LMFDB - Newform orbit 4334.2.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, 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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4334 = 2 \cdot 11 \cdot 197$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} + \cdots - 400$ x^17 - 7x^16 - 7x^15 + 137x^14 - 98x^13 - 1048x^12 + 1313x^11 + 4085x^10 - 6021x^9 - 8879x^8 + 13530x^7 + 11150x^6 - 15676x^5 - 8037x^4 + 8646x^3 + 3040x^2 - 1640x - 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$5$
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 $1,\beta_1,\ldots,\beta_{16}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} + \cdots - 400$ x^17 - 7*x^16 - 7*x^15 + 137*x^14 - 98*x^13 - 1048*x^12 + 1313*x^11 + 4085*x^10 - 6021*x^9 - 8879*x^8 + 13530*x^7 + 11150*x^6 - 15676*x^5 - 8037*x^4 + 8646*x^3 + 3040*x^2 - 1640*x - 400 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - \nu - 3$ v^2 - v - 3
$\beta_{3}$	$=$	$( 917115519 \nu^{16} - 5212700619 \nu^{15} - 13267205103 \nu^{14} + 108089258473 \nu^{13} + \cdots + 275653971112 ) / 100018832$ (917115519v^16 - 5212700619v^15 - 13267205103v^14 + 108089258473v^13 + 52303712884v^12 - 890592629248v^11 + 30754333695v^10 + 3775145083049v^9 - 539070143465v^8 - 8813127289315v^7 + 756086254864v^6 + 11151653716722v^5 + 388493970960v^4 - 6794537459179v^3 - 1076889587164v^2 + 1340650067096v + 275653971112) / 100018832
$\beta_{4}$	$=$	$( 2118122577 \nu^{16} - 11979762949 \nu^{15} - 30898783617 \nu^{14} + 248436835479 \nu^{13} + \cdots + 623351680584 ) / 200037664$ (2118122577v^16 - 11979762949v^15 - 30898783617v^14 + 248436835479v^13 + 126180705388v^12 - 2046326225264v^11 + 27481433841v^10 + 8663175750807v^9 - 1074823822151v^8 - 20165139572429v^7 + 1427318174144v^6 + 25388722755550v^5 + 1130270781664v^4 - 15371342144757v^3 - 2495048512180v^2 + 3024441265352v + 623351680584) / 200037664
$\beta_{5}$	$=$	$( 1190055619 \nu^{16} - 6765107123 \nu^{15} - 17236648943 \nu^{14} + 140423501145 \nu^{13} + \cdots + 363827038920 ) / 100018832$ (1190055619v^16 - 6765107123v^15 - 17236648943v^14 + 140423501145v^13 + 68116872216v^12 - 1158620144112v^11 + 39490811443v^10 + 4920362895953v^9 - 702985657697v^8 - 11514175822331v^7 + 991433484244v^6 + 14611869522794v^5 + 507362826840v^4 - 8931436569023v^3 - 1418774262536v^2 + 1767096237176v + 363827038920) / 100018832
$\beta_{6}$	$=$	$( 3747780633 \nu^{16} - 21227672221 \nu^{15} - 54617142073 \nu^{14} + 440646194303 \nu^{13} + \cdots + 1126215265736 ) / 200037664$ (3747780633v^16 - 21227672221v^15 - 54617142073v^14 + 440646194303v^13 + 221531905052v^12 - 3634702369104v^11 + 67473294937v^10 + 15419948886015v^9 - 1991422490479v^8 - 36001804780101v^7 + 2707553369232v^6 + 45510851426158v^5 + 1893641207264v^4 - 27681098132029v^3 - 4468799873156v^2 + 5461989743560v + 1126215265736) / 200037664
$\beta_{7}$	$=$	$( - 9954510309 \nu^{16} + 56495783281 \nu^{15} + 144505682861 \nu^{14} - 1172270244043 \nu^{13} + \cdots - 3005730994568 ) / 400075328$ (-9954510309v^16 + 56495783281v^15 + 144505682861v^14 - 1172270244043v^13 - 577280311748v^12 + 9666330329520v^11 - 263767090469v^10 - 41006633039419v^9 + 5599662973099v^8 + 95790778468857v^7 - 7743095187784v^6 - 121250407489910v^5 - 4632497428592v^4 + 73887115108921v^3 + 11836149895612v^2 - 14591854456296v - 3005730994568) / 400075328
$\beta_{8}$	$=$	$( 2707538429 \nu^{16} - 15377659701 \nu^{15} - 39287625049 \nu^{14} + 319260268927 \nu^{13} + \cdots + 826000500232 ) / 100018832$ (2707538429v^16 - 15377659701v^15 - 39287625049v^14 + 319260268927v^13 + 156408278384v^12 - 2634675505232v^11 + 78802145853v^10 + 11189636938055v^9 - 1557860920983v^8 - 26180709839949v^7 + 2178587173876v^6 + 33207826728630v^5 + 1215374016040v^4 - 20282706719345v^3 - 3238125022576v^2 + 4010985816024v + 826000500232) / 100018832
$\beta_{9}$	$=$	$( 2837634319 \nu^{16} - 16108690395 \nu^{15} - 41184831247 \nu^{14} + 334303865353 \nu^{13} + \cdots + 858999681160 ) / 100018832$ (2837634319v^16 - 16108690395v^15 - 41184831247v^14 + 334303865353v^13 + 164304960564v^12 - 2757232092864v^11 + 77989138735v^10 + 11700479665545v^9 - 1609771486105v^8 - 27344196747971v^7 + 2235868298784v^6 + 34631577210498v^5 + 1300475342576v^4 - 21117097708267v^3 - 3377448845372v^2 + 4171969878392v + 858999681160) / 100018832
$\beta_{10}$	$=$	$( - 728303375 \nu^{16} + 4131017823 \nu^{15} + 10592530399 \nu^{14} - 85773093645 \nu^{13} + \cdots - 220825196084 ) / 25004708$ (-728303375v^16 + 4131017823v^15 + 10592530399v^14 - 85773093645v^13 - 42576437488v^12 + 707832557708v^11 - 17197982699v^10 - 3005473719013v^9 + 403772089357v^8 + 7027252438155v^7 - 558224580696v^6 - 8902537051546v^5 - 346078010592v^4 + 5428719236991v^3 + 870584102976v^2 - 1072479423632v - 220825196084) / 25004708
$\beta_{11}$	$=$	$( - 11821377509 \nu^{16} + 67109376065 \nu^{15} + 171605413149 \nu^{14} - 1392946997659 \nu^{13} + \cdots - 3585384415688 ) / 400075328$ (-11821377509v^16 + 67109376065v^15 + 171605413149v^14 - 1392946997659v^13 - 684830985492v^12 + 11491009310704v^11 - 324846133733v^10 - 48775644063787v^9 + 6715722446939v^8 + 114025234835017v^7 - 9345082915736v^6 - 144462250580854v^5 - 5396377253456v^4 + 88113783944057v^3 + 14083605139692v^2 - 17407879949608v - 3585384415688) / 400075328
$\beta_{12}$	$=$	$( 6485269861 \nu^{16} - 36814938225 \nu^{15} - 94066526173 \nu^{14} + 763681201211 \nu^{13} + \cdots + 1949477027720 ) / 200037664$ (6485269861v^16 - 36814938225v^15 - 94066526173v^14 + 763681201211v^13 + 374698521364v^12 - 6294804448000v^11 + 181312945125v^10 + 26691174915387v^9 - 3680002902523v^8 - 62314158668361v^7 + 5105021882392v^6 + 78824878470502v^5 + 2946561825840v^4 - 48001960685305v^3 - 7663460213468v^2 + 9475041636312v + 1949477027720) / 200037664
$\beta_{13}$	$=$	$( - 6607866089 \nu^{16} + 37544828365 \nu^{15} + 95697548569 \nu^{14} - 778806039487 \nu^{13} + \cdots - 1994818528520 ) / 200037664$ (-6607866089v^16 + 37544828365v^15 + 95697548569v^14 - 778806039487v^13 - 378710389548v^12 + 6419850927808v^11 - 209553266889v^10 - 27227733919855v^9 + 3846454306575v^8 + 63600536503013v^7 - 5382966652992v^6 - 80523750869182v^5 - 2870160021504v^4 + 49090446693741v^3 + 7805125511684v^2 - 9694113662072v - 1994818528520) / 200037664
$\beta_{14}$	$=$	$( 14135303493 \nu^{16} - 80212263505 \nu^{15} - 205268091629 \nu^{14} + 1664519690699 \nu^{13} + \cdots + 4274328485320 ) / 400075328$ (14135303493v^16 - 80212263505v^15 - 205268091629v^14 + 1664519690699v^13 + 821012038244v^12 - 13726588684816v^11 + 365928497861v^10 + 58235903554779v^9 - 7923182745323v^8 - 136044731616441v^7 + 10945028598376v^6 + 172201964010198v^5 + 6630349736240v^4 - 104929420661913v^3 - 16837827003132v^2 + 20720767080072v + 4274328485320) / 400075328
$\beta_{15}$	$=$	$( - 3708772897 \nu^{16} + 21035949917 \nu^{15} + 53921864145 \nu^{14} - 436648931271 \nu^{13} + \cdots - 1121904694504 ) / 100018832$ (-3708772897v^16 + 21035949917v^15 + 53921864145v^14 - 436648931271v^13 - 216606080428v^12 + 3602065006536v^11 - 87651982529v^10 - 15287425588959v^9 + 2050600042871v^8 + 35724481502701v^7 - 2823772651056v^6 - 45229285226230v^5 - 1777989934800v^4 + 27562259659189v^3 + 4429074853644v^2 - 5442677854832v - 1121904694504) / 100018832
$\beta_{16}$	$=$	$( - 2361132485 \nu^{16} + 13408798839 \nu^{15} + 34223642511 \nu^{14} - 278142921505 \nu^{13} + \cdots - 711627008544 ) / 50009416$ (-2361132485v^16 + 13408798839v^15 + 34223642511v^14 - 278142921505v^13 - 135934710226v^12 + 2292685853680v^11 - 69819436069v^10 - 9722411102661v^9 + 1354208180165v^8 + 22704035998199v^7 - 1884395973086v^6 - 28732594669742v^5 - 1056052223892v^4 + 17507804123377v^3 + 2792122996110v^2 - 3457385494912v - 711627008544) / 50009416

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 3$ b2 + b1 + 3
$\nu^{3}$	$=$	$\beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{4} + 2\beta_{2} + 7\beta_1$ b14 + b12 + b11 - b10 - 2b9 - b4 + 2b2 + 7*b1
$\nu^{4}$	$=$	$\beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 12$ b15 + 2b14 - b13 + b11 - 2b10 - 5b9 + b8 - b7 - b6 + b5 + b4 - b3 + 11b2 + 17*b1 + 12
$\nu^{5}$	$=$	$\beta_{16} + \beta_{15} + 13 \beta_{14} - 4 \beta_{13} + 12 \beta_{12} + 15 \beta_{11} - 13 \beta_{10} + \cdots - 3$ b16 + b15 + 13b14 - 4b13 + 12b12 + 15b11 - 13b10 - 29b9 + 3b8 - 2b7 + b5 - 10b4 - 5b3 + 30b2 + 67b1 - 3
$\nu^{6}$	$=$	$5 \beta_{16} + 15 \beta_{15} + 37 \beta_{14} - 25 \beta_{13} + 11 \beta_{12} + 31 \beta_{11} - 40 \beta_{10} + \cdots + 42$ 5b16 + 15b15 + 37b14 - 25b13 + 11b12 + 31b11 - 40b10 - 89b9 + 16b8 - 18b7 - 16b6 + 15b5 + 8b4 - 29b3 + 125b2 + 214b1 + 42
$\nu^{7}$	$=$	$35 \beta_{16} + 32 \beta_{15} + 166 \beta_{14} - 94 \beta_{13} + 134 \beta_{12} + 198 \beta_{11} + \cdots - 98$ 35b16 + 32b15 + 166b14 - 94b13 + 134b12 + 198b11 - 172b10 - 378b9 + 55b8 - 56b7 - 20b6 + 34b5 - 76b4 - 111b3 + 380b2 + 759b1 - 98
$\nu^{8}$	$=$	$138 \beta_{16} + 211 \beta_{15} + 538 \beta_{14} - 420 \beta_{13} + 241 \beta_{12} + 534 \beta_{11} + \cdots - 95$ 138b16 + 211b15 + 538b14 - 420b13 + 241b12 + 534b11 - 596b10 - 1249b9 + 211b8 - 290b7 - 215b6 + 213b5 + 57b4 - 495b3 + 1446b2 + 2600b1 - 95
$\nu^{9}$	$=$	$672 \beta_{16} + 609 \beta_{15} + 2112 \beta_{14} - 1530 \beta_{13} + 1543 \beta_{12} + 2502 \beta_{11} + \cdots - 1857$ 672b16 + 609b15 + 2112b14 - 1530b13 + 1543b12 + 2502b11 - 2261b10 - 4775b9 + 749b8 - 1022b7 - 470b6 + 640b5 - 519b4 - 1783b3 + 4629b2 + 9047b1 - 1857
$\nu^{10}$	$=$	$2509 \beta_{16} + 2914 \beta_{15} + 7164 \beta_{14} - 6056 \beta_{13} + 3804 \beta_{12} + 7601 \beta_{11} + \cdots - 5269$ 2509b16 + 2914b15 + 7164b14 - 6056b13 + 3804b12 + 7601b11 - 8014b10 - 16229b9 + 2609b8 - 4347b7 - 2773b6 + 2958b5 + 470b4 - 7130b3 + 16889b2 + 31453b1 - 5269
$\nu^{11}$	$=$	$10525 \beta_{16} + 9480 \beta_{15} + 26534 \beta_{14} - 21687 \beta_{13} + 18321 \beta_{12} + \cdots - 28609$ 10525b16 + 9480b15 + 26534b14 - 21687b13 + 18321b12 + 30895b11 - 29035b10 - 59267b9 + 9234b8 - 15727b7 - 7678b6 + 9834b5 - 3181b4 - 25247b3 + 55694b2 + 109151b1 - 28609
$\nu^{12}$	$=$	$38461 \beta_{16} + 39190 \beta_{15} + 91457 \beta_{14} - 81103 \beta_{13} + 53112 \beta_{12} + \cdots - 93861$ 38461b16 + 39190b15 + 91457b14 - 81103b13 + 53112b12 + 99568b11 - 102797b10 - 203738b9 + 31252b8 - 61343b7 - 35157b6 + 39851b5 + 4836b4 - 95326b3 + 198585b2 + 380134b1 - 93861
$\nu^{13}$	$=$	$149613 \beta_{16} + 133692 \beta_{15} + 329175 \beta_{14} - 287489 \beta_{13} + 221332 \beta_{12} + \cdots - 398621$ 149613b16 + 133692b15 + 329175b14 - 287489b13 + 221332b12 + 376455b11 - 365136b10 - 727165b9 + 109437b8 - 221752b7 - 108971b6 + 137481b5 - 15224b4 - 335434b3 + 666836b2 + 1319010b1 - 398621
$\nu^{14}$	$=$	$539818 \beta_{16} + 513491 \beta_{15} + 1140848 \beta_{14} - 1043078 \beta_{13} + 698651 \beta_{12} + \cdots - 1362389$ 539818b16 + 513491b15 + 1140848b14 - 1043078b13 + 698651b12 + 1251606b11 - 1285859b10 - 2511541b9 + 368192b8 - 827494b7 - 440539b6 + 522269b5 + 58370b4 - 1225545b3 + 2346366b2 + 4589246b1 - 1362389
$\nu^{15}$	$=$	$2013096 \beta_{16} + 1783860 \beta_{15} + 4043297 \beta_{14} - 3672837 \beta_{13} + 2692678 \beta_{12} + \cdots - 5256266$ 2013096b16 + 1783860b15 + 4043297b14 - 3672837b13 + 2692678b12 + 4551428b11 - 4522535b10 - 8852491b9 + 1277286b8 - 2969941b7 - 1444382b6 + 1825133b5 - 9068b4 - 4297489b3 + 7967568b2 + 15922649b1 - 5256266
$\nu^{16}$	$=$	$7196125 \beta_{16} + 6583561 \beta_{15} + 14035487 \beta_{14} - 13096144 \beta_{13} + 8890642 \beta_{12} + \cdots - 18209145$ 7196125b16 + 6583561b15 + 14035487b14 - 13096144b13 + 8890642b12 + 15392025b11 - 15848240b10 - 30633520b9 + 4304653b8 - 10806160b7 - 5467646b6 + 6695948b5 + 753267b4 - 15395899b3 + 27819450b2 + 55329711b1 - 18209145

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

3.45762
3.00096
2.69188
2.63673
1.67442
1.32994
1.32368
1.31815
0.577346
−0.216237
−0.673774
−1.02051
−1.23460
−1.29570
−1.87628
−2.06015
−2.63348

1.00000

−3.45762

1.00000

−2.31501

−3.45762

2.74192

1.00000

8.95516

−2.31501

1.2

1.00000

−3.00096

1.00000

4.03995

−3.00096

−2.78062

1.00000

6.00574

4.03995

1.3

1.00000

−2.69188

1.00000

1.31987

−2.69188

2.55905

1.00000

4.24624

1.31987

1.4

1.00000

−2.63673

1.00000

−3.59467

−2.63673

−1.69075

1.00000

3.95235

−3.59467

1.5

1.00000

−1.67442

1.00000

−0.572222

−1.67442

−0.229301

1.00000

−0.196306

−0.572222

1.6

1.00000

−1.32994

1.00000

2.68374

−1.32994

−0.271741

1.00000

−1.23125

2.68374

1.7

1.00000

−1.32368

1.00000

−0.966345

−1.32368

1.91821

1.00000

−1.24787

−0.966345

1.8

1.00000

−1.31815

1.00000

1.46276

−1.31815

−4.53584

1.00000

−1.26249

1.46276

1.9

1.00000

−0.577346

1.00000

0.543375

−0.577346

4.26235

1.00000

−2.66667

0.543375

1.10

1.00000

0.216237

1.00000

0.737633

0.216237

0.342001

1.00000

−2.95324

0.737633

1.11

1.00000

0.673774

1.00000

2.41156

0.673774

−3.55111

1.00000

−2.54603

2.41156

1.12

1.00000

1.02051

1.00000

−4.21122

1.02051

2.47064

1.00000

−1.95855

−4.21122

1.13

1.00000

1.23460

1.00000

−1.34398

1.23460

−1.90837

1.00000

−1.47576

−1.34398

1.14

1.00000

1.29570

1.00000

1.60714

1.29570

−2.75710

1.00000

−1.32117

1.60714

1.15

1.00000

1.87628

1.00000

−2.40821

1.87628

1.70446

1.00000

0.520431

−2.40821

1.16

1.00000

2.06015

1.00000

−2.29448

2.06015

0.0148240

1.00000

1.24422

−2.29448

1.17

1.00000

2.63348

1.00000

−1.09989

2.63348

−3.28860

1.00000

3.93521

−1.09989

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$11$	$+1$
$197$	$-1$

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	4334.2.a.d	✓	17

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4334.2.a.d	✓	17	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{17} + 7 T_{3}^{16} - 7 T_{3}^{15} - 137 T_{3}^{14} - 98 T_{3}^{13} + 1048 T_{3}^{12} + 1313 T_{3}^{11} + \cdots + 400$ T3^17 + 7*T3^16 - 7*T3^15 - 137*T3^14 - 98*T3^13 + 1048*T3^12 + 1313*T3^11 - 4085*T3^10 - 6021*T3^9 + 8879*T3^8 + 13530*T3^7 - 11150*T3^6 - 15676*T3^5 + 8037*T3^4 + 8646*T3^3 - 3040*T3^2 - 1640*T3 + 400 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4334))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 1)^{17}$ (T - 1)^17
$3$	$T^{17} + 7 T^{16} + \cdots + 400$ T^17 + 7T^16 - 7T^15 - 137T^14 - 98T^13 + 1048T^12 + 1313T^11 - 4085T^10 - 6021T^9 + 8879T^8 + 13530T^7 - 11150T^6 - 15676T^5 + 8037T^4 + 8646T^3 - 3040T^2 - 1640T + 400
$5$	$T^{17} + 4 T^{16} + \cdots + 5147$ T^17 + 4T^16 - 36T^15 - 147T^14 + 464T^13 + 1972T^12 - 2823T^11 - 12805T^10 + 8778T^9 + 43928T^8 - 13942T^7 - 81069T^6 + 9737T^5 + 77505T^4 - 1355T^3 - 33995T^2 - 309T + 5147
$7$	$T^{17} + 5 T^{16} + \cdots + 100$ T^17 + 5T^16 - 43T^15 - 228T^14 + 684T^13 + 3965T^12 - 5242T^11 - 34421T^10 + 20072T^9 + 159010T^8 - 32363T^7 - 374123T^6 - 2869T^5 + 362972T^4 + 47538T^3 - 36255T^2 - 6220T + 100
$11$	$(T + 1)^{17}$ (T + 1)^17
$13$	$T^{17} + 18 T^{16} + \cdots + 772532$ T^17 + 18T^16 + 63T^15 - 701T^14 - 5961T^13 - 3641T^12 + 111905T^11 + 383896T^10 - 293369T^9 - 3876685T^8 - 5552804T^7 + 8755840T^6 + 31373574T^5 + 18367264T^4 - 30769050T^3 - 45989697T^2 - 16426056T + 772532
$17$	$T^{17} + \cdots + 140740076$ T^17 + 10T^16 - 92T^15 - 1067T^14 + 2675T^13 + 40201T^12 - 37867T^11 - 755132T^10 + 323750T^9 + 7848649T^8 - 1725467T^7 - 46447793T^6 + 4365316T^5 + 152532229T^4 + 1318925T^3 - 248453291T^2 - 19045134T + 140740076
$19$	$T^{17} + 31 T^{16} + \cdots - 89994944$ T^17 + 31T^16 + 284T^15 - 738T^14 - 28414T^13 - 134714T^12 + 485790T^11 + 6294695T^10 + 12678038T^9 - 65439515T^8 - 340453325T^7 - 220817131T^6 + 1742169472T^5 + 3997043824T^4 + 940074832T^3 - 4870546299T^2 - 3747059376T - 89994944
$23$	$T^{17} + 6 T^{16} + \cdots - 726928$ T^17 + 6T^16 - 179T^15 - 968T^14 + 11777T^13 + 55349T^12 - 355589T^11 - 1389628T^10 + 5096672T^9 + 15465758T^8 - 34352872T^7 - 73513200T^6 + 101481505T^5 + 118350181T^4 - 106205620T^3 - 13004688T^2 + 11586048T - 726928
$29$	$T^{17} + \cdots - 1614622436$ T^17 + 16T^16 - 86T^15 - 2589T^14 - 4409T^13 + 132549T^12 + 531896T^11 - 2790270T^10 - 15958201T^9 + 25038828T^8 + 212193019T^7 - 69670768T^6 - 1375365788T^5 - 240154585T^4 + 4066444343T^3 + 1547191597T^2 - 4048313942T - 1614622436
$31$	$T^{17} + 30 T^{16} + \cdots - 22065715$ T^17 + 30T^16 + 190T^15 - 2575T^14 - 37404T^13 - 60086T^12 + 1362611T^11 + 7898161T^10 + 438913T^9 - 125292978T^8 - 416553446T^7 - 282339886T^6 + 1278680656T^5 + 3462151322T^4 + 3558878247T^3 + 1480411996T^2 + 82685000T - 22065715
$37$	$T^{17} + \cdots + 1740996688$ T^17 + 23T^16 - 78T^15 - 5273T^14 - 23405T^13 + 320777T^12 + 2683992T^11 - 1998654T^10 - 67588221T^9 - 108510989T^8 + 557019514T^7 + 1661679710T^6 - 662270619T^5 - 6676044245T^4 - 6508299510T^3 + 1458450776T^2 + 4758929272T + 1740996688
$41$	$T^{17} + \cdots - 4085969468$ T^17 + 7T^16 - 289T^15 - 2374T^14 + 29343T^13 + 292627T^12 - 1155847T^11 - 16316815T^10 + 7382385T^9 + 423098676T^8 + 488173175T^7 - 4866860355T^6 - 8635045510T^5 + 25583988861T^4 + 49320975441T^3 - 51105495401T^2 - 89724702678T - 4085969468
$43$	$T^{17} + 23 T^{16} + \cdots - 92908$ T^17 + 23T^16 - 65T^15 - 4797T^14 - 17056T^13 + 324501T^12 + 1824110T^11 - 9889137T^10 - 67517804T^9 + 147864588T^8 + 1091271891T^7 - 1108884090T^6 - 6749226990T^5 + 4355191308T^4 + 5505605039T^3 - 3980063769T^2 + 520458468T - 92908
$47$	$T^{17} + 19 T^{16} + \cdots - 10194992$ T^17 + 19T^16 - 122T^15 - 4319T^14 - 11096T^13 + 269980T^12 + 1566186T^11 - 4663529T^10 - 48185582T^9 - 9691391T^8 + 515012044T^7 + 480377044T^6 - 2327576715T^5 - 2236625745T^4 + 4012193074T^3 + 2685938376T^2 - 1381559208T - 10194992
$53$	$T^{17} + \cdots + 365178260368$ T^17 + 30T^16 + 38T^15 - 6977T^14 - 56992T^13 + 487954T^12 + 7182767T^11 - 2695117T^10 - 353288825T^9 - 928884265T^8 + 7110345252T^7 + 33317633043T^6 - 38870062959T^5 - 381313030287T^4 - 264894578878T^3 + 1024066757448T^2 + 1247145903368T + 365178260368
$59$	$T^{17} + 28 T^{16} + \cdots - 7289225$ T^17 + 28T^16 - 14T^15 - 6390T^14 - 35098T^13 + 455928T^12 + 3359210T^11 - 16270775T^10 - 124148511T^9 + 370369325T^8 + 2071552327T^7 - 5400278648T^6 - 13463642513T^5 + 34101750578T^4 + 28864974403T^3 - 70524388204T^2 - 2424887535T - 7289225
$61$	$T^{17} + \cdots - 1699473732784$ T^17 + 19T^16 - 317T^15 - 8002T^14 + 18055T^13 + 1195425T^12 + 3028357T^11 - 75731786T^10 - 392172576T^9 + 1882214626T^8 + 15573628587T^7 - 5573048269T^6 - 234378645434T^5 - 334189225264T^4 + 971763416594T^3 + 2251975789099T^2 - 70484652060T - 1699473732784
$67$	$T^{17} + \cdots - 386291825862400$ T^17 + 35T^16 - 226T^15 - 19443T^14 - 69255T^13 + 4081729T^12 + 26359702T^11 - 445486971T^10 - 3154653464T^9 + 30204541543T^8 + 178526970589T^7 - 1372682214687T^6 - 4552209101647T^5 + 38781818986349T^4 + 16652539629202T^3 - 490563079672484T^2 + 936585416184320T - 386291825862400
$71$	$T^{17} + \cdots + 51038766188513$ T^17 - T^16 - 520T^15 + 1619T^14 + 109193T^13 - 553170T^12 - 11441466T^11 + 81714751T^10 + 580793159T^9 - 6049045668T^8 - 8301632645T^7 + 219692487790T^6 - 375902353003T^5 - 3018915598669T^4 + 12477558406439T^3 - 5624977166980T^2 - 39287292975298*T + 51038766188513
$73$	$T^{17} + \cdots - 1774258011932$ T^17 + 10T^16 - 783T^15 - 7024T^14 + 242438T^13 + 1926826T^12 - 37873744T^11 - 267457225T^10 + 3141273026T^9 + 20258395157T^8 - 133074710116T^7 - 835132599711T^6 + 2504885386548T^5 + 17137600378686T^4 - 10427014860693T^3 - 135501285271871T^2 - 135064675469328T - 1774258011932
$79$	$T^{17} + \cdots - 9468083562500$ T^17 + 27T^16 - 447T^15 - 19076T^14 - 36411T^13 + 4199129T^12 + 40254734T^11 - 217059270T^10 - 5281262805T^9 - 21832622706T^8 + 122236513455T^7 + 1636255398257T^6 + 7396330971859T^5 + 16901556254450T^4 + 17550020746199T^3 - 696716760575T^2 - 16485488941250T - 9468083562500
$83$	$T^{17} + \cdots + 25755751965776$ T^17 + 40T^16 + 145T^15 - 12736T^14 - 142073T^13 + 1244360T^12 + 23903082T^11 - 11641659T^10 - 1669433217T^9 - 4673448624T^8 + 50114809403T^7 + 256633856924T^6 - 411756714481T^5 - 4493584914730T^4 - 5276561726643T^3 + 15658165249951T^2 + 40746489281776T + 25755751965776
$89$	$T^{17} + \cdots + 1574641256000$ T^17 + 17T^16 - 318T^15 - 6632T^14 + 25779T^13 + 876824T^12 - 297725T^11 - 56684101T^10 - 39708881T^9 + 2060097528T^8 + 1486158022T^7 - 43658671004T^6 - 9856936198T^5 + 507697381195T^4 - 231092110662T^3 - 2546003315780T^2 + 2540040607600T + 1574641256000
$97$	$T^{17} + \cdots + 14372578999$ T^17 + 34T^16 - 148T^15 - 15493T^14 - 85082T^13 + 2097002T^12 + 19504803T^11 - 89199823T^10 - 1260473526T^9 + 644946612T^8 + 33170031069T^7 + 35425074449T^6 - 334479641937T^5 - 625963224705T^4 + 617303213685T^3 + 1125805584791T^2 - 353834003096T + 14372578999
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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 1.17
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	4334.2.a.d

Level	$4334$
Weight	$2$
Character orbit	4334.a
Self dual	yes
Analytic conductor	$34.607$
Analytic rank	$1$
Dimension	$17$
CM	no
Inner twists	$1$