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Properties

Label 105.2.q.a
Level $105$
Weight $2$
Character orbit 105.q
Analytic conductor $0.838$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,2,Mod(4,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.q (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: \(0.838429221223\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{6} - \beta_{5}) q^{2} + \beta_{3} q^{3} + (\beta_{10} + \beta_{9} + \beta_{7} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{6} - \beta_{5}) q^{2} + \beta_{3} q^{3} + (\beta_{10} + \beta_{9} + \beta_{7} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{13} + \beta_{12} + \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 2 q^{5} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 2 q^{5} - 8 q^{6} + 8 q^{9} - 4 q^{10} - 24 q^{14} - 4 q^{15} - 24 q^{19} - 8 q^{20} - 4 q^{21} - 12 q^{24} - 4 q^{25} - 12 q^{26} + 24 q^{29} - 12 q^{30} + 16 q^{31} + 16 q^{34} - 10 q^{35} + 16 q^{36} - 4 q^{39} + 32 q^{40} + 16 q^{41} + 20 q^{44} - 2 q^{45} - 32 q^{46} - 40 q^{49} - 40 q^{50} + 4 q^{51} - 4 q^{54} + 8 q^{55} + 84 q^{56} + 4 q^{59} + 16 q^{60} + 16 q^{61} + 16 q^{64} + 30 q^{65} + 28 q^{66} + 40 q^{69} + 16 q^{70} - 56 q^{71} + 40 q^{74} + 8 q^{75} - 64 q^{76} - 16 q^{79} + 52 q^{80} - 8 q^{81} + 12 q^{84} - 64 q^{85} - 48 q^{86} + 16 q^{89} - 8 q^{90} + 8 q^{91} - 32 q^{94} - 22 q^{95} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 38414168495 \nu^{15} + 46999484762 \nu^{14} - 56685616158 \nu^{13} + \cdots - 33746305280705 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 145306983361 \nu^{15} - 177522986678 \nu^{14} + 213949914242 \nu^{13} + \cdots + 31632456480199 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 760172030748 \nu^{15} - 826140524550 \nu^{14} + 311805690170 \nu^{13} - 1979665829518 \nu^{12} + \cdots - 575089767434 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1159675947687 \nu^{15} + 2123382923210 \nu^{14} - 1790860341485 \nu^{13} + \cdots + 29672440598565 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2241001108620 \nu^{15} + 2440547176294 \nu^{14} - 921260080696 \nu^{13} + \cdots + 1695440450074 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4058097157982 \nu^{15} - 7568755097298 \nu^{14} + 7059730047127 \nu^{13} + \cdots - 16137847630635 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2575730834 \nu^{15} + 4699915310 \nu^{14} - 4379029892 \nu^{13} + 9614611682 \nu^{12} + \cdots + 17015828908 ) / 7008735847 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1063280569755 \nu^{15} - 2138087245431 \nu^{14} + 2140603616666 \nu^{13} + \cdots - 8602022998880 ) / 1633035452351 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7502690467678 \nu^{15} - 13694199839130 \nu^{14} + 12758334474904 \nu^{13} + \cdots - 29172119880252 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8183751109447 \nu^{15} + 14956739476040 \nu^{14} - 13922497684986 \nu^{13} + \cdots + 54263991142284 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9494932205673 \nu^{15} + 17203785936492 \nu^{14} - 15830728042254 \nu^{13} + \cdots + 61766681609962 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10583106332007 \nu^{15} - 18562866254784 \nu^{14} + 16212768119001 \nu^{13} + \cdots - 62361843284805 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10663789073454 \nu^{15} - 18661163594876 \nu^{14} + 16331067749359 \nu^{13} + \cdots - 32056382963606 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11015276267169 \nu^{15} - 18856066985592 \nu^{14} + 16454339422594 \nu^{13} + \cdots - 62916861144830 ) / 11431248166457 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13399447651346 \nu^{15} - 23430444102746 \nu^{14} + 20519536606234 \nu^{13} + \cdots - 46488804269431 ) / 11431248166457 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{11} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} - 8\beta_{13} + 3\beta_{12} - 2\beta_{6} + 3\beta_{4} - 2\beta_{3} - 3\beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{10} + \beta_{9} - 8\beta_{7} + 5\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23 \beta_{14} + 11 \beta_{11} - 11 \beta_{10} + 23 \beta_{9} - 13 \beta_{7} - 23 \beta_{5} + \cdots + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{15} - \beta_{13} - 23\beta_{12} + 28\beta_{6} - 22\beta_{4} + 28\beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 103 \beta_{15} - 43 \beta_{14} - 148 \beta_{13} + 43 \beta_{12} - 43 \beta_{11} - 43 \beta_{10} + \cdots + 166 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -10\beta_{14} + 94\beta_{10} + 40\beta_{9} - 10\beta_{8} - 157\beta_{7} - 40\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 459\beta_{15} - 479\beta_{12} - 168\beta_{6} + 171\beta_{4} - 168\beta_{3} - 459\beta_{2} + 171\beta _1 + 339 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 214 \beta_{15} + 469 \beta_{14} - 70 \beta_{13} - 469 \beta_{12} + 399 \beta_{11} - 70 \beta_{8} + \cdots + 70 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 681 \beta_{14} - 681 \beta_{11} - 681 \beta_{10} + 2047 \beta_{9} - 2868 \beta_{8} - 1389 \beta_{7} + \cdots + 2047 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 424\beta_{13} - 424\beta_{12} - 683\beta_{2} - 1693\beta _1 - 4232 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 9149 \beta_{15} + 9997 \beta_{14} - 9997 \beta_{12} - 2701 \beta_{11} + 2701 \beta_{10} - 9149 \beta_{9} + \cdots - 3910 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9573\beta_{14} + 7194\beta_{11} - 2379\beta_{8} + 5603\beta_{5} - 9050\beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 40993 \beta_{15} + 56392 \beta_{13} - 10641 \beta_{12} + 9630 \beta_{6} - 10641 \beta_{4} + \cdots - 36121 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.05078 + 0.281555i
−0.0543258 + 0.202747i
−1.96595 − 0.526774i
−0.556918 + 2.07845i
2.07845 + 0.556918i
0.526774 − 1.96595i
0.202747 + 0.0543258i
−0.281555 + 1.05078i
1.05078 − 0.281555i
−0.0543258 − 0.202747i
−1.96595 + 0.526774i
−0.556918 − 2.07845i
2.07845 − 0.556918i
0.526774 + 1.96595i
0.202747 − 0.0543258i
−0.281555 − 1.05078i
−2.17942 + 1.25829i 0.866025 + 0.500000i 2.16659 − 3.75264i 2.23607 + 0.00136408i −2.51658 1.31340 − 2.29673i 5.87162i 0.500000 + 0.866025i −4.87505 + 2.81065i
4.2 −1.54296 + 0.890827i −0.866025 − 0.500000i 0.587145 − 1.01696i −0.881181 − 2.05512i 1.78165 2.40898 − 1.09398i − 1.47113i 0.500000 + 0.866025i 3.19038 + 2.38598i
4.3 −1.34443 + 0.776205i 0.866025 + 0.500000i 0.204988 − 0.355049i −1.98074 + 1.03763i −1.55241 0.478401 + 2.60214i − 2.46837i 0.500000 + 0.866025i 1.85754 − 2.93248i
4.4 −0.248840 + 0.143668i −0.866025 − 0.500000i −0.958719 + 1.66055i 1.47507 + 1.68052i 0.287336 1.11487 + 2.39939i − 1.12562i 0.500000 + 0.866025i −0.608495 − 0.206261i
4.5 0.248840 − 0.143668i 0.866025 + 0.500000i −0.958719 + 1.66055i 0.717839 + 2.11771i 0.287336 −1.11487 − 2.39939i 1.12562i 0.500000 + 0.866025i 0.482874 + 0.423842i
4.6 1.34443 − 0.776205i −0.866025 − 0.500000i 0.204988 − 0.355049i 1.88899 − 1.19655i −1.55241 −0.478401 − 2.60214i 2.46837i 0.500000 + 0.866025i 1.61083 − 3.07491i
4.7 1.54296 − 0.890827i 0.866025 + 0.500000i 0.587145 − 1.01696i −1.33920 − 1.79069i 1.78165 −2.40898 + 1.09398i 1.47113i 0.500000 + 0.866025i −3.66151 − 1.56996i
4.8 2.17942 − 1.25829i −0.866025 − 0.500000i 2.16659 − 3.75264i −1.11685 + 1.93717i −2.51658 −1.31340 + 2.29673i − 5.87162i 0.500000 + 0.866025i 0.00343282 + 5.62724i
79.1 −2.17942 − 1.25829i 0.866025 − 0.500000i 2.16659 + 3.75264i 2.23607 − 0.00136408i −2.51658 1.31340 + 2.29673i − 5.87162i 0.500000 − 0.866025i −4.87505 − 2.81065i
79.2 −1.54296 − 0.890827i −0.866025 + 0.500000i 0.587145 + 1.01696i −0.881181 + 2.05512i 1.78165 2.40898 + 1.09398i 1.47113i 0.500000 − 0.866025i 3.19038 − 2.38598i
79.3 −1.34443 − 0.776205i 0.866025 − 0.500000i 0.204988 + 0.355049i −1.98074 − 1.03763i −1.55241 0.478401 − 2.60214i 2.46837i 0.500000 − 0.866025i 1.85754 + 2.93248i
79.4 −0.248840 − 0.143668i −0.866025 + 0.500000i −0.958719 − 1.66055i 1.47507 − 1.68052i 0.287336 1.11487 − 2.39939i 1.12562i 0.500000 − 0.866025i −0.608495 + 0.206261i
79.5 0.248840 + 0.143668i 0.866025 − 0.500000i −0.958719 − 1.66055i 0.717839 − 2.11771i 0.287336 −1.11487 + 2.39939i − 1.12562i 0.500000 − 0.866025i 0.482874 − 0.423842i
79.6 1.34443 + 0.776205i −0.866025 + 0.500000i 0.204988 + 0.355049i 1.88899 + 1.19655i −1.55241 −0.478401 + 2.60214i − 2.46837i 0.500000 − 0.866025i 1.61083 + 3.07491i
79.7 1.54296 + 0.890827i 0.866025 − 0.500000i 0.587145 + 1.01696i −1.33920 + 1.79069i 1.78165 −2.40898 − 1.09398i − 1.47113i 0.500000 − 0.866025i −3.66151 + 1.56996i
79.8 2.17942 + 1.25829i −0.866025 + 0.500000i 2.16659 + 3.75264i −1.11685 − 1.93717i −2.51658 −1.31340 − 2.29673i 5.87162i 0.500000 − 0.866025i 0.00343282 − 5.62724i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.q.a ✓ 16
3.b odd 2 1 315.2.bf.b 16
4.b odd 2 1 1680.2.di.d 16
5.b even 2 1 inner 105.2.q.a ✓ 16
5.c odd 4 1 525.2.i.h 8
5.c odd 4 1 525.2.i.k 8
7.b odd 2 1 735.2.q.g 16
7.c even 3 1 inner 105.2.q.a ✓ 16
7.c even 3 1 735.2.d.d 8
7.d odd 6 1 735.2.d.e 8
7.d odd 6 1 735.2.q.g 16
15.d odd 2 1 315.2.bf.b 16
20.d odd 2 1 1680.2.di.d 16
21.g even 6 1 2205.2.d.o 8
21.h odd 6 1 315.2.bf.b 16
21.h odd 6 1 2205.2.d.s 8
28.g odd 6 1 1680.2.di.d 16
35.c odd 2 1 735.2.q.g 16
35.i odd 6 1 735.2.d.e 8
35.i odd 6 1 735.2.q.g 16
35.j even 6 1 inner 105.2.q.a ✓ 16
35.j even 6 1 735.2.d.d 8
35.k even 12 1 3675.2.a.bn 4
35.k even 12 1 3675.2.a.cb 4
35.l odd 12 1 525.2.i.h 8
35.l odd 12 1 525.2.i.k 8
35.l odd 12 1 3675.2.a.bp 4
35.l odd 12 1 3675.2.a.bz 4
105.o odd 6 1 315.2.bf.b 16
105.o odd 6 1 2205.2.d.s 8
105.p even 6 1 2205.2.d.o 8
140.p odd 6 1 1680.2.di.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a ✓ 16 1.a even 1 1 trivial
105.2.q.a ✓ 16 5.b even 2 1 inner
105.2.q.a ✓ 16 7.c even 3 1 inner
105.2.q.a ✓ 16 35.j even 6 1 inner
315.2.bf.b 16 3.b odd 2 1
315.2.bf.b 16 15.d odd 2 1
315.2.bf.b 16 21.h odd 6 1
315.2.bf.b 16 105.o odd 6 1
525.2.i.h 8 5.c odd 4 1
525.2.i.h 8 35.l odd 12 1
525.2.i.k 8 5.c odd 4 1
525.2.i.k 8 35.l odd 12 1
735.2.d.d 8 7.c even 3 1
735.2.d.d 8 35.j even 6 1
735.2.d.e 8 7.d odd 6 1
735.2.d.e 8 35.i odd 6 1
735.2.q.g 16 7.b odd 2 1
735.2.q.g 16 7.d odd 6 1
735.2.q.g 16 35.c odd 2 1
735.2.q.g 16 35.i odd 6 1
1680.2.di.d 16 4.b odd 2 1
1680.2.di.d 16 20.d odd 2 1
1680.2.di.d 16 28.g odd 6 1
1680.2.di.d 16 140.p odd 6 1
2205.2.d.o 8 21.g even 6 1
2205.2.d.o 8 105.p even 6 1
2205.2.d.s 8 21.h odd 6 1
2205.2.d.s 8 105.o odd 6 1
3675.2.a.bn 4 35.k even 12 1
3675.2.a.bp 4 35.l odd 12 1
3675.2.a.bz 4 35.l odd 12 1
3675.2.a.cb 4 35.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 20 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 18 T^{6} + \cdots + 900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 60 T^{6} + \cdots + 16129)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 322417936 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 5143987297296 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - 38 T^{2} + \cdots - 22)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 3721)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 676751377201 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 50 T^{2} + \cdots - 10)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 128 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 207360000 \) Copy content Toggle raw display
$53$ \( T^{16} - 160 T^{14} + \cdots + 84934656 \) Copy content Toggle raw display
$59$ \( (T^{8} - 2 T^{7} + \cdots + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 8 T^{7} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 670801950625 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 3202)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3722279521041 \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + \cdots + 50140561)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 148 T^{6} + \cdots + 131044)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 8 T^{7} + \cdots + 285156)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 20 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
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