LMFDB - Newform orbit 650.2.w.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(193,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([9, 7]))

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

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

chi := DirichletCharacter("650.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	650.w (of order $12$ , degree $4$ , minimal)

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:	no
Analytic conductor:	$5.19027613138$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81$ x^16 + 28x^14 + 294x^12 + 1516x^10 + 4147x^8 + 6012x^6 + 4338x^4 + 1296*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$ -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.

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

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81$ x^16 + 28*x^14 + 294*x^12 + 1516*x^10 + 4147*x^8 + 6012*x^6 + 4338*x^4 + 1296*x^2 + 81 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 4 \nu^{15} - 8 \nu^{14} + 145 \nu^{13} - 221 \nu^{12} + 2010 \nu^{11} - 2295 \nu^{10} + 13435 \nu^{9} + \cdots + 756 ) / 4968$ (4v^15 - 8v^14 + 145v^13 - 221v^12 + 2010v^11 - 2295v^10 + 13435v^9 - 11759v^8 + 45196v^7 - 31121v^6 + 72339v^5 - 36969v^4 + 45468v^3 - 10413v^2 + 4590*v + 756) / 4968
$\beta_{3}$	$=$	$( 53 \nu^{15} - 138 \nu^{14} + 1490 \nu^{13} - 3657 \nu^{12} + 15489 \nu^{11} - 35397 \nu^{10} + \cdots - 9315 ) / 14904$ (53v^15 - 138v^14 + 1490v^13 - 3657v^12 + 15489v^11 - 35397v^10 + 75911v^9 - 163254v^8 + 178775v^7 - 382053v^6 + 168459v^5 - 431388v^4 - 4680v^3 - 181953v^2 - 78597*v - 9315) / 14904
$\beta_{4}$	$=$	$( 37\nu^{14} + 979\nu^{12} + 9381\nu^{10} + 41836\nu^{8} + 89761\nu^{6} + 83688\nu^{4} + 26658\nu^{2} + 4887 ) / 2484$ (37v^14 + 979v^12 + 9381v^10 + 41836v^8 + 89761v^6 + 83688v^4 + 26658*v^2 + 4887) / 2484
$\beta_{5}$	$=$	$( 89 \nu^{15} - 132 \nu^{14} + 2381 \nu^{13} - 3750 \nu^{12} + 23229 \nu^{11} - 39834 \nu^{10} + \cdots - 35964 ) / 14904$ (89v^15 - 132v^14 + 2381v^13 - 3750v^12 + 23229v^11 - 39834v^10 + 106781v^9 - 204270v^8 + 243575v^7 - 532230v^6 + 265785v^5 - 665982v^4 + 135018v^3 - 329238v^2 + 35370*v - 35964) / 14904
$\beta_{6}$	$=$	$( 44 \nu^{15} - 37 \nu^{14} + 1250 \nu^{13} - 979 \nu^{12} + 13278 \nu^{11} - 9381 \nu^{10} + \cdots - 2403 ) / 4968$ (44v^15 - 37v^14 + 1250v^13 - 979v^12 + 13278v^11 - 9381v^10 + 68090v^9 - 41836v^8 + 177410v^7 - 89761v^6 + 221994v^5 - 83688v^4 + 109746v^3 - 26658v^2 + 11988*v - 2403) / 4968
$\beta_{7}$	$=$	$( 89 \nu^{15} + 132 \nu^{14} + 2381 \nu^{13} + 3750 \nu^{12} + 23229 \nu^{11} + 39834 \nu^{10} + \cdots + 35964 ) / 14904$ (89v^15 + 132v^14 + 2381v^13 + 3750v^12 + 23229v^11 + 39834v^10 + 106781v^9 + 204270v^8 + 243575v^7 + 532230v^6 + 265785v^5 + 665982v^4 + 135018v^3 + 329238v^2 + 35370*v + 35964) / 14904
$\beta_{8}$	$=$	$( 29 \nu^{15} - 81 \nu^{14} + 758 \nu^{13} - 2229 \nu^{12} + 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968$ (29v^15 - 81v^14 + 758v^13 - 2229v^12 + 7086v^11 - 22659v^10 + 29870v^9 - 109926v^8 + 54086v^7 - 267171v^6 + 15876v^5 - 305682v^4 - 54963v^3 - 133920v^2 - 33480*v - 6939) / 4968
$\beta_{9}$	$=$	$( 61 \nu^{15} - 77 \nu^{14} + 1711 \nu^{13} - 2015 \nu^{12} + 17991 \nu^{11} - 18924 \nu^{10} + \cdots + 7587 ) / 4968$ (61v^15 - 77v^14 + 1711v^13 - 2015v^12 + 17991v^11 - 18924v^10 + 92638v^9 - 81380v^8 + 250675v^7 - 162704v^6 + 349914v^5 - 125634v^4 + 222876v^3 - 7929v^2 + 38637*v + 7587) / 4968
$\beta_{10}$	$=$	$( - 29 \nu^{15} - 81 \nu^{14} - 758 \nu^{13} - 2229 \nu^{12} - 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968$ (-29v^15 - 81v^14 - 758v^13 - 2229v^12 - 7086v^11 - 22659v^10 - 29870v^9 - 109926v^8 - 54086v^7 - 267171v^6 - 15876v^5 - 305682v^4 + 54963v^3 - 133920v^2 + 33480*v - 6939) / 4968
$\beta_{11}$	$=$	$( - 73 \nu^{15} - 38 \nu^{14} - 1939 \nu^{13} - 998 \nu^{12} - 18639 \nu^{11} - 9504 \nu^{10} + \cdots - 3861 ) / 4968$ (-73v^15 - 38v^14 - 1939v^13 - 998v^12 - 18639v^11 - 9504v^10 - 83056v^9 - 42659v^8 - 176779v^7 - 96230v^6 - 161004v^5 - 106827v^4 - 42984v^3 - 50238v^2 + 2241*v - 3861) / 4968
$\beta_{12}$	$=$	$( -2\nu^{14} - 53\nu^{12} - 510\nu^{10} - 2303\nu^{8} - 5132\nu^{6} - 5331\nu^{4} - 2142\nu^{2} + 36\nu - 162 ) / 72$ (-2v^14 - 53v^12 - 510v^10 - 2303v^8 - 5132v^6 - 5331v^4 - 2142v^2 + 36v - 162) / 72
$\beta_{13}$	$=$	$( - 19 \nu^{15} - 23 \nu^{14} - 522 \nu^{13} - 598 \nu^{12} - 5327 \nu^{11} - 5543 \nu^{10} + \cdots + 2277 ) / 1656$ (-19v^15 - 23v^14 - 522v^13 - 598v^12 - 5327v^11 - 5543v^10 - 26401v^9 - 23207v^8 - 68493v^7 - 43861v^6 - 92525v^5 - 29555v^4 - 59136v^3 + 828v^2 - 11349*v + 2277) / 1656
$\beta_{14}$	$=$	$( - 77 \nu^{15} - 46 \nu^{14} - 2084 \nu^{13} - 1219 \nu^{12} - 20649 \nu^{11} - 11799 \nu^{10} + \cdots - 3105 ) / 4968$ (-77v^15 - 46v^14 - 2084v^13 - 1219v^12 - 20649v^11 - 11799v^10 - 96491v^9 - 54418v^8 - 221975v^7 - 127351v^6 - 233343v^5 - 143796v^4 - 88452v^3 - 60651v^2 - 2349*v - 3105) / 4968
$\beta_{15}$	$=$	$( 2\nu^{15} + 54\nu^{13} + 535\nu^{11} + 2522\nu^{9} + 5991\nu^{7} + 6892\nu^{5} + 3345\nu^{3} + 450\nu + 36 ) / 72$ (2v^15 + 54v^13 + 535v^11 + 2522v^9 + 5991v^7 + 6892v^5 + 3345v^3 + 450v + 36) / 72

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 4$ b10 + b8 + b7 - b5 + b4 - 4
$\nu^{3}$	$=$	$- 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 5 \beta_1$ -2b15 + 2b14 - b13 - b11 - b10 + b9 + b8 + 2b7 + 2b6 + 2b5 + b4 - b3 - 2b2 - 5*b1
$\nu^{4}$	$=$	$2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} + \cdots + 28$ 2b14 + 2b13 - 4b12 + 2b11 - 9b10 + 2b9 - 9b8 - 6b7 + 14b5 - 7b4 + 4b3 - 4b2 + 6*b1 + 28
$\nu^{5}$	$=$	$38 \beta_{15} - 24 \beta_{14} + 21 \beta_{13} + 15 \beta_{11} + 17 \beta_{10} - 21 \beta_{9} - 17 \beta_{8} + \cdots - 7$ 38b15 - 24b14 + 21b13 + 15b11 + 17b10 - 21b9 - 17b8 - 30b7 - 24b6 - 30b5 - 12b4 + 9b3 + 36b2 + 37b1 - 7
$\nu^{6}$	$=$	$- 22 \beta_{14} - 27 \beta_{13} + 62 \beta_{12} - 40 \beta_{11} + 88 \beta_{10} - 27 \beta_{9} + \cdots - 259$ -22b14 - 27b13 + 62b12 - 40b11 + 88b10 - 27b9 + 88b8 + 41b7 - 165b5 + 67b4 - 62b3 + 67b2 - 93*b1 - 259
$\nu^{7}$	$=$	$- 524 \beta_{15} + 265 \beta_{14} - 293 \beta_{13} - 182 \beta_{11} - 222 \beta_{10} + 293 \beta_{9} + \cdots + 128$ -524b15 + 265b14 - 293b13 - 182b11 - 222b10 + 293b9 + 222b8 + 397b7 + 268b6 + 397b5 + 134b4 - 83b3 - 475b2 - 342b1 + 128
$\nu^{8}$	$=$	$234 \beta_{14} + 316 \beta_{13} - 792 \beta_{12} + 546 \beta_{11} - 923 \beta_{10} + 316 \beta_{9} + \cdots + 2713$ 234b14 + 316b13 - 792b12 + 546b11 - 923b10 + 316b9 - 923b8 - 328b7 + 1888b5 - 739b4 + 780b3 - 862b2 + 1176*b1 + 2713
$\nu^{9}$	$=$	$6506 \beta_{15} - 2942 \beta_{14} + 3637 \beta_{13} + 2101 \beta_{11} + 2695 \beta_{10} - 3637 \beta_{9} + \cdots - 1749$ 6506b15 - 2942b14 + 3637b13 + 2101b11 + 2695b10 - 3637b9 - 2695b8 - 4904b7 - 3008b6 - 4904b5 - 1504b4 + 841b3 + 5738b2 + 3554b1 - 1749
$\nu^{10}$	$=$	$- 2594 \beta_{14} - 3605 \beta_{13} + 9514 \beta_{12} - 6680 \beta_{11} + 10101 \beta_{10} + \cdots - 29935$ -2594b14 - 3605b13 + 9514b12 - 6680b11 + 10101b10 - 3605b9 + 10101b8 + 2994b7 - 21542b5 + 8458b4 - 9274b3 + 10285b2 - 14031*b1 - 29935
$\nu^{11}$	$=$	$- 77366 \beta_{15} + 33081 \beta_{14} - 43200 \beta_{13} - 23991 \beta_{11} - 31763 \beta_{10} + \cdots + 21658$ -77366b15 + 33081b14 - 43200b13 - 23991b11 - 31763b10 + 43200b9 + 31763b8 + 58377b7 + 34050b6 + 58377b5 + 17025b4 - 9090b3 - 67191b2 - 39025b1 + 21658
$\nu^{12}$	$=$	$29416 \beta_{14} + 41016 \beta_{13} - 111416 \beta_{12} + 78628 \beta_{11} - 113122 \beta_{10} + \cdots + 337525$ 29416b14 + 41016b13 - 111416b12 + 78628b11 - 113122b10 + 41016b9 - 113122b8 - 30038b7 + 246126b5 - 97402b4 + 108044b3 - 119644b2 + 163752*b1 + 337525
$\nu^{13}$	$=$	$902396 \beta_{15} - 375304 \beta_{14} + 503534 \beta_{13} + 273782 \beta_{11} + 369198 \beta_{10} + \cdots - 257408$ 902396b15 - 375304b14 + 503534b13 + 273782b11 + 369198b10 - 503534b9 - 369198b8 - 681940b7 - 387580b6 - 681940b5 - 193790b4 + 101522b3 + 777316b2 + 439047b1 - 257408
$\nu^{14}$	$=$	$- 336384 \beta_{14} - 467572 \beta_{13} + 1289976 \beta_{12} - 911652 \beta_{11} + 1281923 \beta_{10} + \cdots - 3840844$ -336384b14 - 467572b13 + 1289976b12 - 911652b11 + 1281923b10 - 467572b9 + 1281923b8 + 319945b7 - 2816017b5 + 1120987b4 - 1248036b3 + 1379224b2 - 1893024*b1 - 3840844
$\nu^{15}$	$=$	$- 10431278 \beta_{15} + 4279454 \beta_{14} - 5818687 \beta_{13} - 3128719 \beta_{11} - 4262047 \beta_{10} + \cdots + 3002670$ -10431278b15 + 4279454b14 - 5818687b13 - 3128719b11 - 4262047b10 + 5818687b9 + 4262047b8 + 7891298b7 + 4425938b6 + 7891298b5 + 2212969b4 - 1150735b3 - 8947406b2 - 4991579b1 + 3002670

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$\beta_{5} + \beta_{7}$	$\beta_{7}$

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}

193.1

		3.38585i
	−	0.779057i
		1.77906i
	−	2.38585i
		1.28827i
	−	1.05387i
	−	0.288273i
		2.05387i
	−	3.38585i
		0.779057i
	−	1.77906i
		2.38585i
	−	1.28827i
		1.05387i
		0.288273i
	−	2.05387i

0.500000

0.866025i

−0.876322

3.27048i

−0.500000

0.866025i

−3.27048

0.876322i

3.61605

2.08773i

−1.00000

−7.33001

−

4.23198i

193.2

0.500000

0.866025i

−0.201635

0.752511i

−0.500000

0.866025i

−0.752511

0.201635i

−2.09844

−

1.21153i

−1.00000

2.07246

1.19654i

193.3

0.500000

0.866025i

0.460454

−

1.71844i

−0.500000

0.866025i

1.71844

−

0.460454i

0.670111

0.386889i

−1.00000

−0.142931

−

0.0825211i

193.4

0.500000

0.866025i

0.617503

−

2.30455i

−0.500000

0.866025i

2.30455

−

0.617503i

2.54433

1.46897i

−1.00000

−2.33157

−

1.34613i

293.1

0.500000

−

0.866025i

−1.24438

0.333429i

−0.500000

−

0.866025i

−0.333429

1.24438i

2.21606

−

1.27944i

−1.00000

−1.16078

0.670177i

293.2

0.500000

−

0.866025i

−1.01796

0.272763i

−0.500000

−

0.866025i

−0.272763

1.01796i

0.524968

−

0.303090i

−1.00000

−1.63622

0.944675i

293.3

0.500000

−

0.866025i

0.278450

−

0.0746104i

−0.500000

−

0.866025i

0.0746104

−

0.278450i

−4.15170

2.39698i

−1.00000

−2.52611

1.45845i

293.4

0.500000

−

0.866025i

1.98389

−

0.531582i

−0.500000

−

0.866025i

0.531582

−

1.98389i

2.67861

−

1.54650i

−1.00000

1.05516

−

0.609199i

357.1

0.500000

−

0.866025i

−0.876322

−

3.27048i

−0.500000

−

0.866025i

−3.27048

−

0.876322i

3.61605

−

2.08773i

−1.00000

−7.33001

4.23198i

357.2

0.500000

−

0.866025i

−0.201635

−

0.752511i

−0.500000

−

0.866025i

−0.752511

−

0.201635i

−2.09844

1.21153i

−1.00000

2.07246

−

1.19654i

357.3

0.500000

−

0.866025i

0.460454

1.71844i

−0.500000

−

0.866025i

1.71844

0.460454i

0.670111

−

0.386889i

−1.00000

−0.142931

0.0825211i

357.4

0.500000

−

0.866025i

0.617503

2.30455i

−0.500000

−

0.866025i

2.30455

0.617503i

2.54433

−

1.46897i

−1.00000

−2.33157

1.34613i

457.1

0.500000

0.866025i

−1.24438

−

0.333429i

−0.500000

0.866025i

−0.333429

−

1.24438i

2.21606

1.27944i

−1.00000

−1.16078

−

0.670177i

457.2

0.500000

0.866025i

−1.01796

−

0.272763i

−0.500000

0.866025i

−0.272763

−

1.01796i

0.524968

0.303090i

−1.00000

−1.63622

−

0.944675i

457.3

0.500000

0.866025i

0.278450

0.0746104i

−0.500000

0.866025i

0.0746104

0.278450i

−4.15170

−

2.39698i

−1.00000

−2.52611

−

1.45845i

457.4

0.500000

0.866025i

1.98389

0.531582i

−0.500000

0.866025i

0.531582

1.98389i

2.67861

1.54650i

−1.00000

1.05516

0.609199i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.o	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	650.2.w.h	yes	16
5.b	even	2	1		650.2.w.f	yes	16
5.c	odd	4	1		650.2.t.f	✓	16
5.c	odd	4	1		650.2.t.h	yes	16
13.f	odd	12	1		650.2.t.f	✓	16
65.o	even	12	1	inner	650.2.w.h	yes	16
65.s	odd	12	1		650.2.t.h	yes	16
65.t	even	12	1		650.2.w.f	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
650.2.t.f	✓	16	5.c	odd	4	1
650.2.t.f	✓	16	13.f	odd	12	1
650.2.t.h	yes	16	5.c	odd	4	1
650.2.t.h	yes	16	65.s	odd	12	1
650.2.w.f	yes	16	5.b	even	2	1
650.2.w.f	yes	16	65.t	even	12	1
650.2.w.h	yes	16	1.a	even	1	1	trivial
650.2.w.h	yes	16	65.o	even	12	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{16} + 12 T_{3}^{14} - 12 T_{3}^{13} + 23 T_{3}^{12} - 12 T_{3}^{11} - 228 T_{3}^{10} + 48 T_{3}^{9} + \cdots + 81$ T3^16 + 12*T3^14 - 12*T3^13 + 23*T3^12 - 12*T3^11 - 228*T3^10 + 48*T3^9 + 160*T3^8 + 96*T3^7 + 1818*T3^6 + 2952*T3^5 + 1899*T3^4 + 756*T3^3 - 162*T3^2 - 324*T3 + 81 acting on $S_{2}^{\mathrm{new}}(650, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - T + 1)^{8}$ (T^2 - T + 1)^8
$3$	$T^{16} + 12 T^{14} + \cdots + 81$ T^16 + 12T^14 - 12T^13 + 23T^12 - 12T^11 - 228T^10 + 48T^9 + 160T^8 + 96T^7 + 1818T^6 + 2952T^5 + 1899T^4 + 756T^3 - 162T^2 - 324T + 81
$5$	$T^{16}$ T^16
$7$	$T^{16} - 12 T^{15} + \cdots + 279841$ T^16 - 12T^15 + 36T^14 + 144T^13 - 811T^12 - 2964T^11 + 31140T^10 - 81684T^9 + 14952T^8 + 281688T^7 - 69300T^6 - 2133972T^5 + 5711861T^4 - 7034688T^3 + 4780044T^2 - 1752048*T + 279841
$11$	$T^{16} + 4 T^{15} + \cdots + 81$ T^16 + 4T^15 + 20T^14 + 244T^13 + 199T^12 - 2392T^11 + 68T^10 - 28924T^9 + 101104T^8 - 288T^7 - 113994T^6 - 1980T^5 + 39915T^4 + 13932T^3 + 9234T^2 - 324*T + 81
$13$	$T^{16} + \cdots + 815730721$ T^16 - 8T^15 + 46T^14 - 188T^13 + 570T^12 - 396T^11 - 7068T^10 + 48252T^9 - 208217T^8 + 627276T^7 - 1194492T^6 - 870012T^5 + 16279770T^4 - 69803084T^3 + 222033214T^2 - 501988136*T + 815730721
$17$	$T^{16} + 8 T^{15} + \cdots + 9199089$ T^16 + 8T^15 + 56T^14 + 488T^13 + 2161T^12 + 8176T^11 + 36632T^10 + 41344T^9 + 80008T^8 - 402480T^7 - 1977768T^6 + 2272536T^5 + 6020361T^4 - 8700696T^3 + 6520824T^2 - 12229056*T + 9199089
$19$	$T^{16} - 16 T^{15} + \cdots + 81$ T^16 - 16T^15 + 158T^14 - 1084T^13 + 5122T^12 - 17996T^11 + 45788T^10 - 72764T^9 + 147607T^8 - 132420T^7 - 28260T^6 + 76212T^5 - 8190T^4 - 10476T^3 + 3726T^2 - 648*T + 81
$23$	$T^{16} + 4 T^{15} + \cdots + 11881809$ T^16 + 4T^15 + 56T^14 + 424T^13 + 1222T^12 + 10388T^11 + 20576T^10 - 52036T^9 + 751915T^8 - 5726676T^7 + 18865440T^6 - 64645884T^5 + 190649142T^4 - 309323016T^3 + 248883192T^2 - 89470332*T + 11881809
$29$	$T^{16} + \cdots + 7001170929$ T^16 - 36T^15 + 516T^14 - 3024T^13 - 3141T^12 + 66852T^11 + 1614816T^10 - 31959360T^9 + 274302936T^8 - 1434901392T^7 + 4898840634T^6 - 10712036808T^5 + 13192228215T^4 - 4332884400T^3 - 8016076710T^2 + 3524306760*T + 7001170929
$31$	$T^{16} + 8 T^{15} + \cdots + 20736$ T^16 + 8T^15 + 32T^14 + 176T^13 + 9040T^12 + 82528T^11 + 386432T^10 + 580672T^9 + 1040224T^8 + 5450112T^7 + 24602112T^6 + 40128768T^5 + 28876032T^4 - 16547328T^3 + 4147200T^2 - 414720*T + 20736
$37$	$T^{16} + 36 T^{15} + \cdots + 187489$ T^16 + 36T^15 + 468T^14 + 1296T^13 - 22021T^12 - 95832T^11 + 1950984T^10 + 21090480T^9 + 71052528T^8 - 2338956T^7 - 370379106T^6 + 16054476T^5 + 1715676527T^4 - 850147992T^3 + 158449806T^2 - 8890356*T + 187489
$41$	$T^{16} + \cdots + 5749430625$ T^16 - 32T^15 + 572T^14 - 8000T^13 + 89770T^12 - 806608T^11 + 5858456T^10 - 32154112T^9 + 124543939T^8 - 328525104T^7 + 487579176T^6 + 169975872T^5 - 1639432566T^4 + 2146925520T^3 - 428449500T^2 - 4039956000*T + 5749430625
$43$	$T^{16} + \cdots + 45270647361$ T^16 + 36T^15 + 522T^14 + 4380T^13 + 28691T^12 + 92292T^11 - 1222578T^10 - 17725872T^9 - 91892744T^8 + 42914796T^7 + 5179356324T^6 + 37638493488T^5 + 101270405511T^4 + 235204982388T^3 + 1094063834196T^2 - 106385351076*T + 45270647361
$47$	$T^{16} + \cdots + 4512867171201$ T^16 + 448T^14 + 77862T^12 + 6930448T^10 + 348357835T^8 + 10160604720T^6 + 167657006214T^4 + 1414599237072*T^2 + 4512867171201
$53$	$T^{16} + \cdots + 178024081041$ T^16 - 44T^15 + 968T^14 - 12092T^13 + 93376T^12 - 439756T^11 + 2069528T^10 - 15330556T^9 + 111917086T^8 - 474691332T^7 + 1376185752T^6 - 4726599156T^5 + 29011869792T^4 - 114908505444T^3 + 269592370632T^2 - 309819089268*T + 178024081041
$59$	$T^{16} + 24 T^{15} + \cdots + 3470769$ T^16 + 24T^15 + 198T^14 + 1188T^13 + 9090T^12 + 12636T^11 + 148716T^10 - 347004T^9 - 234009T^8 - 1122660T^7 + 918540T^6 + 6657228T^5 + 1436130T^4 - 3770388T^3 - 695466T^2 - 3621672*T + 3470769
$61$	$T^{16} + 20 T^{15} + \cdots + 42849$ T^16 + 20T^15 + 382T^14 + 2360T^13 + 21346T^12 + 33380T^11 + 835912T^10 + 908360T^9 + 10865047T^8 + 16586460T^7 + 113503224T^6 + 122590800T^5 + 302471010T^4 - 144239400T^3 + 119487906T^2 + 2235600*T + 42849
$67$	$T^{16} + \cdots + 36051756129$ T^16 + 12T^15 + 250T^14 + 2568T^13 + 38538T^12 + 346116T^11 + 3189688T^10 + 19725624T^9 + 127374247T^8 + 634233588T^7 + 3219794664T^6 + 11587278312T^5 + 34233521994T^4 + 64711762488T^3 + 91390062510T^2 + 69256556496*T + 36051756129
$71$	$T^{16} + \cdots + 10\!\cdots\!64$ T^16 - 16T^15 + 176T^14 - 784T^13 - 20168T^12 + 395104T^11 - 4265728T^10 + 16421248T^9 + 443232304T^8 - 10083336576T^7 + 129589926912T^6 - 1221959886336T^5 + 8676774566784T^4 - 50215081545216T^3 + 237504267832320T^2 - 737095318391808*T + 1018337960542464
$73$	$(T^{8} - 304 T^{6} + \cdots + 90000)^{2}$ (T^8 - 304T^6 - 576T^5 + 16024T^4 - 7680T^3 - 90816T^2 + 57600T + 90000)^2
$79$	$T^{16} + \cdots + 211086032481$ T^16 + 340T^14 + 46914T^12 + 3377416T^10 + 136154011T^8 + 3077005752T^6 + 36774155970T^4 + 193983892956*T^2 + 211086032481
$83$	$T^{16} + \cdots + 175707680625$ T^16 + 556T^14 + 122760T^12 + 13811212T^10 + 840750478T^8 + 26823953844T^6 + 388340941896T^4 + 1632521798100*T^2 + 175707680625
$89$	$T^{16} + \cdots + 709755295760289$ T^16 + 28T^15 + 470T^14 + 4624T^13 - 5687T^12 - 530248T^11 - 5970706T^10 - 33424180T^9 + 552677380T^8 + 2343390876T^7 + 30646605726T^6 - 185334946344T^5 - 1123912276839T^4 - 5321131293168T^3 + 9979762438566T^2 + 266866190045388*T + 709755295760289
$97$	$T^{16} + \cdots + 45\!\cdots\!89$ T^16 + 8T^15 + 632T^14 + 4192T^13 + 254753T^12 + 1664968T^11 + 59614248T^10 + 394187328T^9 + 10213239424T^8 + 66025537104T^7 + 1107548623584T^6 + 6643119318736T^5 + 82841023083809T^4 + 412424441551504T^3 + 2484718586853104T^2 + 3611294865915440*T + 4554033974517889
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 193.4
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	650.2.w.h

Level	$650$
Weight	$2$
Character orbit	650.w
Analytic conductor	$5.190$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$