LMFDB - Newform orbit 882.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(293,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("882.293");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$882 = 2 \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	882.m (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:	$7.04280545828$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3^{2}$
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.

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

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
$\beta_{2}$	$=$	$( 1292 \nu^{15} - 10486 \nu^{14} + 25660 \nu^{13} + 10145 \nu^{12} - 192280 \nu^{11} + 408694 \nu^{10} + \cdots - 9270693 ) / 142155$ (1292v^15 - 10486v^14 + 25660v^13 + 10145v^12 - 192280v^11 + 408694v^10 - 51650v^9 - 1459361v^8 + 2979638v^7 - 1138791v^6 - 5534208v^5 + 10987326v^4 - 5516586v^3 - 9628389v^2 + 17565255*v - 9270693) / 142155
$\beta_{3}$	$=$	$( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + \cdots - 20783061 ) / 142155$ (2846v^15 - 22369v^14 + 55246v^13 + 17972v^12 - 402586v^11 + 878875v^10 - 166676v^9 - 3023495v^8 + 6386423v^7 - 2756469v^6 - 11333493v^5 + 23486949v^4 - 12667509v^3 - 19623951v^2 + 37934244*v - 20783061) / 142155
$\beta_{4}$	$=$	$( - 2782 \nu^{15} + 15918 \nu^{14} - 26947 \nu^{13} - 42629 \nu^{12} + 270897 \nu^{11} + \cdots + 7405182 ) / 47385$ (-2782v^15 + 15918v^14 - 26947v^13 - 42629v^12 + 270897v^11 - 425335v^10 - 220488v^9 + 2001225v^8 - 3082271v^7 - 70562v^6 + 7458951v^5 - 11186748v^4 + 2636253v^3 + 12673422v^2 - 17180343*v + 7405182) / 47385
$\beta_{5}$	$=$	$( - 3648 \nu^{15} + 23120 \nu^{14} - 45376 \nu^{13} - 47012 \nu^{12} + 401996 \nu^{11} + \cdots + 14281839 ) / 47385$ (-3648v^15 + 23120v^14 - 45376v^13 - 47012v^12 + 401996v^11 - 719704v^10 - 150629v^9 + 2988436v^8 - 5221301v^7 + 915140v^6 + 11160666v^5 - 19038915v^4 + 7000290v^3 + 19081575v^2 - 29765799*v + 14281839) / 47385
$\beta_{6}$	$=$	$( - 16948 \nu^{15} + 107120 \nu^{14} - 210206 \nu^{13} - 216202 \nu^{12} + 1856546 \nu^{11} + \cdots + 66620394 ) / 142155$ (-16948v^15 + 107120v^14 - 210206v^13 - 216202v^12 + 1856546v^11 - 3329594v^10 - 672599v^9 + 13772146v^8 - 24135496v^7 + 4376130v^6 + 51333696v^5 - 87961680v^4 + 32792850v^3 + 87596640v^2 - 137624994*v + 66620394) / 142155
$\beta_{7}$	$=$	$( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14935023 ) / 28431$ (-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 + 14935023) / 28431
$\beta_{8}$	$=$	$( 24706 \nu^{15} - 155855 \nu^{14} + 305147 \nu^{13} + 316579 \nu^{12} - 2700647 \nu^{11} + \cdots - 96324228 ) / 142155$ (24706v^15 - 155855v^14 + 305147v^13 + 316579v^12 - 2700647v^11 + 4831208v^10 + 1007453v^9 - 20043787v^8 + 35017972v^7 - 6181845v^6 - 74736702v^5 + 127624545v^4 - 47130660v^3 - 127644255v^2 + 199674558*v - 96324228) / 142155
$\beta_{9}$	$=$	$( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + \cdots - 7453296 ) / 10935$ (2015v^15 - 12538v^14 + 24088v^13 + 26576v^12 - 216643v^11 + 381184v^10 + 93322v^9 - 1605386v^8 + 2760692v^7 - 423483v^6 - 5980032v^5 + 10047753v^4 - 3553308v^3 - 10193607v^2 + 15657462*v - 7453296) / 10935
$\beta_{10}$	$=$	$( - 26777 \nu^{15} + 172141 \nu^{14} - 345205 \nu^{13} - 329150 \nu^{12} + 2992915 \nu^{11} + \cdots + 112005018 ) / 142155$ (-26777v^15 + 172141v^14 - 345205v^13 - 329150v^12 + 2992915v^11 - 5473354v^10 - 873115v^9 + 22225271v^8 - 39691043v^7 + 8309796v^6 + 82849923v^5 - 144806481v^4 + 56755161v^3 + 141385419v^2 - 227291265*v + 112005018) / 142155
$\beta_{11}$	$=$	$( 32006 \nu^{15} - 201190 \nu^{14} + 390892 \nu^{13} + 415394 \nu^{12} - 3479542 \nu^{11} + \cdots - 121376313 ) / 142155$ (32006v^15 - 201190v^14 + 390892v^13 + 415394v^12 - 3479542v^11 + 6180118v^10 + 1385833v^9 - 25793657v^8 + 44738537v^7 - 7406040v^6 - 96117777v^5 + 162796500v^4 - 58760640v^3 - 164045655v^2 + 253970478*v - 121376313) / 142155
$\beta_{12}$	$=$	$( - 10990 \nu^{15} + 68842 \nu^{14} - 133062 \nu^{13} - 143724 \nu^{12} + 1190062 \nu^{11} + \cdots + 41433444 ) / 47385$ (-10990v^15 + 68842v^14 - 133062v^13 - 143724v^12 + 1190062v^11 - 2105781v^10 - 489283v^9 + 8822519v^8 - 15253998v^7 + 2459777v^6 + 32873688v^5 - 55540512v^4 + 19941417v^3 + 56061558v^2 - 86668623*v + 41433444) / 47385
$\beta_{13}$	$=$	$( - 35897 \nu^{15} + 219337 \nu^{14} - 409846 \nu^{13} - 492632 \nu^{12} + 3774151 \nu^{11} + \cdots + 122931270 ) / 142155$ (-35897v^15 + 219337v^14 - 409846v^13 - 492632v^12 + 3774151v^11 - 6478657v^10 - 1950109v^9 + 27930713v^8 - 46902542v^7 + 5479452v^6 + 103995882v^5 - 170423757v^4 + 56059857v^3 + 177093783v^2 - 264354354*v + 122931270) / 142155
$\beta_{14}$	$=$	$( - 14807 \nu^{15} + 92828 \nu^{14} - 179757 \nu^{13} - 193044 \nu^{12} + 1605257 \nu^{11} + \cdots + 55960227 ) / 47385$ (-14807v^15 + 92828v^14 - 179757v^13 - 193044v^12 + 1605257v^11 - 2844540v^10 - 652043v^9 + 11900965v^8 - 20604021v^7 + 3361183v^6 + 44347701v^5 - 75011778v^4 + 26997138v^3 + 75667932v^2 - 117064278*v + 55960227) / 47385
$\beta_{15}$	$=$	$( - 68699 \nu^{15} + 433198 \nu^{14} - 844756 \nu^{13} - 889547 \nu^{12} + 7506256 \nu^{11} + \cdots + 264515463 ) / 142155$ (-68699v^15 + 433198v^14 - 844756v^13 - 889547v^12 + 7506256v^11 - 13372516v^10 - 2912329v^9 + 55693424v^8 - 96903605v^7 + 16506963v^6 + 207637920v^5 - 352968678v^4 + 128769183v^3 + 354359367v^2 - 551274174*v + 264515463) / 142155

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

$\nu$	$=$	$( -\beta_{15} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$ (-b15 - b9 - b8 + b7 + b6 - 2b4 + 2b3 - 2*b2 + b1 + 2) / 3
$\nu^{2}$	$=$	$( - \beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 3 ) / 3$ (-b15 + 4b14 + b13 - 2b12 + 2b11 - b10 - b9 - b8 - b6 - b5 - 2b4 - b3 - 2b2 + 2b1 + 3) / 3
$\nu^{3}$	$=$	$( - 3 \beta_{15} - \beta_{14} + 5 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{8} + \cdots - 4 ) / 3$ (-3b15 - b14 + 5b13 + 2b12 + b11 + b10 - 3b8 - 2b7 - b6 - 2b5 - 3b3 - 6b2 - b1 - 4) / 3
$\nu^{4}$	$=$	$( - \beta_{15} + 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} + \cdots + 7 ) / 3$ (-b15 + 2b14 + 11b13 - 4b12 + 7b11 + 4b10 + 8b9 - 4b8 - b7 - 6b6 - 2b5 + 7b4 - 10b3 - 2b2 - 15*b1 + 7) / 3
$\nu^{5}$	$=$	$( - 11 \beta_{15} - 6 \beta_{14} + 9 \beta_{13} + 12 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots + 10 ) / 3$ (-11b15 - 6b14 + 9b13 + 12b11 + 6b10 - 2b9 + b8 + 23b7 + 17b6 + 3b5 + 8b4 - 5b3 - 16b2 - 31*b1 + 10) / 3
$\nu^{6}$	$=$	$( - 5 \beta_{15} + 26 \beta_{14} + 2 \beta_{13} - 31 \beta_{12} + 28 \beta_{11} + 13 \beta_{10} + \cdots + 15 ) / 3$ (-5b15 + 26b14 + 2b13 - 31b12 + 28b11 + 13b10 - 14b9 + 4b8 + 39b7 - 20b6 - 5b5 - b4 - 2b3 - 10b2 - 20b1 + 15) / 3
$\nu^{7}$	$=$	$( - 15 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} + 16 \beta_{12} + 29 \beta_{11} - 13 \beta_{10} + \cdots - 11 ) / 3$ (-15b15 + 16b14 - 20b13 + 16b12 + 29b11 - 13b10 - 60b9 + 6b8 + 26b7 - 5b6 - 10b5 - 3b4 - 3b3 - 30b2 - 14*b1 - 11) / 3
$\nu^{8}$	$=$	$( 43 \beta_{15} + 28 \beta_{14} - 8 \beta_{13} + 10 \beta_{12} + 44 \beta_{11} - 16 \beta_{10} + 25 \beta_{9} + \cdots - 1 ) / 3$ (43b15 + 28b14 - 8b13 + 10b12 + 44b11 - 16b10 + 25b9 - 92b8 - 65b7 - 114b6 - 52b5 - 7b4 + 31b3 + 26b2 - 27*b1 - 1) / 3
$\nu^{9}$	$=$	$( 50 \beta_{15} - 12 \beta_{14} - 18 \beta_{13} + 102 \beta_{12} + 102 \beta_{11} - 144 \beta_{10} + \cdots + 134 ) / 3$ (50b15 - 12b14 - 18b13 + 102b12 + 102b11 - 144b10 + 80b9 - 106b8 - 104b7 + 229b6 - 84b5 + 34b4 + 44b3 - 8b2 - 194*b1 + 134) / 3
$\nu^{10}$	$=$	$( 140 \beta_{15} + 139 \beta_{14} + 31 \beta_{13} - 224 \beta_{12} + 95 \beta_{11} - 46 \beta_{10} + \cdots - 109 \beta_1 ) / 3$ (140b15 + 139b14 + 31b13 - 224b12 + 95b11 - 46b10 + 320b9 - 220b8 - 117b7 + 251b6 - 337b5 + 10b4 + 224b3 - 80b2 - 109*b1) / 3
$\nu^{11}$	$=$	$( 60 \beta_{15} + 503 \beta_{14} + 23 \beta_{13} - 313 \beta_{12} + 133 \beta_{11} - 218 \beta_{10} + \cdots - 193 ) / 3$ (60b15 + 503b14 + 23b13 - 313b12 + 133b11 - 218b10 + 114b9 + 171b8 - 482b7 + 779b6 - 524b5 + 270b4 - 12b3 - 399b2 - 166*b1 - 193) / 3
$\nu^{12}$	$=$	$( - 94 \beta_{15} + 722 \beta_{14} + 191 \beta_{13} - 343 \beta_{12} - 182 \beta_{11} + 238 \beta_{10} + \cdots - 1112 ) / 3$ (-94b15 + 722b14 + 191b13 - 343b12 - 182b11 + 238b10 + 569b9 - 454b8 - 1069b7 - 192b6 - 674b5 + 382b4 + 398b3 - 719b2 - 84*b1 - 1112) / 3
$\nu^{13}$	$=$	$( - 425 \beta_{15} + 1353 \beta_{14} + 255 \beta_{13} + 1086 \beta_{12} + 1068 \beta_{11} - 588 \beta_{10} + \cdots + 727 ) / 3$ (-425b15 + 1353b14 + 255b13 + 1086b12 + 1068b11 - 588b10 + 691b9 - 122b8 - 1675b7 + 662b6 + 498b5 + 707b4 - 206b3 - 454b2 - 1759*b1 + 727) / 3
$\nu^{14}$	$=$	$( - 953 \beta_{15} + 446 \beta_{14} + 35 \beta_{13} + 2231 \beta_{12} + 1294 \beta_{11} + 82 \beta_{10} + \cdots + 1896 ) / 3$ (-953b15 + 446b14 + 35b13 + 2231b12 + 1294b11 + 82b10 + 1990b9 - 1151b8 + 1755b7 + 91b6 + 688b5 - 970b4 + 1378b3 - 484b2 - 1505*b1 + 1896) / 3
$\nu^{15}$	$=$	$( 1563 \beta_{15} + 1963 \beta_{14} - 725 \beta_{13} + 2308 \beta_{12} + 3086 \beta_{11} - 1306 \beta_{10} + \cdots + 6694 ) / 3$ (1563b15 + 1963b14 - 725b13 + 2308b12 + 3086b11 - 1306b10 + 714b9 + 4854b8 + 2906b7 - 1550b6 + 32b5 - 1380b4 - 2391b3 + 1551b2 + 1375*b1 + 6694) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$ .

$n$	$199$	$785$
$\chi(n)$	$-1$	$-\beta_{7}$

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}

293.1

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

−0.866025

0.500000i

−1.62418

−

0.601709i

0.500000

−

0.866025i

0.0338034

−

0.0585493i

1.70743

−

0.290993i

1.00000i

2.27589

1.95456i

0.0676069i

293.2

−0.866025

0.500000i

0.0569233

−

1.73112i

0.500000

−

0.866025i

1.82207

−

3.15592i

0.816261

1.52765i

1.00000i

−2.99352

−

0.197082i

3.64414i

293.3

−0.866025

0.500000i

0.840895

1.51423i

0.500000

−

0.866025i

−1.14095

1.97618i

−1.48535

−

0.890915i

1.00000i

−1.58579

2.54662i

−

2.28190i

293.4

−0.866025

0.500000i

1.59238

−

0.681407i

0.500000

−

0.866025i

−0.714925

1.23829i

−1.03834

1.38631i

1.00000i

2.07137

−

2.17012i

−

1.42985i

293.5

0.866025

−

0.500000i

−1.71170

0.264701i

0.500000

−

0.866025i

−1.77612

3.07634i

−1.35003

1.08509i

−

1.00000i

2.85987

−

0.906179i

3.55225i

293.6

0.866025

−

0.500000i

−1.43162

0.974922i

0.500000

−

0.866025i

1.80966

−

3.13442i

−0.752355

1.56012i

−

1.00000i

1.09905

−

2.79143i

−

3.61932i

293.7

0.866025

−

0.500000i

0.991125

1.42045i

0.500000

−

0.866025i

−0.483662

0.837727i

1.56856

0.734581i

−

1.00000i

−1.03534

2.81568i

0.967324i

293.8

0.866025

−

0.500000i

1.28617

−

1.16007i

0.500000

−

0.866025i

0.450129

−

0.779646i

0.533822

−

1.64774i

−

1.00000i

0.308473

−

2.98410i

−

0.900258i

587.1

−0.866025

−

0.500000i

−1.62418

0.601709i

0.500000

0.866025i

0.0338034

0.0585493i

1.70743

0.290993i

−

1.00000i

2.27589

−

1.95456i

−

0.0676069i

587.2

−0.866025

−

0.500000i

0.0569233

1.73112i

0.500000

0.866025i

1.82207

3.15592i

0.816261

−

1.52765i

−

1.00000i

−2.99352

0.197082i

−

3.64414i

587.3

−0.866025

−

0.500000i

0.840895

−

1.51423i

0.500000

0.866025i

−1.14095

−

1.97618i

−1.48535

0.890915i

−

1.00000i

−1.58579

−

2.54662i

2.28190i

587.4

−0.866025

−

0.500000i

1.59238

0.681407i

0.500000

0.866025i

−0.714925

−

1.23829i

−1.03834

−

1.38631i

−

1.00000i

2.07137

2.17012i

1.42985i

587.5

0.866025

0.500000i

−1.71170

−

0.264701i

0.500000

0.866025i

−1.77612

−

3.07634i

−1.35003

−

1.08509i

1.00000i

2.85987

0.906179i

−

3.55225i

587.6

0.866025

0.500000i

−1.43162

−

0.974922i

0.500000

0.866025i

1.80966

3.13442i

−0.752355

−

1.56012i

1.00000i

1.09905

2.79143i

3.61932i

587.7

0.866025

0.500000i

0.991125

−

1.42045i

0.500000

0.866025i

−0.483662

−

0.837727i

1.56856

−

0.734581i

1.00000i

−1.03534

−

2.81568i

−

0.967324i

587.8

0.866025

0.500000i

1.28617

1.16007i

0.500000

0.866025i

0.450129

0.779646i

0.533822

1.64774i

1.00000i

0.308473

2.98410i

0.900258i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.o	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.2.m.a		16
3.b	odd	2	1		2646.2.m.a		16
7.b	odd	2	1		882.2.m.b		16
7.c	even	3	1		126.2.t.a	yes	16
7.c	even	3	1		882.2.l.b		16
7.d	odd	6	1		126.2.l.a	✓	16
7.d	odd	6	1		882.2.t.a		16
9.c	even	3	1		2646.2.m.b		16
9.d	odd	6	1		882.2.m.b		16
21.c	even	2	1		2646.2.m.b		16
21.g	even	6	1		378.2.l.a		16
21.g	even	6	1		2646.2.t.b		16
21.h	odd	6	1		378.2.t.a		16
21.h	odd	6	1		2646.2.l.a		16
28.f	even	6	1		1008.2.ca.c		16
28.g	odd	6	1		1008.2.df.c		16
63.g	even	3	1		378.2.l.a		16
63.h	even	3	1		1134.2.k.b		16
63.h	even	3	1		2646.2.t.b		16
63.i	even	6	1		126.2.t.a	yes	16
63.j	odd	6	1		882.2.t.a		16
63.j	odd	6	1		1134.2.k.a		16
63.k	odd	6	1		1134.2.k.a		16
63.k	odd	6	1		2646.2.l.a		16
63.l	odd	6	1		2646.2.m.a		16
63.n	odd	6	1		126.2.l.a	✓	16
63.o	even	6	1	inner	882.2.m.a		16
63.s	even	6	1		882.2.l.b		16
63.s	even	6	1		1134.2.k.b		16
63.t	odd	6	1		378.2.t.a		16
84.j	odd	6	1		3024.2.ca.c		16
84.n	even	6	1		3024.2.df.c		16
252.o	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.bl	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	7.d	odd	6	1
126.2.l.a	✓	16	63.n	odd	6	1
126.2.t.a	yes	16	7.c	even	3	1
126.2.t.a	yes	16	63.i	even	6	1
378.2.l.a		16	21.g	even	6	1
378.2.l.a		16	63.g	even	3	1
378.2.t.a		16	21.h	odd	6	1
378.2.t.a		16	63.t	odd	6	1
882.2.l.b		16	7.c	even	3	1
882.2.l.b		16	63.s	even	6	1
882.2.m.a		16	1.a	even	1	1	trivial
882.2.m.a		16	63.o	even	6	1	inner
882.2.m.b		16	7.b	odd	2	1
882.2.m.b		16	9.d	odd	6	1
882.2.t.a		16	7.d	odd	6	1
882.2.t.a		16	63.j	odd	6	1
1008.2.ca.c		16	28.f	even	6	1
1008.2.ca.c		16	252.o	even	6	1
1008.2.df.c		16	28.g	odd	6	1
1008.2.df.c		16	252.r	odd	6	1
1134.2.k.a		16	63.j	odd	6	1
1134.2.k.a		16	63.k	odd	6	1
1134.2.k.b		16	63.h	even	3	1
1134.2.k.b		16	63.s	even	6	1
2646.2.l.a		16	21.h	odd	6	1
2646.2.l.a		16	63.k	odd	6	1
2646.2.m.a		16	3.b	odd	2	1
2646.2.m.a		16	63.l	odd	6	1
2646.2.m.b		16	9.c	even	3	1
2646.2.m.b		16	21.c	even	2	1
2646.2.t.b		16	21.g	even	6	1
2646.2.t.b		16	63.h	even	3	1
3024.2.ca.c		16	84.j	odd	6	1
3024.2.ca.c		16	252.bl	odd	6	1
3024.2.df.c		16	84.n	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 $S_{2}^{\mathrm{new}}(882, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - T^{2} + 1)^{4}$ (T^4 - T^2 + 1)^4
$3$	$T^{16} - 3 T^{14} + \cdots + 6561$ T^16 - 3T^14 + 12T^13 + 9T^12 - 36T^11 + 63T^10 + 90T^9 - 207T^8 + 270T^7 + 567T^6 - 972T^5 + 729T^4 + 2916T^3 - 2187*T^2 + 6561
$5$	$T^{16} + 24 T^{14} + \cdots + 81$ 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
$7$	$T^{16}$ T^16
$11$	$T^{16} + 12 T^{15} + \cdots + 61732449$ 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 + 6T^15 - 57T^14 - 414T^13 + 2682T^12 + 19584T^11 - 40167T^10 - 423144T^9 + 452916T^8 + 6581682T^7 + 2433294T^6 - 46095318T^5 - 17730819T^4 + 226594584T^3 + 88818120T^2 - 543480678*T + 390971529
$17$	$(T^{8} + 18 T^{7} + \cdots + 7488)^{2}$ (T^8 + 18T^7 + 93T^6 - 60T^5 - 1764T^4 - 4032T^3 + 576T^2 + 9792*T + 7488)^2
$19$	$T^{16} + 144 T^{14} + \cdots + 9199089$ T^16 + 144T^14 + 7218T^12 + 163890T^10 + 1833507T^8 + 10310598T^6 + 27294651T^4 + 27924102*T^2 + 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
$29$	$T^{16} - 6 T^{15} + \cdots + 1108809$ T^16 - 6T^15 - 36T^14 + 288T^13 + 1197T^12 - 7938T^11 - 19278T^10 + 121986T^9 + 299376T^8 - 912708T^7 - 1095687T^6 + 3639168T^5 + 3174066T^4 - 9950850T^3 + 2880279T^2 + 4264650*T + 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
$37$	$(T^{8} - 2 T^{7} + \cdots - 180959)^{2}$ (T^8 - 2T^7 - 176T^6 + 70T^5 + 10390T^4 + 11170T^3 - 198872T^2 - 469766*T - 180959)^2
$41$	$T^{16} - 6 T^{15} + \cdots + 81$ T^16 - 6T^15 + 105T^14 + 210T^13 + 4086T^12 + 10656T^11 + 114903T^10 + 435366T^9 + 1500462T^8 + 2659284T^7 + 3629772T^6 + 2008098T^5 + 789021T^4 + 157140T^3 + 22518T^2 + 1620*T + 81
$43$	$T^{16} + \cdots + 2999643361$ T^16 + 2T^15 + 135T^14 - 86T^13 + 11744T^12 - 12744T^11 + 594199T^10 - 1079260T^9 + 21475854T^8 - 31678138T^7 + 414160000T^6 - 541864944T^5 + 5436205979T^4 - 4349128328T^3 + 7799247270T^2 + 3327326288*T + 2999643361
$47$	$T^{16} + \cdots + 588203099136$ 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 + 612T^14 + 143694T^12 + 16315506T^10 + 929001879T^8 + 24405305994T^6 + 223877138571T^4 + 573665404626*T^2 + 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
$73$	$T^{16} + 300 T^{14} + \cdots + 71115489$ T^16 + 300T^14 + 29682T^12 + 1322766T^10 + 28536831T^8 + 291169458T^6 + 1203388407T^4 + 1383448410*T^2 + 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
$83$	$T^{16} + \cdots + 953512641$ T^16 + 177T^14 - 312T^13 + 22680T^12 - 35334T^11 + 1367865T^10 - 1501128T^9 + 59460768T^8 - 16891146T^7 + 898947882T^6 + 2738647692T^5 + 8139710007T^4 + 10462495572T^3 + 10746004968T^2 + 3606173136T + 953512641
$89$	$(T^{8} + 24 T^{7} + \cdots + 11451861)^{2}$ (T^8 + 24T^7 - 126T^6 - 6102T^5 - 24705T^4 + 310878T^3 + 3105783T^2 + 10082070*T + 11451861)^2
$97$	$T^{16} + \cdots + 9120206721024$ T^16 + 6T^15 - 381T^14 - 2358T^13 + 102717T^12 + 354204T^11 - 15309036T^10 - 28723248T^9 + 1709674848T^8 - 1664083008T^7 - 99905635008T^6 + 297673761024T^5 + 3933195148800T^4 - 29552832820224T^3 + 71879395230720T^2 - 42337728663552*T + 9120206721024
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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 293.8
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	882.2.m.a

Level	$882$
Weight	$2$
Character orbit	882.m
Analytic conductor	$7.043$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$