LMFDB - Newform orbit 1134.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(647,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("1134.647");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1134 = 2 \cdot 3^{4} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1134.k (of order $6$ , degree $2$ , 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:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561$ x^16 - 8x^15 + 23x^14 - 8x^13 - 131x^12 + 380x^11 - 289x^10 - 880x^9 + 2785x^8 - 2640x^7 - 2601x^6 + 10260x^5 - 10611x^4 - 1944x^3 + 16767x^2 - 17496*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{4}$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

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

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561$ x^16 - 8*x^15 + 23*x^14 - 8*x^13 - 131*x^12 + 380*x^11 - 289*x^10 - 880*x^9 + 2785*x^8 - 2640*x^7 - 2601*x^6 + 10260*x^5 - 10611*x^4 - 1944*x^3 + 16767*x^2 - 17496*x + 6561 :

$\beta_{1}$	$=$	$( 50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + \cdots + 2825604 ) / 142155$ (50v^15 + 1352v^14 - 6827v^13 + 7676v^12 + 27422v^11 - 107246v^10 + 107467v^9 + 206194v^8 - 757363v^7 + 724572v^6 + 756198v^5 - 2730942v^4 + 2372247v^3 + 1318518v^2 - 4444713*v + 2825604) / 142155
$\beta_{2}$	$=$	$( 169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + \cdots - 244944 ) / 47385$ (169v^15 - 866v^14 + 1319v^13 + 2308v^12 - 13199v^11 + 19055v^10 + 12131v^9 - 90010v^8 + 128722v^7 + 13734v^6 - 313347v^5 + 432351v^4 - 55431v^3 - 524799v^2 + 629856*v - 244944) / 47385
$\beta_{3}$	$=$	$( 154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} + \cdots - 1285227 ) / 47385$ (154v^15 - 1325v^14 + 3608v^13 - 224v^12 - 22478v^11 + 55022v^10 - 23518v^9 - 159688v^8 + 382978v^7 - 226785v^6 - 566733v^5 + 1355805v^4 - 871830v^3 - 949725v^2 + 2135727*v - 1285227) / 47385
$\beta_{4}$	$=$	$( 1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + \cdots - 7117227 ) / 47385$ (1445v^15 - 9836v^14 + 21081v^13 + 15627v^12 - 172766v^11 + 334353v^10 + 13019v^9 - 1283242v^8 + 2419149v^7 - 693301v^6 - 4779069v^5 + 8827731v^4 - 3909951v^3 - 8165529v^2 + 13935564*v - 7117227) / 47385
$\beta_{5}$	$=$	$( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + \cdots - 14041269 ) / 47385$ (2858v^15 - 19265v^14 + 40866v^13 + 31392v^12 - 338066v^11 + 649239v^10 + 37829v^9 - 2521996v^8 + 4717386v^7 - 1293175v^6 - 9430341v^5 + 17293545v^4 - 7519770v^3 - 16197165v^2 + 27464589*v - 14041269) / 47385
$\beta_{6}$	$=$	$( - 11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} + \cdots + 42998607 ) / 142155$ (-11192v^15 + 70123v^14 - 136087v^13 - 145859v^12 + 1215277v^11 - 2154880v^10 - 494788v^9 + 9027170v^8 - 15633236v^7 + 2544783v^6 + 33710346v^5 - 57047193v^4 + 20576268v^3 + 57644217v^2 - 89475273*v + 42998607) / 142155
$\beta_{7}$	$=$	$( 16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + \cdots - 69445998 ) / 142155$ (16898v^15 - 108472v^14 + 217033v^13 + 208526v^12 - 1883968v^11 + 3436840v^10 + 565132v^9 - 13978340v^8 + 24892859v^7 - 5100702v^6 - 52089894v^5 + 90692622v^4 - 35165097v^3 - 88915158v^2 + 142069707*v - 69445998) / 142155
$\beta_{8}$	$=$	$( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14963454 ) / 28431$ (-4120v^15 + 25571v^14 - 48788v^13 - 55006v^12 + 441224v^11 - 771188v^10 - 200900v^9 + 3269857v^8 - 5585140v^7 + 796053v^6 + 12187116v^5 - 20322468v^4 + 7028856v^3 + 20790351v^2 - 31653180*v + 14963454) / 28431
$\beta_{9}$	$=$	$( 8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + \cdots - 30865131 ) / 47385$ (8586v^15 - 53254v^14 + 101261v^13 + 115567v^12 - 919066v^11 + 1601225v^10 + 428149v^9 - 6806990v^8 + 11591968v^7 - 1602394v^6 - 25345758v^5 + 42144849v^4 - 14468679v^3 - 43148781v^2 + 65494089*v - 30865131) / 47385
$\beta_{10}$	$=$	$( - 26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} + \cdots + 102032298 ) / 142155$ (-26068v^15 + 165707v^14 - 325403v^13 - 334861v^12 + 2873363v^11 - 5145350v^10 - 1063457v^9 + 21324130v^8 - 37270684v^7 + 6581337v^6 + 79507809v^5 - 135722682v^4 + 50026572v^3 + 135776493v^2 - 212064642*v + 102032298) / 142155
$\beta_{11}$	$=$	$( - 31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} + \cdots + 117273501 ) / 142155$ (-31130v^15 + 194263v^14 - 374983v^13 - 405716v^12 + 3354958v^11 - 5936959v^10 - 1366837v^9 + 24841391v^8 - 43003982v^7 + 7063563v^6 + 92428227v^5 - 156501153v^4 + 56669868v^3 + 157282722v^2 - 244082322*v + 117273501) / 142155
$\beta_{12}$	$=$	$( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + \cdots - 121811526 ) / 142155$ (33311v^15 - 207064v^14 + 396466v^13 + 441842v^12 - 3574291v^11 + 6268915v^10 + 1580329v^9 - 26474600v^8 + 45384563v^7 - 6735609v^6 - 98586783v^5 + 165029724v^4 - 57791394v^3 - 167924421v^2 + 256798269*v - 121811526) / 142155
$\beta_{13}$	$=$	$( 5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + \cdots - 18826182 ) / 15795$ (5368v^15 - 32959v^14 + 62025v^13 + 72930v^12 - 567820v^11 + 981351v^10 + 280045v^9 - 4204439v^8 + 7108452v^7 - 906329v^6 - 15657912v^5 + 25851399v^4 - 8698959v^3 - 26672436v^2 + 40175190*v - 18826182) / 15795
$\beta_{14}$	$=$	$( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} + \cdots + 244346949 ) / 142155$ (-62668v^15 + 397070v^14 - 778856v^13 - 802747v^12 + 6878696v^11 - 12322064v^10 - 2527469v^9 + 51016561v^8 - 89230126v^7 + 15905550v^6 + 190088901v^5 - 324842130v^4 + 120144465v^3 + 324340605v^2 - 507326409*v + 244346949) / 142155
$\beta_{15}$	$=$	$( 5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + \cdots - 18816948 ) / 10935$ (5105v^15 - 31804v^14 + 61039v^13 + 67583v^12 - 549514v^11 + 965497v^10 + 239686v^9 - 4072523v^8 + 6992981v^7 - 1054329v^6 - 15172956v^5 + 25447554v^4 - 8949609v^3 - 25866621v^2 + 39638646*v - 18816948) / 10935

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

$\nu$	$=$	$( -\beta_{15} + 2\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 3$ (-b15 + 2b9 + b8 + b7 - 2b6 - b4 - b3 + 2*b2 - b1 + 2) / 3
$\nu^{2}$	$=$	$( \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 3 ) / 3$ (b14 - 2b13 - b11 + b10 + 3b9 - 2b8 + 2b7 - b6 - 2b4 - b3 - 2b1 + 3) / 3
$\nu^{3}$	$=$	$( - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + \cdots - 4 ) / 3$ (-2b13 + b12 + 2b11 + 3b10 + 6b9 + 2b8 + 5b7 + 2b5 - 3b4 - 5b3 + 8b2 + b1 - 4) / 3
$\nu^{4}$	$=$	$( 5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \cdots + 8 ) / 3$ (5b15 - 3b13 - b12 + 3b11 + 6b10 + 2b9 - 3b8 + b7 + 7b6 + 4b5 + 2b4 + 4b3 - 2b2 + 11b1 + 8) / 3
$\nu^{5}$	$=$	$( - 10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} + \cdots - 1 ) / 3$ (-10b14 + 3b13 + 3b12 + 9b11 + 5b10 + 6b9 + 30b8 - 3b7 - 2b6 + 6b5 - 7b4 + b3 + 16b2 + 21*b1 - 1) / 3
$\nu^{6}$	$=$	$( - 7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + \cdots + 20 ) / 3$ (-7b15 - 2b14 + 7b13 - 2b12 + 11b11 - 2b10 + 8b9 + 15b8 + 30b7 - 3b6 - 4b5 - 12b4 + 32b3 - 21b2 + 51*b1 + 20) / 3
$\nu^{7}$	$=$	$( - 36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + \cdots - 3 ) / 3$ (-36b15 + 3b14 + 6b13 + 37b12 + 24b11 - 21b10 + 33b9 + 49b8 + 93b7 - b5 - 39b4 - 17b3 + 46b2 + 90*b1 - 3) / 3
$\nu^{8}$	$=$	$( - 42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + \cdots + 94 ) / 3$ (-42b15 + 68b14 + 10b13 + 66b12 + 50b11 - 25b10 + 42b9 - 71b8 + 179b7 + 61b6 + 18b5 + 50b4 + 31b3 - 16b2 + 163*b1 + 94) / 3
$\nu^{9}$	$=$	$( - 44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + \cdots + 168 ) / 3$ (-44b15 + 32b14 + 216b12 + 72b11 + 14b10 + 40b9 + 50b8 - 19b7 + 66b6 + 126b5 + 66b4 - 64b3 + 110b2 + 67b1 + 168) / 3
$\nu^{10}$	$=$	$( - 58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + \cdots + 188 ) / 3$ (-58b15 - 18b14 + 90b13 + 172b12 + 126b11 + 162b10 + 62b9 + 14b8 - 161b7 - 8b6 + 167b5 + 158b4 + 153b3 - 144b2 - 61*b1 + 188) / 3
$\nu^{11}$	$=$	$( - 126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + \cdots + 217 ) / 3$ (-126b15 - 351b14 - 100b13 + 231b12 + 13b11 + 333b10 + 48b9 + 80b8 - 482b7 - 81b6 + 141b5 - 219b4 - 138b3 + 86b2 - 46*b1 + 217) / 3
$\nu^{12}$	$=$	$( - 209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + \cdots - 97 ) / 3$ (-209b15 - 418b14 - 251b13 - 170b12 + 68b11 + 416b10 + 301b9 - 320b8 + 325b7 - 36b6 - 298b5 + 153b4 - 357b3 + 376b2 + 1031*b1 - 97) / 3
$\nu^{13}$	$=$	$( 385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} + \cdots + 1243 ) / 3$ (385b15 - 609b14 - 1440b13 + 402b12 - 186b11 + 414b10 - 155b9 - 478b8 - 220b7 + 98b6 - 597b5 + 277b4 - 1199b3 + 1540b2 + 2320*b1 + 1243) / 3
$\nu^{14}$	$=$	$( 1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} + \cdots - 1641 ) / 3$ (1281b15 - 511b14 - 835b13 + 960b12 + 1042b11 + 812b10 - 27b9 + 3809b8 + 541b7 - 1481b6 - 882b5 + 1178b4 - 1454b3 + 2715b2 + 2486*b1 - 1641) / 3
$\nu^{15}$	$=$	$( 2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} + \cdots + 208 ) / 3$ (2826b15 - 1179b14 - 1012b13 + 2828b12 + 1522b11 + 2838b10 - 2730b9 + 6361b8 - 11b7 - 2559b6 - 431b5 - 2124b4 + 809b3 + 757b2 + 1787*b1 + 208) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$ .

$n$	$325$	$407$
$\chi(n)$	$1 - \beta_{8}$	$-1$

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}

647.1

1.73109	+	0.0577511i
−1.68301	−	0.409224i
0.320287	−	1.70218i
0.765614	+	1.55365i
1.27866	+	1.16834i
−1.70672	+	0.295146i
1.58110	−	0.707199i
1.71298	−	0.256290i
1.73109	−	0.0577511i
−1.68301	+	0.409224i
0.320287	+	1.70218i
0.765614	−	1.55365i
1.27866	−	1.16834i
−1.70672	−	0.295146i
1.58110	+	0.707199i
1.71298	+	0.256290i

−0.866025

0.500000i

0.500000

−

0.866025i

−1.14095

−

1.97618i

−2.64314

−

0.117551i

1.00000i

1.97618

1.14095i

647.2

−0.866025

0.500000i

0.500000

−

0.866025i

−0.714925

−

1.23829i

2.10995

−

1.59628i

1.00000i

1.23829

0.714925i

647.3

−0.866025

0.500000i

0.500000

−

0.866025i

0.0338034

0.0585493i

1.44425

−

2.21679i

1.00000i

−0.0585493

−

0.0338034i

647.4

−0.866025

0.500000i

0.500000

−

0.866025i

1.82207

3.15592i

−1.04503

2.43062i

1.00000i

−3.15592

−

1.82207i

647.5

0.866025

−

0.500000i

0.500000

−

0.866025i

−1.77612

−

3.07634i

−1.14420

−

2.38554i

−

1.00000i

−3.07634

−

1.77612i

647.6

0.866025

−

0.500000i

0.500000

−

0.866025i

−0.483662

−

0.837727i

−0.238876

2.63495i

−

1.00000i

−0.837727

−

0.483662i

647.7

0.866025

−

0.500000i

0.500000

−

0.866025i

0.450129

0.779646i

−2.62906

−

0.296732i

−

1.00000i

0.779646

0.450129i

647.8

0.866025

−

0.500000i

0.500000

−

0.866025i

1.80966

3.13442i

2.14611

1.54733i

−

1.00000i

3.13442

1.80966i

971.1

−0.866025

−

0.500000i

0.500000

0.866025i

−1.14095

1.97618i

−2.64314

0.117551i

−

1.00000i

1.97618

−

1.14095i

971.2

−0.866025

−

0.500000i

0.500000

0.866025i

−0.714925

1.23829i

2.10995

1.59628i

−

1.00000i

1.23829

−

0.714925i

971.3

−0.866025

−

0.500000i

0.500000

0.866025i

0.0338034

−

0.0585493i

1.44425

2.21679i

−

1.00000i

−0.0585493

0.0338034i

971.4

−0.866025

−

0.500000i

0.500000

0.866025i

1.82207

−

3.15592i

−1.04503

−

2.43062i

−

1.00000i

−3.15592

1.82207i

971.5

0.866025

0.500000i

0.500000

0.866025i

−1.77612

3.07634i

−1.14420

2.38554i

1.00000i

−3.07634

1.77612i

971.6

0.866025

0.500000i

0.500000

0.866025i

−0.483662

0.837727i

−0.238876

−

2.63495i

1.00000i

−0.837727

0.483662i

971.7

0.866025

0.500000i

0.500000

0.866025i

0.450129

−

0.779646i

−2.62906

0.296732i

1.00000i

0.779646

−

0.450129i

971.8

0.866025

0.500000i

0.500000

0.866025i

1.80966

−

3.13442i

2.14611

−

1.54733i

1.00000i

3.13442

−

1.80966i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1134.2.k.b		16
3.b	odd	2	1		1134.2.k.a		16
7.d	odd	6	1		1134.2.k.a		16
9.c	even	3	1		126.2.t.a	yes	16
9.c	even	3	1		378.2.l.a		16
9.d	odd	6	1		126.2.l.a	✓	16
9.d	odd	6	1		378.2.t.a		16
21.g	even	6	1	inner	1134.2.k.b		16
36.f	odd	6	1		1008.2.df.c		16
36.f	odd	6	1		3024.2.ca.c		16
36.h	even	6	1		1008.2.ca.c		16
36.h	even	6	1		3024.2.df.c		16
63.g	even	3	1		882.2.l.b		16
63.g	even	3	1		2646.2.m.b		16
63.h	even	3	1		882.2.m.a		16
63.h	even	3	1		2646.2.t.b		16
63.i	even	6	1		126.2.t.a	yes	16
63.i	even	6	1		2646.2.m.b		16
63.j	odd	6	1		882.2.t.a		16
63.j	odd	6	1		2646.2.m.a		16
63.k	odd	6	1		126.2.l.a	✓	16
63.k	odd	6	1		2646.2.m.a		16
63.l	odd	6	1		882.2.t.a		16
63.l	odd	6	1		2646.2.l.a		16
63.n	odd	6	1		882.2.m.b		16
63.n	odd	6	1		2646.2.l.a		16
63.o	even	6	1		882.2.l.b		16
63.o	even	6	1		2646.2.t.b		16
63.s	even	6	1		378.2.l.a		16
63.s	even	6	1		882.2.m.a		16
63.t	odd	6	1		378.2.t.a		16
63.t	odd	6	1		882.2.m.b		16
252.n	even	6	1		1008.2.ca.c		16
252.r	odd	6	1		1008.2.df.c		16
252.bj	even	6	1		3024.2.df.c		16
252.bn	odd	6	1		3024.2.ca.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.l.a	✓	16	9.d	odd	6	1
126.2.l.a	✓	16	63.k	odd	6	1
126.2.t.a	yes	16	9.c	even	3	1
126.2.t.a	yes	16	63.i	even	6	1
378.2.l.a		16	9.c	even	3	1
378.2.l.a		16	63.s	even	6	1
378.2.t.a		16	9.d	odd	6	1
378.2.t.a		16	63.t	odd	6	1
882.2.l.b		16	63.g	even	3	1
882.2.l.b		16	63.o	even	6	1
882.2.m.a		16	63.h	even	3	1
882.2.m.a		16	63.s	even	6	1
882.2.m.b		16	63.n	odd	6	1
882.2.m.b		16	63.t	odd	6	1
882.2.t.a		16	63.j	odd	6	1
882.2.t.a		16	63.l	odd	6	1
1008.2.ca.c		16	36.h	even	6	1
1008.2.ca.c		16	252.n	even	6	1
1008.2.df.c		16	36.f	odd	6	1
1008.2.df.c		16	252.r	odd	6	1
1134.2.k.a		16	3.b	odd	2	1
1134.2.k.a		16	7.d	odd	6	1
1134.2.k.b		16	1.a	even	1	1	trivial
1134.2.k.b		16	21.g	even	6	1	inner
2646.2.l.a		16	63.l	odd	6	1
2646.2.l.a		16	63.n	odd	6	1
2646.2.m.a		16	63.j	odd	6	1
2646.2.m.a		16	63.k	odd	6	1
2646.2.m.b		16	63.g	even	3	1
2646.2.m.b		16	63.i	even	6	1
2646.2.t.b		16	63.h	even	3	1
2646.2.t.b		16	63.o	even	6	1
3024.2.ca.c		16	36.f	odd	6	1
3024.2.ca.c		16	252.bn	odd	6	1
3024.2.df.c		16	36.h	even	6	1
3024.2.df.c		16	252.bj	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{16} + 24 T_{5}^{14} + 24 T_{5}^{13} + 423 T_{5}^{12} + 450 T_{5}^{11} + 3582 T_{5}^{10} + \cdots + 81$ T5^16 + 24*T5^14 + 24*T5^13 + 423*T5^12 + 450*T5^11 + 3582*T5^10 + 5814*T5^9 + 22536*T5^8 + 25002*T5^7 + 42201*T5^6 + 19494*T5^5 + 32724*T5^4 + 11826*T5^3 + 16929*T5^2 - 1134*T5 + 81 acting on $S_{2}^{\mathrm{new}}(1134, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - T^{2} + 1)^{4}$ (T^4 - T^2 + 1)^4
$3$	$T^{16}$ T^16
$5$	$T^{16} + 24 T^{14} + \cdots + 81$ T^16 + 24T^14 + 24T^13 + 423T^12 + 450T^11 + 3582T^10 + 5814T^9 + 22536T^8 + 25002T^7 + 42201T^6 + 19494T^5 + 32724T^4 + 11826T^3 + 16929T^2 - 1134T + 81
$7$	$T^{16} + 4 T^{15} + \cdots + 5764801$ T^16 + 4T^15 + 9T^14 + 32T^13 + 83T^12 + 450T^11 + 1333T^10 + 2668T^9 + 8577T^8 + 18676T^7 + 65317T^6 + 154350T^5 + 199283T^4 + 537824T^3 + 1058841T^2 + 3294172*T + 5764801
$11$	$T^{16} - 12 T^{15} + \cdots + 61732449$ T^16 - 12T^15 + 18T^14 + 360T^13 - 1017T^12 - 7776T^11 + 30132T^10 + 92826T^9 - 395118T^8 - 868968T^7 + 3600531T^6 + 5642460T^5 - 16460820T^4 - 28680318T^3 + 43505991T^2 + 105645222*T + 61732449
$13$	$T^{16} + \cdots + 390971529$ T^16 + 150T^14 + 8919T^12 + 272682T^10 + 4674987T^8 + 45543438T^6 + 239352246T^4 + 577843956*T^2 + 390971529
$17$	$T^{16} - 18 T^{15} + \cdots + 56070144$ T^16 - 18T^15 + 231T^14 - 1794T^13 + 11493T^12 - 53892T^11 + 241380T^10 - 875952T^9 + 3161232T^8 - 8227008T^7 + 19253376T^6 - 31774464T^5 + 53021952T^4 - 66023424T^3 + 91570176T^2 - 73322496*T + 56070144
$19$	$T^{16} - 72 T^{14} + \cdots + 9199089$ T^16 - 72T^14 + 4167T^12 - 10116T^11 - 55782T^10 + 162594T^9 + 634680T^8 - 1418796T^7 - 3302019T^6 + 6237810T^5 + 14697288T^4 - 9389520T^3 - 11629251T^2 + 6551280*T + 9199089
$23$	$T^{16} + \cdots + 187388721$ T^16 + 6T^15 - 54T^14 - 396T^13 + 2763T^12 + 11340T^11 - 54108T^10 - 187596T^9 + 815022T^8 + 1751058T^7 - 6579225T^6 - 10493226T^5 + 39946284T^4 + 27232524T^3 - 100363617T^2 - 48787596*T + 187388721
$29$	$T^{16} + 108 T^{14} + \cdots + 1108809$ T^16 + 108T^14 + 4518T^12 + 92502T^10 + 966411T^8 + 4946994T^6 + 11762415T^4 + 10641942*T^2 + 1108809
$31$	$T^{16} + 6 T^{15} + \cdots + 65610000$ T^16 + 6T^15 - 84T^14 - 576T^13 + 6705T^12 + 13608T^11 - 196938T^10 - 178524T^9 + 4847445T^8 - 10540368T^7 - 6430023T^6 + 36006768T^5 + 12764061T^4 - 80525340T^3 - 14652900T^2 + 91854000*T + 65610000
$37$	$T^{16} + \cdots + 32746159681$ T^16 + 2T^15 + 180T^14 - 212T^13 + 20726T^12 - 40410T^11 + 1458136T^10 - 4587118T^9 + 74228751T^8 - 226939114T^7 + 2287630168T^6 - 7527670110T^5 + 46677522614T^4 - 97465928012T^3 + 184692416508T^2 - 85008385594*T + 32746159681
$41$	$(T^{8} + 6 T^{7} - 69 T^{6} + \cdots + 9)^{2}$ (T^8 + 6T^7 - 69T^6 - 102T^5 + 1287T^4 - 2250T^3 + 1098T^2 - 180*T + 9)^2
$43$	$(T^{8} - 2 T^{7} + \cdots + 54769)^{2}$ (T^8 - 2T^7 - 131T^6 + 88T^5 + 5593T^4 + 1900T^3 - 75014T^2 - 60752*T + 54769)^2
$47$	$T^{16} + \cdots + 588203099136$ T^16 + 18T^15 + 321T^14 + 3354T^13 + 38619T^12 + 335628T^11 + 2972988T^10 + 20639754T^9 + 140431797T^8 + 797520384T^7 + 4357561698T^6 + 19327462218T^5 + 74568403089T^4 + 211321078200T^3 + 470997221472T^2 + 638729369856*T + 588203099136
$53$	$T^{16} + \cdots + 36759242529$ T^16 - 36T^15 + 342T^14 + 3240T^13 - 59697T^12 - 312498T^11 + 9260244T^10 - 18378090T^9 - 432814590T^8 + 1559379114T^7 + 18102553515T^6 - 151911504432T^5 + 467177028516T^4 - 623279673504T^3 + 216276115095T^2 + 192322120608*T + 36759242529
$59$	$T^{16} + \cdots + 216504090000$ T^16 - 30T^15 + 672T^14 - 9348T^13 + 113769T^12 - 1072944T^11 + 9838458T^10 - 72646884T^9 + 493635321T^8 - 2348646408T^7 + 9371184921T^6 - 20166063156T^5 + 52632256221T^4 - 90386999280T^3 + 199283579100T^2 - 210725064000*T + 216504090000
$61$	$T^{16} + \cdots + 547560000$ T^16 - 60T^15 + 1548T^14 - 20880T^13 + 135891T^12 - 67572T^11 - 3676878T^10 - 7743330T^9 + 418107141T^8 - 3086545284T^7 + 9700556031T^6 - 7525757628T^5 - 25483947759T^4 + 23542982100T^3 + 86065788600T^2 + 11814660000*T + 547560000
$67$	$T^{16} + \cdots + 2603856784$ T^16 - 14T^15 + 297T^14 - 526T^13 + 18443T^12 + 38232T^11 + 1101268T^10 + 3752662T^9 + 34299369T^8 + 125698780T^7 + 776010586T^6 + 2376076410T^5 + 7750191281T^4 + 10814306060T^3 + 14528452572T^2 - 5000131664*T + 2603856784
$71$	$T^{16} + 486 T^{14} + \cdots + 65610000$ T^16 + 486T^14 + 77553T^12 + 4631418T^10 + 118814850T^8 + 1363509936T^6 + 5897814849T^4 + 1520839800*T^2 + 65610000
$73$	$T^{16} - 150 T^{14} + \cdots + 71115489$ T^16 - 150T^14 + 18909T^12 - 99126T^11 - 293184T^10 + 2724732T^9 + 5074506T^8 - 52748334T^7 + 2268459T^6 + 381956796T^5 + 184027950T^4 - 1251964674T^3 + 804430197T^2 + 461301966*T + 71115489
$79$	$T^{16} + \cdots + 970422010000$ T^16 + 16T^15 + 405T^14 + 2228T^13 + 50093T^12 + 147744T^11 + 4930570T^10 + 3465268T^9 + 235975761T^8 + 420147388T^7 + 7055668672T^6 + 13192279974T^5 + 140363929769T^4 + 339772788080T^3 + 1444878963900T^2 + 1233975664000*T + 970422010000
$83$	$(T^{8} - 177 T^{6} + \cdots - 30879)^{2}$ (T^8 - 177T^6 - 156T^5 + 8649T^4 + 7722T^3 - 93672T^2 + 116784T - 30879)^2
$89$	$T^{16} + \cdots + 131145120363321$ T^16 - 24T^15 + 702T^14 - 9180T^13 + 187029T^12 - 2265570T^11 + 32872068T^10 - 313712028T^9 + 3371130738T^8 - 29227238586T^7 + 232008422067T^6 - 1393795430352T^5 + 6794512511634T^4 - 24192458322894T^3 + 66081140272737T^2 - 115458464232270*T + 131145120363321
$97$	$T^{16} + \cdots + 9120206721024$ T^16 + 798T^14 + 257913T^12 + 43557768T^10 + 4109294448T^8 + 212849641728T^6 + 5464804071168T^4 + 52780972640256*T^2 + 9120206721024
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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 647.8
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	1134.2.k.b

Level	$1134$
Weight	$2$
Character orbit	1134.k
Analytic conductor	$9.055$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$