LMFDB - Newform orbit 114.2.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [114,2,Mod(29,114)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 17])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("114.29"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$114 = 2 \cdot 3 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	114.l (of order $18$ , degree $6$ , minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.910294583043$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$3$ over $\Q(\zeta_{18})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683$ x^18 - x^15 - 18x^14 + 36x^13 + 10x^12 + 18x^11 + 90x^10 - 567x^9 + 270x^8 + 162x^7 + 270x^6 + 2916x^5 - 4374x^4 - 729x^3 + 19683
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$ -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.

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

Basis of coefficient ring in terms of a root $\nu$ of $x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683$ x^18 - x^15 - 18*x^14 + 36*x^13 + 10*x^12 + 18*x^11 + 90*x^10 - 567*x^9 + 270*x^8 + 162*x^7 + 270*x^6 + 2916*x^5 - 4374*x^4 - 729*x^3 + 19683 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - \nu^{17} + \nu^{14} + 18 \nu^{13} - 36 \nu^{12} - 10 \nu^{11} - 18 \nu^{10} - 90 \nu^{9} + \cdots + 729 \nu^{2} ) / 6561$ (-v^17 + v^14 + 18v^13 - 36v^12 - 10v^11 - 18v^10 - 90v^9 + 567v^8 - 270v^7 - 162v^6 - 270v^5 - 2916v^4 + 4374v^3 + 729v^2) / 6561
$\beta_{3}$	$=$	$( 2 \nu^{17} + 21 \nu^{16} - 81 \nu^{15} + 97 \nu^{14} - 57 \nu^{13} - 198 \nu^{12} + 1568 \nu^{11} + \cdots - 65610 ) / 4374$ (2v^17 + 21v^16 - 81v^15 + 97v^14 - 57v^13 - 198v^12 + 1568v^11 - 3075v^10 + 2313v^9 - 117v^8 - 9261v^7 + 29268v^6 - 31374v^5 + 11421v^4 + 11421v^3 - 82377v^2 + 150903*v - 65610) / 4374
$\beta_{4}$	$=$	$( - 5 \nu^{17} + 99 \nu^{16} - 153 \nu^{15} - 22 \nu^{14} + 207 \nu^{13} - 1485 \nu^{12} + \cdots + 334611 ) / 13122$ (-5v^17 + 99v^16 - 153v^15 - 22v^14 + 207v^13 - 1485v^12 + 4675v^11 - 2637v^10 - 4167v^9 + 10098v^8 - 38475v^7 + 51273v^6 + 24003v^5 - 60507v^4 + 91125v^3 - 237654v^2 + 63423*v + 334611) / 13122
$\beta_{5}$	$=$	$( \nu^{17} + 2 \nu^{16} - 27 \nu^{15} + 50 \nu^{14} - 47 \nu^{13} + 9 \nu^{12} + 409 \nu^{11} + \cdots - 72171 ) / 1458$ (v^17 + 2v^16 - 27v^15 + 50v^14 - 47v^13 + 9v^12 + 409v^11 - 1366v^10 + 1629v^9 - 822v^8 - 1809v^7 + 11025v^6 - 19359v^5 + 12420v^4 - 81v^3 - 24786v^2 + 75087v - 72171) / 1458
$\beta_{6}$	$=$	$( - 16 \nu^{17} + 81 \nu^{16} + 36 \nu^{15} - 335 \nu^{14} + 477 \nu^{13} - 1422 \nu^{12} + \cdots + 603612 ) / 13122$ (-16v^17 + 81v^16 + 36v^15 - 335v^14 + 477v^13 - 1422v^12 + 1406v^11 + 6165v^10 - 12420v^9 + 12609v^8 - 20493v^7 - 25272v^6 + 127062v^5 - 103761v^4 + 67554v^3 - 41553v^2 - 373977*v + 603612) / 13122
$\beta_{7}$	$=$	$( - 11 \nu^{17} - 3 \nu^{16} - 6 \nu^{15} + 92 \nu^{14} + 48 \nu^{13} - 255 \nu^{12} - 137 \nu^{11} + \cdots - 225261 ) / 4374$ (-11v^17 - 3v^16 - 6v^15 + 92v^14 + 48v^13 - 255v^12 - 137v^11 - 1425v^10 + 3108v^9 + 1350v^8 - 504v^7 + 3645v^6 - 36045v^5 + 26001v^4 + 10854v^3 + 8262v^2 + 74358*v - 225261) / 4374
$\beta_{8}$	$=$	$( - 5 \nu^{17} - 105 \nu^{16} + 90 \nu^{15} + 275 \nu^{14} - 318 \nu^{13} + 1296 \nu^{12} + \cdots - 787320 ) / 13122$ (-5v^17 - 105v^16 + 90v^15 + 275v^14 - 318v^13 + 1296v^12 - 4181v^11 - 2247v^10 + 12492v^9 - 11367v^8 + 36126v^7 - 25110v^6 - 114507v^5 + 125469v^4 - 81648v^3 + 210681v^2 + 183708*v - 787320) / 13122
$\beta_{9}$	$=$	$( - 11 \nu^{17} + 147 \nu^{16} - 216 \nu^{15} - 43 \nu^{14} + 375 \nu^{13} - 2259 \nu^{12} + \cdots + 452709 ) / 13122$ (-11v^17 + 147v^16 - 216v^15 - 43v^14 + 375v^13 - 2259v^12 + 6451v^11 - 3183v^10 - 6660v^9 + 15525v^8 - 52947v^7 + 64233v^6 + 40041v^5 - 88857v^4 + 125388v^3 - 292329v^2 + 50301*v + 452709) / 13122
$\beta_{10}$	$=$	$( - 17 \nu^{17} - 5 \nu^{16} + 99 \nu^{15} - 136 \nu^{14} + 284 \nu^{13} - 405 \nu^{12} - 1655 \nu^{11} + \cdots + 63423 ) / 4374$ (-17v^17 - 5v^16 + 99v^15 - 136v^14 + 284v^13 - 405v^12 - 1655v^11 + 4369v^10 - 4167v^9 + 5472v^8 + 5508v^7 - 41229v^6 + 46683v^5 - 25569v^4 + 13851v^3 + 103518v^2 - 237654*v + 63423) / 4374
$\beta_{11}$	$=$	$( - 4 \nu^{17} + 17 \nu^{16} - 10 \nu^{15} - 23 \nu^{14} + 82 \nu^{13} - 332 \nu^{12} + 536 \nu^{11} + \cdots + 56862 ) / 1458$ (-4v^17 + 17v^16 - 10v^15 - 23v^14 + 82v^13 - 332v^12 + 536v^11 + 197v^10 - 1234v^9 + 2619v^8 - 5760v^7 + 2430v^6 + 9666v^5 - 12825v^4 + 18198v^3 - 24057v^2 - 20412*v + 56862) / 1458
$\beta_{12}$	$=$	$( - 23 \nu^{17} + 10 \nu^{16} + 6 \nu^{15} + 86 \nu^{14} + 161 \nu^{13} - 717 \nu^{12} - 41 \nu^{11} + \cdots - 247131 ) / 4374$ (-23v^17 + 10v^16 + 6v^15 + 86v^14 + 161v^13 - 717v^12 - 41v^11 - 908v^10 + 2814v^9 + 4716v^8 - 4941v^7 - 1377v^6 - 32373v^5 + 23436v^4 + 35640v^3 + 7290v^2 + 26973*v - 247131) / 4374
$\beta_{13}$	$=$	$( - 9 \nu^{17} - 4 \nu^{16} + 39 \nu^{15} - 21 \nu^{14} + 94 \nu^{13} - 174 \nu^{12} - 717 \nu^{11} + \cdots - 34992 ) / 1458$ (-9v^17 - 4v^16 + 39v^15 - 21v^14 + 94v^13 - 174v^12 - 717v^11 + 1220v^10 - 393v^9 + 1797v^8 + 2520v^7 - 14598v^6 + 6345v^5 + 270v^4 + 5913v^3 + 41553v^2 - 67068*v - 34992) / 1458
$\beta_{14}$	$=$	$( - 76 \nu^{17} - 108 \nu^{16} + 459 \nu^{15} - 194 \nu^{14} + 747 \nu^{13} - 522 \nu^{12} + \cdots - 551124 ) / 13122$ (-76v^17 - 108v^16 + 459v^15 - 194v^14 + 747v^13 - 522v^12 - 9724v^11 + 13104v^10 - 1521v^9 + 9774v^8 + 50193v^7 - 167832v^6 + 45090v^5 + 39366v^4 - 13851v^3 + 546750v^2 - 649539*v - 551124) / 13122
$\beta_{15}$	$=$	$( 11 \nu^{17} - 17 \nu^{16} - 3 \nu^{15} + 13 \nu^{14} - 145 \nu^{13} + 525 \nu^{12} - 283 \nu^{11} + \cdots + 10935 ) / 1458$ (11v^17 - 17v^16 - 3v^15 + 13v^14 - 145v^13 + 525v^12 - 283v^11 - 413v^10 + 807v^9 - 4125v^8 + 5787v^7 + 2817v^6 - 5103v^5 + 7695v^4 - 26811v^3 + 7047v^2 + 37179*v + 10935) / 1458
$\beta_{16}$	$=$	$( 103 \nu^{17} - 99 \nu^{16} - 27 \nu^{15} - 157 \nu^{14} - 1026 \nu^{13} + 4140 \nu^{12} + \cdots + 669222 ) / 13122$ (103v^17 - 99v^16 - 27v^15 - 157v^14 - 1026v^13 + 4140v^12 - 1265v^11 + 621v^10 - 3555v^9 - 30429v^8 + 39960v^7 + 12150v^6 + 60615v^5 - 24057v^4 - 216513v^3 + 22599v^2 + 74358*v + 669222) / 13122
$\beta_{17}$	$=$	$( - 17 \nu^{17} + 10 \nu^{16} + 27 \nu^{15} - 10 \nu^{14} + 188 \nu^{13} - 576 \nu^{12} - 269 \nu^{11} + \cdots - 78732 ) / 1458$ (-17v^17 + 10v^16 + 27v^15 - 10v^14 + 188v^13 - 576v^12 - 269v^11 + 874v^10 - 351v^9 + 4680v^8 - 3078v^7 - 10746v^6 + 1161v^5 - 702v^4 + 26973v^3 + 20412v^2 - 55404*v - 78732) / 1458

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{16} + \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{4}$ -b16 + b15 + b9 - b8 - 2*b4
$\nu^{3}$	$=$	$-\beta_{15} + \beta_{14} - 4\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2}$ -b15 + b14 - 4b7 - 2b6 + b5 + b3 + 2*b2
$\nu^{4}$	$=$	$- \beta_{17} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + \cdots + 2$ -b17 + b13 + b12 + 2b11 + 2b10 - 3b7 - 3b6 + 2b5 - 3b2 + b1 + 2
$\nu^{5}$	$=$	$- 3 \beta_{17} - 2 \beta_{16} + \beta_{15} + 3 \beta_{13} + 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + \cdots - 12$ -3b17 - 2b16 + b15 + 3b13 + 3b10 - b9 - 2b8 + 3b5 + 2b4 + b3 + 6b1 - 12
$\nu^{6}$	$=$	$- 9 \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \cdots - 6$ -9b16 + 10b15 + 2b14 - b13 - 3b12 + 3b11 + 3b10 - 2b7 + 2b6 + 5b3 + b2 - 9b1 - 6
$\nu^{7}$	$=$	$- 4 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} + 15 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \cdots - 2$ -4b17 + 9b16 - 9b15 + 15b14 - 7b13 + 2b12 + 4b11 + b10 - 9b9 + 18b8 - 12b7 + 7b5 + 36b4 + 15b2 - 3b1 - 2
$\nu^{8}$	$=$	$- 18 \beta_{17} - \beta_{16} + 15 \beta_{15} - 9 \beta_{14} + 21 \beta_{12} + 24 \beta_{11} + \cdots + 9 \beta_1$ -18b17 - b16 + 15b15 - 9b14 + 21b12 + 24b11 + 21b10 - 11b9 + 5b8 + 36b7 + 18b6 + 3b5 + 7b4 - 14b3 - 18b2 + 9*b1
$\nu^{9}$	$=$	$- 27 \beta_{17} - 27 \beta_{16} + 11 \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 30 \beta_{11} + \cdots - 30$ -27b17 - 27b16 + 11b15 + b14 - 4b13 + 3b12 - 30b11 + 6b10 - 27b9 - 36b8 + 74b7 + 76b6 - 7b5 + 13b3 + 44b2 + 27*b1 - 30
$\nu^{10}$	$=$	$33 \beta_{17} - 54 \beta_{16} + 63 \beta_{15} + 24 \beta_{14} - 89 \beta_{13} + \beta_{12} + 2 \beta_{11} + \cdots + 140$ 33b17 - 54b16 + 63b15 + 24b14 - 89b13 + b12 + 2b11 - 10b10 - 27b9 + 18b8 + 9b7 + 66b6 - 76b5 - 18b4 + 63b3 + 9b2 - 67b1 + 140
$\nu^{11}$	$=$	$- 15 \beta_{17} + 100 \beta_{16} - 103 \beta_{15} + 126 \beta_{14} - 129 \beta_{13} + 102 \beta_{12} + \cdots + 120$ -15b17 + 100b16 - 103b15 + 126b14 - 129b13 + 102b12 - 30b11 - 63b10 - b9 - 2b8 - 18b7 + 18b6 - 9b5 + 68b4 - 51b3 + 63b2 + 66b1 + 120
$\nu^{12}$	$=$	$- 90 \beta_{17} - 81 \beta_{16} + 136 \beta_{15} - 154 \beta_{14} + 33 \beta_{13} + 249 \beta_{12} + \cdots - 51$ -90b17 - 81b16 + 136b15 - 154b14 + 33b13 + 249b12 + 129b11 + 111b10 + 36b9 - 342b8 + 148b7 + 2b6 - 34b5 - 612b4 + 17b3 - 155b2 + 81*b1 - 51
$\nu^{13}$	$=$	$- 62 \beta_{17} - 171 \beta_{16} - 189 \beta_{15} + 153 \beta_{14} - 19 \beta_{13} - 100 \beta_{12} + \cdots - 2$ -62b17 - 171b16 - 189b15 + 153b14 - 19b13 - 100b12 - 362b11 - 101b10 - 99b9 - 648b8 - 96b7 - 6b6 - 29b5 - 486b4 + 432b3 + 255b2 + 44*b1 - 2
$\nu^{14}$	$=$	$372 \beta_{17} - 106 \beta_{16} + 206 \beta_{15} + 243 \beta_{14} - 426 \beta_{13} + 9 \beta_{12} + \cdots + 354$ 372b17 - 106b16 + 206b15 + 243b14 - 426b13 + 9b12 + 450b11 - 39b10 - 80b9 + 380b8 - 1026b7 - 756b6 - 525b5 - 29b4 + 710b3 - 810b2 - 393*b1 + 354
$\nu^{15}$	$=$	$- 351 \beta_{17} + 765 \beta_{16} - 802 \beta_{15} + 916 \beta_{14} + 46 \beta_{13} + 327 \beta_{12} + \cdots - 2208$ -351b17 + 765b16 - 802b15 + 916b14 + 46b13 + 327b12 + 402b11 - 624b10 + 351b9 - 378b8 - 376b7 - 974b6 + 522b5 + 270b4 - 752b3 - 298b2 + 279*b1 - 2208
$\nu^{16}$	$=$	$- 878 \beta_{17} - 630 \beta_{16} + 927 \beta_{15} - 1554 \beta_{14} + 1726 \beta_{13} + 430 \beta_{12} + \cdots - 4210$ -878b17 - 630b16 + 927b15 - 1554b14 + 1726b13 + 430b12 + 2318b11 + 998b10 - 423b9 - 558b8 + 390b7 - 351b6 + 326b5 - 3060b4 + 729b3 - 852b2 - 1284*b1 - 4210
$\nu^{17}$	$=$	$- 918 \beta_{17} + 406 \beta_{16} - 3786 \beta_{15} + 1656 \beta_{14} - 378 \beta_{13} - 3126 \beta_{12} + \cdots + 54$ -918b17 + 406b16 - 3786b15 + 1656b14 - 378b13 - 3126b12 - 1536b11 - 1074b10 - 3040b9 + 1750b8 + 504b7 + 738b6 - 192b5 + 5150b4 + 1436b3 + 1125b2 - 1566*b1 + 54

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$ .

$n$	$77$	$97$
$\chi(n)$	$-1$	$-\beta_{4} - \beta_{8}$

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}

29.1

−1.72388	+	0.168030i
1.47158	−	0.913487i
0.0786547	+	1.73026i
−0.442647	+	1.67453i
−1.73189	−	0.0237018i
1.40849	−	1.00804i
−0.363139	−	1.69356i
1.69944	−	0.334495i
−0.396613	+	1.68603i
−1.72388	−	0.168030i
1.47158	+	0.913487i
0.0786547	−	1.73026i
−0.363139	+	1.69356i
1.69944	+	0.334495i
−0.396613	−	1.68603i
−0.442647	−	1.67453i
−1.73189	+	0.0237018i
1.40849	+	1.00804i

−0.173648

−

0.984808i

−0.716422

1.57694i

−0.939693

0.342020i

−1.14133

3.13578i

1.67739

0.431705i

−1.07356

1.85947i

0.500000

0.866025i

−1.97348

−

2.25951i

3.28633

0.579469i

29.2

−0.173648

−

0.984808i

−0.0553136

−

1.73117i

−0.939693

0.342020i

0.882820

−

2.42553i

−1.69526

0.355087i

−1.58376

2.74316i

0.500000

0.866025i

−2.99388

0.191514i

−2.54198

−

0.448219i

29.3

−0.173648

−

0.984808i

1.53778

0.797015i

−0.939693

0.342020i

0.258510

−

0.710252i

0.517874

−

1.65282i

0.777943

−

1.34744i

0.500000

0.866025i

1.72953

2.45127i

−0.744351

−

0.131249i

41.1

−0.766044

−

0.642788i

−1.67151

0.453924i

0.173648

0.984808i

1.96615

0.346685i

1.57223

0.726702i

0.910931

1.57778i

0.500000

−

0.866025i

2.58791

−

1.51748i

−1.28331

−

1.52939i

41.2

−0.766044

−

0.642788i

−0.845418

−

1.51171i

0.173648

0.984808i

−2.22841

−

0.392929i

−0.324081

1.70146i

−1.16829

−

2.02354i

0.500000

−

0.866025i

−1.57054

2.55605i

1.45449

1.73339i

41.3

−0.766044

−

0.642788i

1.57724

0.715766i

0.173648

0.984808i

0.262261

0.0462437i

−0.748148

−

1.56214i

0.604656

1.04730i

0.500000

−

0.866025i

1.97536

2.25787i

−0.171179

−

0.204003i

53.1

0.939693

0.342020i

−1.64823

−

0.532290i

0.766044

0.642788i

2.20556

2.62849i

−1.36678

−

1.06392i

1.68651

−

2.92113i

0.500000

0.866025i

2.43333

1.75467i

1.17355

3.22432i

53.2

0.939693

0.342020i

0.560041

−

1.63901i

0.766044

0.642788i

−0.343148

−

0.408948i

1.08684

−

1.34862i

−0.716507

1.24103i

0.500000

0.866025i

−2.37271

−

1.83583i

−0.182585

−

0.501649i

53.3

0.939693

0.342020i

1.26184

1.18649i

0.766044

0.642788i

−1.86241

−

2.21954i

0.779936

1.54651i

0.562083

−

0.973556i

0.500000

0.866025i

0.184473

2.99432i

−0.990970

−

2.72267i

59.1

−0.173648

0.984808i

−0.716422

−

1.57694i

−0.939693

−

0.342020i

−1.14133

−

3.13578i

1.67739

−

0.431705i

−1.07356

−

1.85947i

0.500000

−

0.866025i

−1.97348

2.25951i

3.28633

−

0.579469i

59.2

−0.173648

0.984808i

−0.0553136

1.73117i

−0.939693

−

0.342020i

0.882820

2.42553i

−1.69526

−

0.355087i

−1.58376

−

2.74316i

0.500000

−

0.866025i

−2.99388

−

0.191514i

−2.54198

0.448219i

59.3

−0.173648

0.984808i

1.53778

−

0.797015i

−0.939693

−

0.342020i

0.258510

0.710252i

0.517874

1.65282i

0.777943

1.34744i

0.500000

−

0.866025i

1.72953

−

2.45127i

−0.744351

0.131249i

71.1

0.939693

−

0.342020i

−1.64823

0.532290i

0.766044

−

0.642788i

2.20556

−

2.62849i

−1.36678

1.06392i

1.68651

2.92113i

0.500000

−

0.866025i

2.43333

−

1.75467i

1.17355

−

3.22432i

71.2

0.939693

−

0.342020i

0.560041

1.63901i

0.766044

−

0.642788i

−0.343148

0.408948i

1.08684

1.34862i

−0.716507

−

1.24103i

0.500000

−

0.866025i

−2.37271

1.83583i

−0.182585

0.501649i

71.3

0.939693

−

0.342020i

1.26184

−

1.18649i

0.766044

−

0.642788i

−1.86241

2.21954i

0.779936

−

1.54651i

0.562083

0.973556i

0.500000

−

0.866025i

0.184473

−

2.99432i

−0.990970

2.72267i

89.1

−0.766044

0.642788i

−1.67151

−

0.453924i

0.173648

−

0.984808i

1.96615

−

0.346685i

1.57223

−

0.726702i

0.910931

−

1.57778i

0.500000

0.866025i

2.58791

1.51748i

−1.28331

1.52939i

89.2

−0.766044

0.642788i

−0.845418

1.51171i

0.173648

−

0.984808i

−2.22841

0.392929i

−0.324081

−

1.70146i

−1.16829

2.02354i

0.500000

0.866025i

−1.57054

−

2.55605i

1.45449

−

1.73339i

89.3

−0.766044

0.642788i

1.57724

−

0.715766i

0.173648

−

0.984808i

0.262261

−

0.0462437i

−0.748148

1.56214i

0.604656

−

1.04730i

0.500000

0.866025i

1.97536

−

2.25787i

−0.171179

0.204003i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
57.j	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	114.2.l.a	✓	18
3.b	odd	2	1		114.2.l.b	yes	18
4.b	odd	2	1		912.2.cc.d		18
12.b	even	2	1		912.2.cc.c		18
19.f	odd	18	1		114.2.l.b	yes	18
57.j	even	18	1	inner	114.2.l.a	✓	18
76.k	even	18	1		912.2.cc.c		18
228.u	odd	18	1		912.2.cc.d		18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
114.2.l.a	✓	18	1.a	even	1	1	trivial
114.2.l.a	✓	18	57.j	even	18	1	inner
114.2.l.b	yes	18	3.b	odd	2	1
114.2.l.b	yes	18	19.f	odd	18	1
912.2.cc.c		18	12.b	even	2	1
912.2.cc.c		18	76.k	even	18	1
912.2.cc.d		18	4.b	odd	2	1
912.2.cc.d		18	228.u	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{18} + 9 T_{5}^{16} + 108 T_{5}^{14} + 162 T_{5}^{13} + 46 T_{5}^{12} - 324 T_{5}^{11} + \cdots + 1728$ T5^18 + 9*T5^16 + 108*T5^14 + 162*T5^13 + 46*T5^12 - 324*T5^11 - 1458*T5^10 + 1458*T5^9 - 30258*T5^8 - 5202*T5^7 + 141193*T5^6 - 60552*T5^5 + 70344*T5^4 - 72*T5^3 + 11232*T5^2 - 10368*T5 + 1728 acting on $S_{2}^{\mathrm{new}}(114, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{6} - T^{3} + 1)^{3}$ (T^6 - T^3 + 1)^3
$3$	$T^{18} + T^{15} + \cdots + 19683$ T^18 + T^15 + 18T^13 + 10T^12 + 18T^11 + 90T^10 - 81T^9 + 270T^8 + 162T^7 + 270T^6 + 1458T^5 + 729T^3 + 19683
$5$	$T^{18} + 9 T^{16} + \cdots + 1728$ T^18 + 9T^16 + 108T^14 + 162T^13 + 46T^12 - 324T^11 - 1458T^10 + 1458T^9 - 30258T^8 - 5202T^7 + 141193T^6 - 60552T^5 + 70344T^4 - 72T^3 + 11232T^2 - 10368*T + 1728
$7$	$T^{18} + 21 T^{16} + \cdots + 87616$ T^18 + 21T^16 - 4T^15 + 297T^14 - 90T^13 + 2222T^12 - 1494T^11 + 11985T^10 - 9622T^9 + 43740T^8 - 41064T^7 + 117161T^6 - 100062T^5 + 195300T^4 - 148720T^3 + 216720T^2 - 113664T + 87616
$11$	$T^{18} - 51 T^{16} + \cdots + 35769627$ T^18 - 51T^16 + 1884T^14 + 3933T^13 - 27296T^12 - 95760T^11 + 285702T^10 + 2063799T^9 + 3068313T^8 - 4590198T^7 - 15944255T^6 + 15125949T^5 + 131458797T^4 + 263848509T^3 + 272171448T^2 + 149822217*T + 35769627
$13$	$T^{18} + 12 T^{17} + \cdots + 2365632$ T^18 + 12T^17 + 27T^16 - 84T^15 + 324T^14 + 3708T^13 - 360T^12 - 42480T^11 - 36207T^10 + 175320T^9 - 1056429T^8 - 1225044T^7 + 14891337T^6 - 4914324T^5 + 27392256T^4 - 26484192T^3 + 22638528T^2 - 7384608*T + 2365632
$17$	$T^{18} + 6 T^{17} + \cdots + 3878307$ T^18 + 6T^17 + 21T^16 + 309T^15 + 1224T^14 + 9255T^13 + 45985T^12 + 32520T^11 + 1118904T^10 - 255264T^9 + 3836457T^8 + 9106179T^7 - 1848509T^6 + 9789921T^5 + 29829231T^4 + 14004081T^3 - 5531679T^2 - 11297232*T + 3878307
$19$	$T^{18} + \cdots + 322687697779$ T^18 + 6T^17 - 30T^16 - 275T^15 + 306T^14 + 6408T^13 - 4006T^12 - 139704T^11 + 47316T^10 + 2942011T^9 + 899004T^8 - 50433144T^7 - 27477154T^6 + 835096968T^5 + 757686294T^4 - 12937617275T^3 - 26816152170T^2 + 101901378246*T + 322687697779
$23$	$T^{18} + \cdots + 123187392$ T^18 - 105T^16 + 108T^15 + 3864T^14 - 15768T^13 - 93302T^12 + 516060T^11 + 2793426T^10 + 10843848T^9 + 71012562T^8 + 273220578T^7 + 689100193T^6 + 860428764T^5 + 1062671292T^4 + 696487032T^3 + 617992416T^2 + 294127200T + 123187392
$29$	$T^{18} + \cdots + 52719833664$ T^18 - 6T^17 + 111T^16 - 606T^15 + 3798T^14 - 9324T^13 + 16692T^12 + 424692T^11 + 188226T^10 - 3826152T^9 - 18102744T^8 - 72681030T^7 + 1080612909T^6 + 6298040142T^5 + 12691761048T^4 - 3628092384T^3 - 31128894864T^2 - 16391255904*T + 52719833664
$31$	$T^{18} + \cdots + 6231379854528$ T^18 - 165T^16 + 18621T^14 - 25614T^13 - 1030362T^12 + 1667142T^11 + 41003091T^10 - 54695574T^9 - 1103329728T^8 + 850196898T^7 + 22269166245T^6 + 4904234262T^5 - 278203568436T^4 - 347326742880T^3 + 2135586670416T^2 + 6774606004608*T + 6231379854528
$37$	$T^{18} + \cdots + 7868768303808$ T^18 + 342T^16 + 44991T^14 + 3146832T^12 + 131833359T^10 + 3468414222T^8 + 57645174609T^6 + 585487601676T^4 + 3301605756576T^2 + 7868768303808
$41$	$T^{18} + \cdots + 1846709769969$ T^18 + 3T^17 - 81T^16 - 237T^15 + 567T^14 - 6291T^13 + 578637T^12 + 6446142T^11 + 35530407T^10 + 183830688T^9 + 1414453590T^8 + 5050712961T^7 + 24457418397T^6 + 212090870691T^5 + 936326856420T^4 + 2055695641578T^3 + 3442117775661T^2 + 3383671593780*T + 1846709769969
$43$	$T^{18} + \cdots + 390621250009$ T^18 + 6T^17 - 21T^16 + 1233T^15 - 2391T^14 - 201903T^13 + 1635219T^12 + 4253289T^11 - 85583703T^10 + 259072400T^9 + 834745182T^8 - 13788062184T^7 + 125197505055T^6 - 642244530381T^5 + 1945100254026T^4 - 3193760135025T^3 + 2502920005845T^2 - 141475570914*T + 390621250009
$47$	$T^{18} + \cdots + 3499077312$ T^18 + 30T^17 + 513T^16 + 3696T^15 - 9561T^14 - 526350T^13 - 4210880T^12 - 436206T^11 + 224832402T^10 + 1305742308T^9 - 819624165T^8 - 20956649406T^7 - 20048430563T^6 + 41796892746T^5 + 228877829676T^4 + 235677357072T^3 + 234020746080T^2 + 22163691744*T + 3499077312
$53$	$T^{18} + \cdots + 3426463296$ T^18 - 60T^17 + 1650T^16 - 27750T^15 + 325836T^14 - 2895858T^13 + 20675985T^12 - 122186682T^11 + 599567868T^10 - 2415368574T^9 + 8030520198T^8 - 22246047792T^7 + 51044044113T^6 - 95821382268T^5 + 145477389636T^4 - 162987312888T^3 + 116975876544T^2 - 33524972064*T + 3426463296
$59$	$T^{18} + \cdots + 38983402581561$ T^18 + 3T^17 + 147T^16 + 879T^15 + 15030T^14 + 280800T^13 + 3016905T^12 + 29118996T^11 + 244996146T^10 + 1300488633T^9 + 4427702217T^8 + 7047029133T^7 - 8041532733T^6 - 8631478017T^5 + 775696949469T^4 + 2532697771320T^3 - 5011803296289T^2 - 19432108592010*T + 38983402581561
$61$	$T^{18} + \cdots + 65033160256$ T^18 - 54T^17 + 1419T^16 - 23830T^15 + 285744T^14 - 2581020T^13 + 18467003T^12 - 120249426T^11 + 892275015T^10 - 7207731646T^9 + 50528749842T^8 - 274879113276T^7 + 1082915269121T^6 - 2696150385294T^5 + 3334220844156T^4 - 151176769792T^3 + 2998086559584T^2 + 837712259040*T + 65033160256
$67$	$T^{18} + \cdots + 56\!\cdots\!23$ T^18 + 15T^17 + 261T^16 + 2271T^15 + 51813T^14 + 347733T^13 + 1625472T^12 + 45891909T^11 + 122075937T^10 + 676511640T^9 + 14609136726T^8 - 14656134762T^7 + 452104179777T^6 - 1657979849619T^5 - 7956330419556T^4 + 114565691996118T^3 - 290433671288019T^2 - 1581247049832693*T + 5641379998084323
$71$	$T^{18} + \cdots + 404099233344$ T^18 + 36T^17 + 783T^16 + 12294T^15 + 163431T^14 + 2019816T^13 + 26973540T^12 + 360192096T^11 + 4239884412T^10 + 43380742482T^9 + 379169768331T^8 + 2655253780470T^7 + 14509501175097T^6 + 60645595428594T^5 + 184787209022256T^4 + 367155605037456T^3 + 367044508201824T^2 + 22573947081024*T + 404099233344
$73$	$T^{18} + \cdots + 192753487369$ T^18 + 42T^17 + 987T^16 + 16047T^15 + 167232T^14 + 926085T^13 + 1456731T^12 - 27864642T^11 - 225519060T^10 - 260359090T^9 + 8018230623T^8 + 64815302400T^7 + 270776321586T^6 + 701270458347T^5 + 1256598051213T^4 + 1263361473954T^3 + 379924057347T^2 - 791600833443*T + 192753487369
$79$	$T^{18} + \cdots + 21259626441408$ T^18 + 6T^17 + 105T^16 + 12T^15 + 15705T^14 + 51786T^13 + 468732T^12 + 3097980T^11 - 37037088T^10 + 446160798T^9 - 2597238567T^8 - 11521887120T^7 + 190286398989T^6 - 1218226937454T^5 + 4243049245944T^4 - 5052547513800T^3 - 962723212560T^2 - 7167426056640*T + 21259626441408
$83$	$T^{18} + \cdots + 176145902499843$ T^18 + 36T^17 + 186T^16 - 8856T^15 - 83529T^14 + 2042253T^13 + 36121867T^12 - 13537359T^11 - 3658557183T^10 - 9426572775T^9 + 355076915166T^8 + 3730676567724T^7 + 13153839521878T^6 - 1882524564045T^5 - 93320828111484T^4 + 13302831266856T^3 + 535595856380577T^2 - 534178903806279*T + 176145902499843
$89$	$T^{18} + \cdots + 381874169643201$ T^18 - 60T^17 + 1776T^16 - 32745T^15 + 410364T^14 - 3384864T^13 + 14602713T^12 - 1600794T^11 + 4114944T^10 - 3747657384T^9 + 29902263615T^8 - 47606747436T^7 - 341040944421T^6 + 2162662551999T^5 - 3165140950848T^4 - 42655301651403T^3 + 237935902096233T^2 + 44793101479392*T + 381874169643201
$97$	$T^{18} + \cdots + 17447631785307$ T^18 - 9T^17 - 144T^16 + 4230T^15 - 41472T^14 + 3240T^13 + 5172174T^12 - 93383199T^11 + 1116199845T^10 - 10544774805T^9 + 80427211497T^8 - 461848162446T^7 + 2166057349236T^6 - 8236950703911T^5 + 29093451072057T^4 - 70055543626245T^3 + 112977824953914T^2 - 76196333175057*T + 17447631785307
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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 29.3
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

Character values

Embeddings

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	114.2.l.a

Level	$114$
Weight	$2$
Character orbit	114.l
Analytic conductor	$0.910$
Analytic rank	$0$
Dimension	$18$
Inner twists	$2$