↑

Properties

Label 1143.2.a.k
Level $1143$
Weight $2$
Character orbit 1143.a
Self dual yes
Analytic conductor $9.127$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1143,2,Mod(1,1143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, 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("1143.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.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: \(9.12690095103\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{14} q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{14} q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{5} + 1) q^{10} + ( - \beta_{15} + \beta_{13}) q^{11} + ( - \beta_{10} + 1) q^{13} + (\beta_{15} - \beta_{14} + \cdots + \beta_1) q^{14}+ \cdots + (3 \beta_{15} - 2 \beta_{14} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{14} - 127\nu^{12} + 1288\nu^{10} - 6632\nu^{8} + 18204\nu^{6} - 25503\nu^{4} + 15677\nu^{2} - 2676 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 25\nu^{12} + 248\nu^{10} - 1240\nu^{8} + 3276\nu^{6} - 4363\nu^{4} + 2499\nu^{2} - 388 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{14} - 185\nu^{12} + 1928\nu^{10} - 10024\nu^{8} + 27060\nu^{6} - 35781\nu^{4} + 19675\nu^{2} - 2892 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{15} - 127\nu^{13} + 1276\nu^{11} - 6416\nu^{9} + 16812\nu^{7} - 21711\nu^{5} + 11813\nu^{3} - 1728\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{15} - 127\nu^{13} + 1276\nu^{11} - 6416\nu^{9} + 16812\nu^{7} - 21735\nu^{5} + 12005\nu^{3} - 1992\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{14} - 25\nu^{12} + 248\nu^{10} - 1236\nu^{8} + 3224\nu^{6} - 4163\nu^{4} + 2287\nu^{2} - 360 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{14} + 26\nu^{12} - 268\nu^{10} + 1388\nu^{8} - 3772\nu^{6} + 5111\nu^{4} - 2974\nu^{2} + 488 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\nu^{14} - 335\nu^{12} + 3416\nu^{10} - 17464\nu^{8} + 46764\nu^{6} - 62487\nu^{4} + 36205\nu^{2} - 6036 ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\nu^{15} - 641\nu^{13} + 6500\nu^{11} - 32968\nu^{9} + 87012\nu^{7} - 112827\nu^{5} + 61555\nu^{3} - 9504\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 59 \nu^{15} - 1525 \nu^{13} + 15592 \nu^{11} - 79832 \nu^{9} + 213444 \nu^{7} - 282849 \nu^{5} + \cdots - 26028 \nu ) / 144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\nu^{15} - 283\nu^{13} + 2884\nu^{11} - 14744\nu^{9} + 39444\nu^{7} - 52377\nu^{5} + 29657\nu^{3} - 4680\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47 \nu^{15} - 1213 \nu^{13} + 12400 \nu^{11} - 63584 \nu^{9} + 170604 \nu^{7} - 227373 \nu^{5} + \cdots - 21060 \nu ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{6} + \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{8} + \beta_{7} + 8\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 11\beta_{10} + 10\beta_{6} - \beta_{5} + 11\beta_{4} + 45\beta_{2} + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{15} - 3\beta_{14} - \beta_{12} - 11\beta_{8} + 13\beta_{7} + 57\beta_{3} + 176\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{11} + 93\beta_{10} + \beta_{9} + 80\beta_{6} - 15\beta_{5} + 93\beta_{4} + 288\beta_{2} + 520 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 29\beta_{15} - 43\beta_{14} - 2\beta_{13} - 11\beta_{12} - 93\beta_{8} + 119\beta_{7} + 393\beta_{3} + 1098\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 117\beta_{11} + 712\beta_{10} + 17\beta_{9} + 594\beta_{6} - 154\beta_{5} + 717\beta_{4} + 1855\beta_{2} + 3236 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 294 \beta_{15} - 426 \beta_{14} - 40 \beta_{13} - 86 \beta_{12} - 714 \beta_{8} + 952 \beta_{7} + \cdots + 6985 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 912 \beta_{11} + 5188 \beta_{10} + 192 \beta_{9} + 4252 \beta_{6} - 1348 \beta_{5} + 5284 \beta_{4} + \cdots + 20521 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2572 \beta_{15} - 3632 \beta_{14} - 504 \beta_{13} - 596 \beta_{12} - 5216 \beta_{8} + \cdots + 45071 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6628 \beta_{11} + 36771 \beta_{10} + 1824 \beta_{9} + 29791 \beta_{6} - 10824 \beta_{5} + 37931 \beta_{4} + \cdots + 131897 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20788 \beta_{15} - 28628 \beta_{14} - 5160 \beta_{13} - 3944 \beta_{12} - 36967 \beta_{8} + \cdots + 293859 \beta_1 \) Copy content Toggle raw display

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
−2.59858
−2.57927
−2.28010
−2.01982
−1.36017
−0.979723
−0.518678
−0.487100
0.487100
0.518678
0.979723
1.36017
2.01982
2.28010
2.57927
2.59858
−2.59858 0 4.75262 −0.560562 0 4.70134 −7.15292 0 1.45667
1.2 −2.57927 0 4.65263 −1.53253 0 −2.06948 −6.84184 0 3.95280
1.3 −2.28010 0 3.19885 3.42195 0 0.843402 −2.73351 0 −7.80238
1.4 −2.01982 0 2.07968 −3.33517 0 −0.226097 −0.160938 0 6.73645
1.5 −1.36017 0 −0.149936 −3.81085 0 4.20131 2.92428 0 5.18341
1.6 −0.979723 0 −1.04014 1.37010 0 3.49660 2.97850 0 −1.34231
1.7 −0.518678 0 −1.73097 4.37543 0 −5.00960 1.93517 0 −2.26944
1.8 −0.487100 0 −1.76273 −2.22708 0 −0.937481 1.83283 0 1.08481
1.9 0.487100 0 −1.76273 2.22708 0 −0.937481 −1.83283 0 1.08481
1.10 0.518678 0 −1.73097 −4.37543 0 −5.00960 −1.93517 0 −2.26944
1.11 0.979723 0 −1.04014 −1.37010 0 3.49660 −2.97850 0 −1.34231
1.12 1.36017 0 −0.149936 3.81085 0 4.20131 −2.92428 0 5.18341
1.13 2.01982 0 2.07968 3.33517 0 −0.226097 0.160938 0 6.73645
1.14 2.28010 0 3.19885 −3.42195 0 0.843402 2.73351 0 −7.80238
1.15 2.57927 0 4.65263 1.53253 0 −2.06948 6.84184 0 3.95280
1.16 2.59858 0 4.75262 0.560562 0 4.70134 7.15292 0 1.45667
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(127\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.2.a.k ✓ 16
3.b odd 2 1 inner 1143.2.a.k ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1143.2.a.k ✓ 16 1.a even 1 1 trivial
1143.2.a.k ✓ 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1143))\):

\( T_{2}^{16} - 26T_{2}^{14} + 269T_{2}^{12} - 1408T_{2}^{10} + 3924T_{2}^{8} - 5655T_{2}^{6} + 3886T_{2}^{4} - 1107T_{2}^{2} + 108 \) Copy content Toggle raw display
\( T_{5}^{16} - 66 T_{5}^{14} + 1742 T_{5}^{12} - 23541 T_{5}^{10} + 173128 T_{5}^{8} - 682800 T_{5}^{6} + \cdots + 248832 \) Copy content Toggle raw display
\( T_{7}^{8} - 5T_{7}^{7} - 29T_{7}^{6} + 158T_{7}^{5} + 112T_{7}^{4} - 824T_{7}^{3} - 288T_{7}^{2} + 544T_{7} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 26 T^{14} + \cdots + 108 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 66 T^{14} + \cdots + 248832 \) Copy content Toggle raw display
$7$ \( (T^{8} - 5 T^{7} + \cdots + 128)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 101059248 \) Copy content Toggle raw display
$13$ \( (T^{8} - 10 T^{7} + \cdots - 6902)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 212487168 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots - 2624)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 28390195200 \) Copy content Toggle raw display
$29$ \( T^{16} - 214 T^{14} + \cdots + 1108992 \) Copy content Toggle raw display
$31$ \( (T^{8} - 9 T^{7} + \cdots - 74656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 4799818)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 256633651200 \) Copy content Toggle raw display
$43$ \( (T^{8} - 13 T^{7} + \cdots + 538624)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 4140380534832 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 40368000000 \) Copy content Toggle raw display
$59$ \( T^{16} - 421 T^{14} + \cdots + 50331648 \) Copy content Toggle raw display
$61$ \( (T^{8} - 18 T^{7} + \cdots + 532930)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 13 T^{7} + \cdots + 304768)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 220283977728 \) Copy content Toggle raw display
$73$ \( (T^{8} - 30 T^{7} + \cdots + 55534)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} + \cdots - 798208)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6491311177728 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 2174768483328 \) Copy content Toggle raw display
$97$ \( (T^{8} - 54 T^{7} + \cdots - 12139520)^{2} \) Copy content Toggle raw display
show more
show less