17 + 3b16 + 2b15 + 3b14 + 2b13 - 4b11 + 2b9 + b8 - b6 - b3 + 2b1 - 1) * q^42 + (-b17 + b16 + b15 + b13 - b11 - b10 - b9 - 2b8 - b7 - b6 - b5 - b4 - 2b2 + b1) * q^43 + (b17 + b13 - b12 - 2b11 + 2b10 + b9 + b8 - b7 - b6 + b5 - b4 - 2b3 + 1) q^44 + (-b15 + b7 - 2) * q^45 + (b17 - 2b16 - 3b15 - 2b14 - 2b13 - 2b12 + 3b11 + 3b10 - 2b9 - 2b8 + b6 + 2b3 - 3b2 - b1) q^46 + (-b16 - b15 + b14 - b10 + b7 + b5 - b1 - 1) * q^47 + (-b17 + 2b15 - 2b13 + 2b12 - 2b11 - 5b10 - 2b9 + b7 - b6 - 3b5 + b4 - 2b3 + b2 - 4) * q^48 + (-b16 - b15 + b14 + b10 - b7 - b5 - 2b4 + b3 - b2 + 3) q^49 - b1 * q^50 + (-b17 + 2b16 - b14 - b13 - b11 + 2b10 - b9 - b6 - 2b5 - b4 - 2b2 + 4b1 - 1) q^51 + (3b17 - b16 - b14 - 3b12 + 3b11 + 3b10 + 2b6 + b5 + b4 - 2b1 + 3) * q^52 + (b15 + b13 - b12 + b11 - b9 - 2b8 + b6 + b4 + b3 - 2b2) * q^53 + (3b17 - b16 + b14 - 4b12 + 2b11 + 4b10 - 5b8 - 2b7 + 2b6 + 2b5 - 2b4 + 2b3 - 5b2 - b1 + 2) q^54 + (-b17 - b5 - 1) * q^55 + (-2b17 - b15 + b12 + 3b11 + b10 + 2b7 + b6 + b5 + 2b3 + b2 - b1 + 2) * q^56 + (-b17 + b16 + b15 + 2b13 + b12 - b11 - b10 + 2b9 + 2b8 + b7 + b5 + 2b4 - b3 + 2b2 + 1) q^57 + (-b17 - b15 + b13 + b12 + b11 + b10 - b9 - b8 - b7 - b6 + b5 - b4 + 2b3 - b2 + 2) q^58 + (-b16 + b15 + 2b14 + b13 + b11 - b10 + b9 + b7 + b6 + 2b5 + b4 - b3 + 2b2 - 2b1 + 1) * q^59 + (b17 + b13 - b12 + b11 + 2b10 + b9 - b8 + b6 + 2b5 + b3 - b2 + 2) * q^60 + (-b15 - b12 + b10 - b8 - b7 - b2 + 2b1 + 1) q^61 + (-b17 + b16 + 2b15 + b14 + b13 + b12 + 2b10 + b9 - 2b8 - b7 + b5 - b4 + b3 - b2 - b1 - 2) q^62 + (-b17 + 3b16 + b14 + b12 - 5b11 - b10 + 2b9 + b8 - b7 - b6 - 2b5 - 2b4 + 6b1 - 2) * q^63 + (-b17 - b14 + 2b13 + 2b12 + b10 + b8 - b7 + b3 + 2b2 + 6) q^64 + (-b17 + b16 + b12 - b11 - b9 - b6 - b5 + b1 - 2) * q^65 + (b17 + b16 + 4b15 + 3b14 - 2b11 - 4b10 + 2b9 + b8 - 2b3 + 3b2 + 5b1 - 2) * q^66 + (-2b15 - 2b13 + b12 - b11 - b10 - b9 + b7 - b5 - 2b4 + b3 + b2 + 2b1 - 1) * q^67 + (b17 - 2b16 - 2b15 - b14 + b10 + 2b9 + 3b8 + b7 + 2b6 + 2b5 - b3 + b2 - 2b1) q^68 + (-b17 + b15 - 2b14 + b13 + b11 - b9 - 2b5 + 3b4 - b3 - 2b1 + 1) * q^69 + (b17 - b16 + b15 - b12 + b11 - b7 + b5 - b3 + 1) * q^70 + (b16 + b14 + 2b13 - b12 + 2b11 - b8 + 2b7 + b5 + 2b4 + b2 - b1 + 2) * q^71 + (b17 - 3b16 - 5b15 - 3b14 - b13 - 3b12 + 6b11 + 5b10 - 3b9 - 5b8 + b6 + 2b5 + b4 + 2b3 - 5b2 - 7b1 - 4) * q^72 + (-b17 + b16 - b15 + b14 - 2b12 + b11 + 2b10 + b9 + b7 + b4 + b3 - b2 + b1 + 1) * q^73 + (-b17 - b16 - b15 - b13 - 2b12 + 2b11 + b10 - b9 - b8 + 2b7 + 3b4 - b3 - 5b1 - 1) q^74 - b10 * q^75 + (-2b17 + 2b16 - b15 + b14 - 2b13 + b12 - 6b11 - 3b10 + 3b8 + 3b7 - 2b6 - 3b5 - 2b3 + 2b2 + 7b1 - 7) * q^76 + (b17 - 2b14 - 3b13 + 3b10 - b8 - b7 - b6 - b5 - 2b4 + 2b3 + 2b1 + 1) * q^77 + (4b17 - 2b16 + 2b15 + b14 + b13 - 3b12 + 3b11 + b10 + b9 - b8 - b7 + 2b6 + 2b5 + b4 + b3 - 2b1 + 6) * q^78 + (b16 - b15 - b14 - 3b13 + b12 - 2b11 - b9 + b7 - b6 - 3b5 - b4 - b3 + 2b1 - 2) * q^79 + (-b15 - b13 - b11 - b9 - b5 - b4 - 2b2 + b1 - 5) q^80 + (2b15 + 2b14 + b13 - 2b11 - b10 + b9 + 2b8 - b5 + b4 - 2b3 + 3b2 + 2b1 + 3) q^81 + (2b17 - b16 + b15 + b14 + b13 - b12 + 3b11 + 2b10 - b9 - 5b8 - 2b7 + 2b5 - b4 + 3b3 - 2b2 - b1 + 1) * q^82 + (-b16 - b15 + b14 - 2b11 + b9 + b8 - b7 - 2b4 - b3 + 2b1 - 2) q^83 + (-4b17 + 2b16 - 3b15 - 2b14 + 4b12 - 2b11 - 3b10 + 3b8 + 3b7 - b6 - b5 + b3 + 2b2 + 3b1 - 4) q^84 + (-b16 + b14 - b8 + b6 - b1 + 1) * q^85 + (-2b17 + b16 + b14 + b13 + 2b12 - 2b11 - 3b10 - b9 - b8 - 2b7 - 3b6 - b5 - b4 + b3 + b2 + 2b1 - 1) q^86 + (-b17 + b16 + b13 + b12 - b11 - 4b10 - b8 + b7 - b5 + b3 + b2 + 2b1 - 3) * q^87 + (-b17 + b16 - b15 - 3b14 + 3b13 + 3b12 - b10 + b9 + 5b8 - b6 + 3b4 - b3 + 5b2 - b1 + 4) * q^88 + (b16 + b14 + b12 - 2b11 + 2b9 + 3b8 - b5 - 2b3 + 3b2 + b1) q^89 + (b16 + 2b15 - b13 + b12 - 2b11 - b10 + b9 + b8 - b5 - b4 - b3 + 2b2 + 3b1) * q^90 + (b17 + b14 + 2b13 - b12 + b11 + b8 + b7 + 2b6 + 2b5 + 2b1 + 3) * q^91 + (3b17 - 2b16 + 3b15 - b13 - b10 + b9 + b8 - 2b7 + 2b6 - b5 - b3 + 2b2 - b1) * q^92 + (2b15 - 2b13 + 2b12 - b11 - 2b10 + b9 + b8 + b5 - b4 - b3 + 2b2 + 4b1 + 3) * q^93 + (-b17 + 2b16 + 3b15 + b14 - 2b13 + 2b12 - b11 - b8 - b7 - 2b5 - 2b4 + b3 + b2 + 3b1 + 3) q^94 + (b11 + b10 - b7 - b1) * q^95 + (b17 - b16 - 3b15 + 3b14 - 3b13 - 3b12 + 6b11 + 4b10 - b9 - 8b8 + b7 + 4b6 + b5 - 3b4 + 7b3 - 5b2 + 1) q^96 + (b17 - b16 - b15 + b13 - 2b12 + 3b11 + 3b10 - b9 + b7 + b6 + b5 + 3b4 - b3 - 4b1) q^97 + (-b17 - b14 - b13 - 2b11 - 2b10 - 3b9 - b8 - b7 - 2b6 - 3b5 + b4 - 2b2 - 4) * q^98 + (3b17 - 2b16 + 2b15 + 2b14 + b13 - 3b12 + 4b11 + 3b10 + 3b9 - 2b8 - b7 + 2b6 + 6b5 - b4 + b3 - b2 - 3b1 + 7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$18 q - 2 q^{2} + 2 q^{3} + 32 q^{4} - 18 q^{5} + 10 q^{6} + 4 q^{7} + 32 q^{9} + 2 q^{10} + 11 q^{11} + 12 q^{13} + q^{14} - 2 q^{15} + 60 q^{16} - 25 q^{17} - 9 q^{18} + 8 q^{19} - 32 q^{20} + 20 q^{21}+ \cdots + 11 q^{99}+O(q^{100})$ 18 * q - 2 * q^2 + 2 * q^3 + 32 * q^4 - 18 * q^5 + 10 * q^6 + 4 * q^7 + 32 * q^9 + 2 * q^10 + 11 * q^11 + 12 * q^13 + q^14 - 2 * q^15 + 60 * q^16 - 25 * q^17 - 9 * q^18 + 8 * q^19 - 32 * q^20 + 20 * q^21 + 29 * q^22 + 12 * q^23 + 15 * q^24 + 18 * q^25 + 5 * q^26 + 5 * q^27 + 15 * q^28 + 24 * q^29 - 10 * q^30 + 11 * q^31 - 17 * q^32 - 33 * q^33 + 17 * q^34 - 4 * q^35 + 90 * q^36 + 45 * q^37 - 28 * q^38 + 38 * q^39 + 12 * q^41 - 9 * q^42 + 19 * q^43 + 6 * q^44 - 32 * q^45 - 5 * q^46 - 16 * q^47 - 3 * q^48 + 62 * q^49 - 2 * q^50 + 7 * q^51 + q^52 - 4 * q^53 - 6 * q^54 - 11 * q^55 + 2 * q^56 + 5 * q^57 + 28 * q^58 - 8 * q^59 + 27 * q^61 - 57 * q^62 + 10 * q^63 + 98 * q^64 - 12 * q^65 - 32 * q^66 + 9 * q^67 - 26 * q^68 + 31 * q^69 - q^70 + 9 * q^71 - 102 * q^72 + 2 * q^73 - 24 * q^74 + 2 * q^75 - 43 * q^76 + 11 * q^77 + 45 * q^78 + 20 * q^79 - 60 * q^80 + 66 * q^81 - 19 * q^82 - 20 * q^83 - 33 * q^84 + 25 * q^85 + 21 * q^86 - 33 * q^87 + 62 * q^88 + 13 * q^89 + 9 * q^90 + 22 * q^91 - 6 * q^92 + 58 * q^93 + 70 * q^94 - 8 * q^95 - 17 * q^96 - 27 * q^97 - 6 * q^98 + 11 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{18} - 2 x^{17} - 32 x^{16} + 64 x^{15} + 417 x^{14} - 839 x^{13} - 2829 x^{12} + 5789 x^{11} + \cdots + 576$ x^18 - 2*x^17 - 32*x^16 + 64*x^15 + 417*x^14 - 839*x^13 - 2829*x^12 + 5789*x^11 + 10562*x^10 - 22471*x^9 - 20867*x^8 + 48611*x^7 + 18052*x^6 - 54316*x^5 - 788*x^4 + 25424*x^3 - 5248*x^2 - 2240*x + 576 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 4$ v^2 - 4
$\beta_{3}$	$=$	$\nu^{3} - 6\nu$ v^3 - 6*v
$\beta_{4}$	$=$	$( 291189 \nu^{17} - 89672 \nu^{16} - 13773536 \nu^{15} + 3015680 \nu^{14} + 241721301 \nu^{13} + \cdots + 1759381728 ) / 246482048$ (291189v^17 - 89672v^16 - 13773536v^15 + 3015680v^14 + 241721301v^13 - 29004529v^12 - 2101387035v^11 + 13988179v^10 + 9831071400v^9 + 1235471181v^8 - 24633951085v^7 - 6926683683v^6 + 30309466422v^5 + 14084959504v^4 - 13489670468v^3 - 10384980120v^2 - 845664432*v + 1759381728) / 246482048
$\beta_{5}$	$=$	$( 6611 \nu^{17} - 102932 \nu^{16} - 147400 \nu^{15} + 3254088 \nu^{14} + 1129363 \nu^{13} + \cdots - 103616736 ) / 3624736$ (6611v^17 - 102932v^16 - 147400v^15 + 3254088v^14 + 1129363v^13 - 42236619v^12 - 2822985v^11 + 289518113v^10 - 4294492v^9 - 1121210837v^8 + 28099529v^7 + 2427522935v^6 - 33779234v^5 - 2703485456v^4 + 31118900v^3 + 1236419928v^2 - 50661968*v - 103616736) / 3624736
$\beta_{6}$	$=$	$( - 352200 \nu^{17} - 1531101 \nu^{16} + 12859718 \nu^{15} + 49775228 \nu^{14} - 191128800 \nu^{13} + \cdots - 1731721392 ) / 61620512$ (-352200v^17 - 1531101v^16 + 12859718v^15 + 49775228v^14 - 191128800v^13 - 661936197v^12 + 1484418879v^11 + 4619137905v^10 - 6424547681v^9 - 18025276554v^8 + 15275558127v^7 + 38756665459v^6 - 18382385095v^5 - 42201966684v^4 + 9329836696v^3 + 18805330628v^2 - 1446508480*v - 1731721392) / 61620512
$\beta_{7}$	$=$	$( - 200609 \nu^{17} - 111305 \nu^{16} + 6840426 \nu^{15} + 2808640 \nu^{14} - 94256757 \nu^{13} + \cdots - 109530144 ) / 30810256$ (-200609v^17 - 111305v^16 + 6840426v^15 + 2808640v^14 - 94256757v^13 - 28454812v^12 + 671692170v^11 + 156972966v^10 - 2629804125v^9 - 568694535v^8 + 5554043316v^7 + 1471808570v^6 - 5788174893v^5 - 2267394640v^4 + 2256700072v^3 + 1424151932v^2 + 35152960*v - 109530144) / 30810256
$\beta_{8}$	$=$	$( 4207833 \nu^{17} - 3035060 \nu^{16} - 136531352 \nu^{15} + 90560592 \nu^{14} + 1814253113 \nu^{13} + \cdots - 3045233376 ) / 246482048$ (4207833v^17 - 3035060v^16 - 136531352v^15 + 90560592v^14 + 1814253113v^13 - 1100805073v^12 - 12685910227v^11 + 7024994955v^10 + 49903604140v^9 - 25360311015v^8 - 109468293869v^7 + 52246949845v^6 + 124263292874v^5 - 58938073536v^4 - 59351846516v^3 + 30948041592v^2 + 4832513168*v - 3045233376) / 246482048
$\beta_{9}$	$=$	$( - 376637 \nu^{17} + 816775 \nu^{16} + 12308969 \nu^{15} - 26268086 \nu^{14} - 165237543 \nu^{13} + \cdots + 1047452304 ) / 15405128$ (-376637v^17 + 816775v^16 + 12308969v^15 - 26268086v^14 - 165237543v^13 + 346295680v^12 + 1171980331v^11 - 2404823799v^10 - 4703791424v^9 + 9405730202v^8 + 10618599184v^7 - 20533946607v^6 - 12554740588v^5 + 23209216847v^4 + 6312897100v^3 - 11076391000v^2 - 519286068*v + 1047452304) / 15405128
$\beta_{10}$	$=$	$( - 6655037 \nu^{17} + 7124952 \nu^{16} + 218103744 \nu^{15} - 221147168 \nu^{14} + \cdots + 5512649248 ) / 246482048$ (-6655037v^17 + 7124952v^16 + 218103744v^15 - 221147168v^14 - 2924863261v^13 + 2801100009v^12 + 20597530787v^11 - 18591914171v^10 - 81271079608v^9 + 69101023915v^8 + 177488178533v^7 - 142608410453v^6 - 198018901654v^5 + 152085774992v^4 + 90814696516v^3 - 68344091432v^2 - 7328973520*v + 5512649248) / 246482048
$\beta_{11}$	$=$	$( 7671053 \nu^{17} - 3019592 \nu^{16} - 252716832 \nu^{15} + 90268544 \nu^{14} + 3409619053 \nu^{13} + \cdots - 2534526752 ) / 246482048$ (7671053v^17 - 3019592v^16 - 252716832v^15 + 90268544v^14 + 3409619053v^13 - 1097511881v^12 - 24189086083v^11 + 6979891291v^10 + 96348202408v^9 - 24951985435v^8 - 213092104933v^7 + 50517379029v^6 + 242047726534v^5 - 55828060688v^4 - 115073529156v^3 + 28410058728v^2 + 12314914640*v - 2534526752) / 246482048
$\beta_{12}$	$=$	$( 2012075 \nu^{17} - 240291 \nu^{16} - 66473910 \nu^{15} + 4644444 \nu^{14} + 901092691 \nu^{13} + \cdots + 402009264 ) / 61620512$ (2012075v^17 - 240291v^16 - 66473910v^15 + 4644444v^14 + 901092691v^13 - 21137330v^12 - 6436376484v^11 - 119181332v^10 + 25857518169v^9 + 1398983949v^8 - 57649069074v^7 - 4612290952v^6 + 65404036993v^5 + 5921044300v^4 - 29876625124v^3 - 2384895196v^2 + 2363279920*v + 402009264) / 61620512
$\beta_{13}$	$=$	$( 1080450 \nu^{17} - 770753 \nu^{16} - 35433032 \nu^{15} + 24019784 \nu^{14} + 475927878 \nu^{13} + \cdots - 857558448 ) / 30810256$ (1080450v^17 - 770753v^16 - 35433032v^15 + 24019784v^14 + 475927878v^13 - 305627919v^12 - 3362050773v^11 + 2039560729v^10 + 13338636489v^9 - 7638540586v^8 - 29395429395v^7 + 15994153823v^6 + 33291594815v^5 - 17660256118v^4 - 15839932236v^3 + 8623626588v^2 + 1722003656*v - 857558448) / 30810256
$\beta_{14}$	$=$	$( 2667907 \nu^{17} + 32580 \nu^{16} - 87752232 \nu^{15} - 4559040 \nu^{14} + 1182540091 \nu^{13} + \cdots + 516806112 ) / 61620512$ (2667907v^17 + 32580v^16 - 87752232v^15 - 4559040v^14 + 1182540091v^13 + 107107077v^12 - 8382337897v^11 - 1067960743v^10 + 33353509092v^9 + 5361225739v^8 - 73532894247v^7 - 13693267065v^6 + 82563558630v^5 + 15982177480v^4 - 37665905116v^3 - 6234728264v^2 + 3062244880*v + 516806112) / 61620512
$\beta_{15}$	$=$	$( - 5514599 \nu^{17} + 1603132 \nu^{16} + 181517160 \nu^{15} - 43215552 \nu^{14} + \cdots - 176155936 ) / 123241024$ (-5514599v^17 + 1603132v^16 + 181517160v^15 - 43215552v^14 - 2445879687v^13 + 451449487v^12 + 17313578493v^11 - 2260208101v^10 - 68667838740v^9 + 5287746313v^8 + 150540568131v^7 - 4036169899v^6 - 167758557078v^5 - 2119413184v^4 + 76080109356v^3 + 2079876600v^2 - 6622603248*v - 176155936) / 123241024
$\beta_{16}$	$=$	$( 19147913 \nu^{17} - 20921528 \nu^{16} - 620919424 \nu^{15} + 652093696 \nu^{14} + \cdots - 18920576800 ) / 246482048$ (19147913v^17 - 20921528v^16 - 620919424v^15 + 652093696v^14 + 8252539369v^13 - 8309014789v^12 - 57763757399v^11 + 55597804399v^10 + 227622823352v^9 - 208805051551v^8 - 500298836081v^7 + 436330711297v^6 + 568461065054v^5 - 471540118064v^4 - 271276764948v^3 + 215172645384v^2 + 24987628688*v - 18920576800) / 246482048
$\beta_{17}$	$=$	$( 5467460 \nu^{17} - 2351091 \nu^{16} - 180215470 \nu^{15} + 71663140 \nu^{14} + 2434217780 \nu^{13} + \cdots - 1937384048 ) / 61620512$ (5467460v^17 - 2351091v^16 - 180215470v^15 + 71663140v^14 + 2434217780v^13 - 889910055v^12 - 17304097363v^11 + 5782165259v^10 + 69133812585v^9 - 21044852890v^8 - 153450076947v^7 + 42849697649v^6 + 174548342207v^5 - 46343193588v^4 - 82033141208v^3 + 22406476636v^2 + 7488185504*v - 1937384048) / 61620512

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 4$ b2 + 4
$\nu^{3}$	$=$	$\beta_{3} + 6\beta_1$ b3 + 6*b1
$\nu^{4}$	$=$	$\beta_{15} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{5} + \beta_{4} + 8\beta_{2} - \beta _1 + 25$ b15 + b13 + b11 + b9 + b5 + b4 + 8*b2 - b1 + 25
$\nu^{5}$	$=$	$\beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 10\beta_{3} + 38\beta _1 + 1$ b11 - b8 + b7 + b6 - b5 + 10b3 + 38b1 + 1
$\nu^{6}$	$=$	$- \beta_{17} + 10 \beta_{15} - \beta_{14} + 12 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} + \beta_{10} + \cdots + 168$ -b17 + 10b15 - b14 + 12b13 + 2b12 + 10b11 + b10 + 10b9 + b8 - b7 + 10b5 + 10b4 + b3 + 58b2 - 10*b1 + 168
$\nu^{7}$	$=$	$- \beta_{16} - \beta_{15} + \beta_{14} - 3 \beta_{13} + 19 \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots + 16$ -b16 - b15 + b14 - 3b13 + 19b11 + 2b10 - b9 - 15b8 + 14b7 + 17b6 - 13b5 + b4 + 84b3 + b2 + 245*b1 + 16
$\nu^{8}$	$=$	$- 14 \beta_{17} + 78 \beta_{15} - 20 \beta_{14} + 109 \beta_{13} + 32 \beta_{12} + 82 \beta_{11} + \cdots + 1160$ -14b17 + 78b15 - 20b14 + 109b13 + 32b12 + 82b11 + 17b10 + 81b9 + 16b8 - 16b7 + 81b5 + 79b4 + 18b3 + 412b2 - 78*b1 + 1160
$\nu^{9}$	$=$	$- 4 \beta_{17} - 14 \beta_{16} - 19 \beta_{15} + 16 \beta_{14} - 51 \beta_{13} + 231 \beta_{11} + \cdots + 181$ -4b17 - 14b16 - 19b15 + 16b14 - 51b13 + 231b11 + 36b10 - 19b9 - 162b8 + 142b7 + 190b6 - 129b5 + 15b4 + 675b3 + 13b2 + 1601b1 + 181
$\nu^{10}$	$=$	$- 138 \beta_{17} - 6 \beta_{16} + 559 \beta_{15} - 254 \beta_{14} + 905 \beta_{13} + 350 \beta_{12} + \cdots + 8140$ -138b17 - 6b16 + 559b15 - 254b14 + 905b13 + 350b12 + 654b11 + 206b10 + 621b9 + 177b8 - 173b7 + 7b6 + 628b5 + 587b4 + 221b3 + 2920b2 - 573*b1 + 8140
$\nu^{11}$	$=$	$- 81 \beta_{17} - 144 \beta_{16} - 247 \beta_{15} + 181 \beta_{14} - 587 \beta_{13} - 2 \beta_{12} + \cdots + 1774$ -81b17 - 144b16 - 247b15 + 181b14 - 587b13 - 2b12 + 2350b11 + 437b10 - 241b9 - 1552b8 + 1278b7 + 1813b6 - 1150b5 + 153b4 + 5347b3 + 110b2 + 10579*b1 + 1774
$\nu^{12}$	$=$	$- 1185 \beta_{17} - 137 \beta_{16} + 3835 \beta_{15} - 2662 \beta_{14} + 7237 \beta_{13} + 3268 \beta_{12} + \cdots + 57823$ -1185b17 - 137b16 + 3835b15 - 2662b14 + 7237b13 + 3268b12 + 5260b11 + 2179b10 + 4681b9 + 1669b8 - 1584b7 + 176b6 + 4842b5 + 4309b4 + 2319b3 + 20753b2 - 4187*b1 + 57823
$\nu^{13}$	$=$	$- 1081 \beta_{17} - 1319 \beta_{16} - 2721 \beta_{15} + 1786 \beta_{14} - 5742 \beta_{13} + \cdots + 16118$ -1081b17 - 1319b16 - 2721b15 + 1786b14 - 5742b13 - 50b12 + 21825b11 + 4536b10 - 2574b9 - 14012b8 + 10855b7 + 16001b6 - 9680b5 + 1344b4 + 42117b3 + 721b2 + 70587*b1 + 16118
$\nu^{14}$	$=$	$- 9476 \beta_{17} - 1993 \beta_{16} + 25488 \beta_{15} - 25193 \beta_{14} + 56809 \beta_{13} + \cdots + 414993$ -9476b17 - 1993b16 + 25488b15 - 25193b14 + 56809b13 + 28046b12 + 42834b11 + 21395b10 + 35099b9 + 14367b8 - 13252b7 + 2763b6 + 37463b5 + 31771b4 + 22401b3 + 148088b2 - 30992*b1 + 414993
$\nu^{15}$	$=$	$- 12018 \beta_{17} - 11392 \beta_{16} - 27312 \beta_{15} + 16418 \beta_{14} - 51517 \beta_{13} + \cdots + 140033$ -12018b17 - 11392b16 - 27312b15 + 16418b14 - 51517b13 - 810b12 + 192225b11 + 43493b10 - 25013b9 - 122107b8 + 89253b7 + 135097b6 - 78624b5 + 10995b4 + 330784b3 + 3455b2 + 475182*b1 + 140033
$\nu^{16}$	$=$	$- 72575 \beta_{17} - 23670 \beta_{16} + 164341 \beta_{15} - 224285 \beta_{14} + 441297 \beta_{13} + \cdots + 3005281$ -72575b17 - 23670b16 + 164341b15 - 224285b14 + 441297b13 + 228694b12 + 351787b11 + 200203b10 + 262791b9 + 116441b8 - 104871b7 + 34862b6 + 291173b5 + 236399b4 + 205841b3 + 1061007b2 - 233359*b1 + 3005281
$\nu^{17}$	$=$	$- 120606 \beta_{17} - 95053 \beta_{16} - 258403 \beta_{15} + 144563 \beta_{14} - 438785 \beta_{13} + \cdots + 1182797$ -120606b17 - 95053b16 - 258403b15 + 144563b14 - 438785b13 - 10764b12 + 1637180b11 + 398144b10 - 229043b9 - 1039044b8 + 719287b7 + 1109796b6 - 623270b5 + 86643b4 + 2592961b3 + 4705b2 + 3225463*b1 + 1182797

Display $\beta_i$ in terms of $\nu^j$

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

2.78505
2.67329
2.53901
2.45014
1.76982
1.47536
1.43785
0.899027
0.348479
0.305904
−0.335211
−1.01684
−1.43913
−1.63907
−2.41658
−2.50213
−2.58952
−2.74545

−2.78505

−1.67120

5.75652

−1.00000

4.65438

−4.20569

−10.4621

−0.207087

2.78505

1.2

−2.67329

3.37901

5.14647

−1.00000

−9.03307

0.692364

−8.41144

8.41770

2.67329

1.3

−2.53901

−3.30364

4.44659

−1.00000

8.38798

3.67389

−6.21194

7.91401

2.53901

1.4

−2.45014

−0.0247552

4.00319

−1.00000

0.0606537

1.10013

−4.90809

−2.99939

2.45014

1.5

−1.76982

2.26705

1.13225

−1.00000

−4.01226

3.82507

1.53576

2.13951

1.76982

1.6

−1.47536

−2.78489

0.176690

−1.00000

4.10872

−0.173538

2.69004

4.75562

1.47536

1.7

−1.43785

−0.254287

0.0674170

−1.00000

0.365627

−3.46667

2.77877

−2.93534

1.43785

1.8

−0.899027

−0.0342120

−1.19175

−1.00000

0.0307575

4.96999

2.86947

−2.99883

0.899027

1.9

−0.348479

1.65774

−1.87856

−1.00000

−0.577689

−0.779308

1.35160

−0.251882

0.348479

1.10

−0.305904

−0.840395

−1.90642

−1.00000

0.257080

−4.91901

1.19499

−2.29374

0.305904

1.11

0.335211

2.61395

−1.88763

−1.00000

0.876225

−0.699553

−1.30318

3.83275

−0.335211

1.12

1.01684

−3.08977

−0.966037

−1.00000

−3.14180

−3.86206

−3.01598

6.54668

−1.01684

1.13

1.43913

2.89473

0.0711026

−1.00000

4.16590

4.37243

−2.77594

5.37946

−1.43913

1.14

1.63907

−1.56822

0.686561

−1.00000

−2.57043

3.96416

−2.15282

−0.540673

−1.63907

1.15

2.41658

0.530330

3.83984

−1.00000

1.28158

1.66759

4.44611

−2.71875

−2.41658

1.16

2.50213

2.91576

4.26066

−1.00000

7.29562

−4.35529

5.65646

5.50167

−2.50213

1.17

2.58952

1.68398

4.70561

−1.00000

4.36069

2.60507

7.00623

−0.164218

−2.58952

1.18

2.74545

−2.37118

5.53751

−1.00000

−6.50997

−0.409554

9.71206

2.62250

−2.74545

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$151$	$-1$

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	755.2.a.k	✓	18
3.b	odd	2	1		6795.2.a.bi		18
5.b	even	2	1		3775.2.a.r		18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
755.2.a.k	✓	18	1.a	even	1	1	trivial
3775.2.a.r		18	5.b	even	2	1
6795.2.a.bi		18	3.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 $S_{2}^{\mathrm{new}}(\Gamma_0(755))$ :

T_{2}^{18} + 2 T_{2}^{17} - 32 T_{2}^{16} - 64 T_{2}^{15} + 417 T_{2}^{14} + 839 T_{2}^{13} - 2829 T_{2}^{12} + \cdots + 576

T_{3}^{18} - 2 T_{3}^{17} - 41 T_{3}^{16} + 79 T_{3}^{15} + 681 T_{3}^{14} - 1245 T_{3}^{13} - 5903 T_{3}^{12} + \cdots + 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{18} + 2 T^{17} + \cdots + 576$ T^18 + 2T^17 - 32T^16 - 64T^15 + 417T^14 + 839T^13 - 2829T^12 - 5789T^11 + 10562T^10 + 22471T^9 - 20867T^8 - 48611T^7 + 18052T^6 + 54316T^5 - 788T^4 - 25424T^3 - 5248T^2 + 2240*T + 576
$3$	$T^{18} - 2 T^{17} + \cdots + 8$ T^18 - 2T^17 - 41T^16 + 79T^15 + 681T^14 - 1245T^13 - 5903T^12 + 9978T^11 + 28723T^10 - 42888T^9 - 78731T^8 + 95263T^7 + 115239T^6 - 93909T^5 - 75829T^4 + 22944T^3 + 10988T^2 + 580*T + 8
$5$	$(T + 1)^{18}$ (T + 1)^18
$7$	$T^{18} - 4 T^{17} + \cdots - 187232$ T^18 - 4T^17 - 86T^16 + 358T^15 + 2881T^14 - 12769T^13 - 46273T^12 + 228979T^11 + 336324T^10 - 2118099T^9 - 519773T^8 + 9015703T^7 - 3960242T^6 - 11773400T^5 + 4865800T^4 + 6758416T^3 - 993584T^2 - 1419632*T - 187232
$11$	$T^{18} - 11 T^{17} + \cdots + 15945984$ T^18 - 11T^17 - 77T^16 + 1048T^15 + 2422T^14 - 41002T^13 - 43790T^12 + 831087T^11 + 553465T^10 - 9009991T^9 - 4389617T^8 + 49474254T^7 + 10230340T^6 - 134105332T^5 + 14598652T^4 + 149211419T^3 - 39005796T^2 - 54177280*T + 15945984
$13$	$T^{18} - 12 T^{17} + \cdots + 6024352$ T^18 - 12T^17 - 76T^16 + 1382T^15 + 206T^14 - 57697T^13 + 106891T^12 + 1104605T^11 - 3237771T^10 - 9730094T^9 + 37420322T^8 + 33048860T^7 - 178808080T^6 - 24993269T^5 + 327835951T^4 - 5902359T^3 - 207891988T^2 - 8703512*T + 6024352
$17$	$T^{18} + 25 T^{17} + \cdots + 8306688$ T^18 + 25T^17 + 137T^16 - 1555T^15 - 20833T^14 - 38702T^13 + 575812T^12 + 3425636T^11 + 1326520T^10 - 44785888T^9 - 146392128T^8 - 49605504T^7 + 709116928T^6 + 1930957312T^5 + 2457801728T^4 + 1739039744T^3 + 676550656T^2 + 128327680*T + 8306688
$19$	$T^{18} - 8 T^{17} + \cdots - 324288$ T^18 - 8T^17 - 164T^16 + 1454T^15 + 8854T^14 - 96872T^13 - 141195T^12 + 2890012T^11 - 2109515T^10 - 36753688T^9 + 72576222T^8 + 135465142T^7 - 403449804T^6 - 2707384T^5 + 499525777T^4 - 304604016T^3 + 36505856T^2 + 3237168*T - 324288
$23$	$T^{18} + \cdots + 143258712$ T^18 - 12T^17 - 180T^16 + 2442T^15 + 12262T^14 - 198741T^13 - 394845T^12 + 8288815T^11 + 6445763T^10 - 187970342T^9 - 75884330T^8 + 2249130280T^7 + 1203627760T^6 - 12173604221T^5 - 10647244341T^4 + 13061143327T^3 + 5183279294T^2 - 2012277356*T + 143258712
$29$	$T^{18} + \cdots - 3481410264$ T^18 - 24T^17 + 25T^16 + 3684T^15 - 28040T^14 - 107999T^13 + 1959132T^12 - 4293349T^11 - 34567057T^10 + 178608486T^9 + 5183617T^8 - 1579090902T^7 + 2305265890T^6 + 3966931869T^5 - 8927765934T^4 - 4009116889T^3 + 11398794950T^2 + 2430536612*T - 3481410264
$31$	$T^{18} + \cdots + 12920205824$ T^18 - 11T^17 - 255T^16 + 2753T^15 + 27261T^14 - 273887T^13 - 1632571T^12 + 13725197T^11 + 61602027T^10 - 358140881T^9 - 1492892029T^8 + 4307166259T^7 + 20742071999T^6 - 9464233157T^5 - 117328600201T^4 - 139122598337T^3 - 26679983080T^2 + 32901774144*T + 12920205824
$37$	$T^{18} + \cdots + 27363905232896$ T^18 - 45T^17 + 563T^16 + 3721T^15 - 141849T^14 + 728292T^13 + 8473512T^12 - 100604376T^11 + 14311536T^10 + 4240975824T^9 - 15543312144T^8 - 59783245248T^7 + 463909410368T^6 - 236251387136T^5 - 4464639624448T^4 + 10110693956608T^3 + 5665350761472T^2 - 36430023598080*T + 27363905232896
$41$	$T^{18} + \cdots - 789639168$ T^18 - 12T^17 - 320T^16 + 4457T^15 + 33045T^14 - 606076T^13 - 973736T^12 + 38868156T^11 - 42087448T^10 - 1207508784T^9 + 3190780304T^8 + 15718131392T^7 - 60373540672T^6 - 41679203904T^5 + 281239040384T^4 - 29252832000T^3 - 291763197440T^2 + 48145307648*T - 789639168
$43$	$T^{18} + \cdots + 123008974848$ T^18 - 19T^17 - 332T^16 + 7352T^15 + 38281T^14 - 1076290T^13 - 1886056T^12 + 76054512T^11 + 44736896T^10 - 2724639936T^9 - 1045812864T^8 + 47694677504T^7 + 24040108800T^6 - 348749611520T^5 - 124587986944T^4 + 1003586891776T^3 + 14024458240T^2 - 745697574912*T + 123008974848
$47$	$T^{18} + \cdots + 268399804416$ T^18 + 16T^17 - 282T^16 - 5305T^15 + 28207T^14 + 711996T^13 - 915620T^12 - 49409248T^11 - 32317264T^10 + 1878192064T^9 + 3228773440T^8 - 38288867072T^7 - 87417735680T^6 + 386067189760T^5 + 950197108736T^4 - 1626162511872T^3 - 3473962500096T^2 + 1858254405632*T + 268399804416
$53$	$T^{18} + \cdots + 41277800448$ T^18 + 4T^17 - 439T^16 - 884T^15 + 73849T^14 - 16076T^13 - 6010104T^12 + 14146452T^11 + 226493440T^10 - 997194736T^9 - 2577490992T^8 + 21089795968T^7 - 19489026112T^6 - 82893356480T^5 + 133889908480T^4 + 95582102528T^3 - 186658746368T^2 + 2023460864*T + 41277800448
$59$	$T^{18} + \cdots - 322195609536$ T^18 + 8T^17 - 509T^16 - 4004T^15 + 95544T^14 + 744479T^13 - 8133252T^12 - 62970723T^11 + 315957965T^10 + 2397993230T^9 - 5514532701T^8 - 37125814986T^7 + 58434154894T^6 + 230885074155T^5 - 365398003666T^4 - 406099178559T^3 + 700914311132T^2 + 142182273520*T - 322195609536
$61$	$T^{18} + \cdots + 49054901641216$ T^18 - 27T^17 - 186T^16 + 10808T^15 - 35005T^14 - 1532270T^13 + 11343044T^12 + 91408092T^11 - 1083154648T^10 - 1573029104T^9 + 47374103056T^8 - 56616266336T^7 - 989075892672T^6 + 2612110724288T^5 + 8462903952512T^4 - 31501847779072T^3 - 13853487150592T^2 + 74370697412608*T + 49054901641216
$67$	$T^{18} + \cdots - 22497872398488$ T^18 - 9T^17 - 525T^16 + 3494T^15 + 115930T^14 - 452300T^13 - 13957074T^12 + 16205029T^11 + 959379387T^10 + 1203298543T^9 - 35255313301T^8 - 120158530184T^7 + 529450609276T^6 + 3415413765946T^5 + 1553308325524T^4 - 28000957926131T^3 - 78284186288410T^2 - 77948174330292*T - 22497872398488
$71$	$T^{18} + \cdots - 190086356140032$ T^18 - 9T^17 - 696T^16 + 6204T^15 + 200921T^14 - 1768916T^13 - 31006336T^12 + 267851620T^11 + 2749477856T^10 - 22961668912T^9 - 140642123184T^8 + 1097026216896T^7 + 4013713565056T^6 - 26925790821568T^5 - 60777568952832T^4 + 295172526731008T^3 + 376449732333568T^2 - 1091231865626624*T - 190086356140032
$73$	$T^{18} + \cdots + 11\!\cdots\!44$ T^18 - 2T^17 - 787T^16 + 972T^15 + 253468T^14 - 140783T^13 - 43429688T^12 - 3980907T^11 + 4301362333T^10 + 3126567644T^9 - 249960469211T^8 - 336613042538T^7 + 8224427706582T^6 + 15964874452941T^5 - 137537294364386T^4 - 347819101670007T^3 + 832514518756760T^2 + 2767510630886712*T + 1134367140516544
$79$	$T^{18} + \cdots - 14\!\cdots\!84$ T^18 - 20T^17 - 729T^16 + 16690T^15 + 187427T^14 - 5545128T^13 - 15960756T^12 + 933322588T^11 - 1240343504T^10 - 83199711440T^9 + 338242532720T^8 + 3630273643456T^7 - 24230954410112T^6 - 49297998747456T^5 + 686567178427136T^4 - 935679050651136T^3 - 5158392530851840T^2 + 17377259283677184*T - 14982704922230784
$83$	$T^{18} + \cdots + 40860420133992$ T^18 + 20T^17 - 360T^16 - 9128T^15 + 33864T^14 + 1605077T^13 + 1878655T^12 - 134773071T^11 - 510139165T^10 + 5346991158T^9 + 30914489962T^8 - 81302885510T^7 - 721428142102T^6 + 174867305277T^5 + 6895527120055T^4 + 3199075883621T^3 - 27865255159398T^2 - 12708706070276*T + 40860420133992
$89$	$T^{18} + \cdots + 309732808310784$ T^18 - 13T^17 - 652T^16 + 6576T^15 + 192893T^14 - 1264656T^13 - 32975464T^12 + 101831244T^11 + 3401342264T^10 - 445141424T^9 - 202650601040T^8 - 455771729344T^7 + 6014675990464T^6 + 26240502366912T^5 - 50679959644800T^4 - 438808798648832T^3 - 526775790855168T^2 + 447478996156416*T + 309732808310784
$97$	$T^{18} + \cdots - 70\!\cdots\!28$ T^18 + 27T^17 - 708T^16 - 24812T^15 + 136681T^14 + 9071484T^13 + 13786144T^12 - 1679042236T^11 - 9021702488T^10 + 162783082656T^9 + 1388002565296T^8 - 7160977012544T^7 - 96897013869568T^6 + 22871306745024T^5 + 2884950065478528T^4 + 6413832731109376T^3 - 19022792131469312T^2 - 57231091162939392*T - 7095906339913728
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.18
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	755.2.a.k

Level	$755$
Weight	$2$
Character orbit	755.a
Self dual	yes
Analytic conductor	$6.029$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$