↑

Properties

Label 2023.2.a.k
Level $2023$
Weight $2$
Character orbit 2023.a
Self dual yes
Analytic conductor $16.154$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,2,Mod(1,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, 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("2023.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2023.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: \(16.1537363289\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: 9.9.22384826361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 9x^{7} - x^{6} + 24x^{5} + 3x^{4} - 22x^{3} + 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} - \beta_{4} q^{3} + (\beta_{8} - \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 1) q^{2} - \beta_{4} q^{3} + (\beta_{8} - \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - 2 \beta_{8} - 3 \beta_{7} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 6 q^{4} - 6 q^{6} + 9 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 6 q^{4} - 6 q^{6} + 9 q^{7} - 6 q^{8} - 3 q^{9} - 3 q^{11} - 18 q^{13} - 6 q^{14} + 3 q^{15} + 3 q^{18} - 12 q^{19} + 6 q^{20} + 6 q^{22} - 21 q^{23} + 21 q^{24} - 3 q^{25} + 6 q^{26} + 12 q^{27} + 6 q^{28} - 6 q^{29} - 9 q^{30} + 9 q^{31} - 18 q^{32} - 24 q^{33} - 24 q^{36} - 12 q^{37} + 9 q^{38} - 9 q^{39} - 45 q^{40} - 3 q^{41} - 6 q^{42} - 3 q^{43} + 12 q^{44} + 3 q^{45} + 18 q^{46} - 24 q^{47} - 15 q^{48} + 9 q^{49} - 33 q^{50} - 27 q^{52} - 15 q^{53} - 18 q^{54} - 33 q^{55} - 6 q^{56} - 30 q^{57} + 48 q^{58} + 6 q^{59} - 3 q^{60} + 39 q^{61} + 21 q^{62} - 3 q^{63} + 24 q^{64} + 18 q^{66} - 18 q^{67} - 15 q^{69} - 39 q^{71} + 33 q^{72} + 15 q^{73} - 12 q^{74} - 9 q^{75} + 30 q^{76} - 3 q^{77} + 18 q^{78} + 12 q^{79} + 87 q^{80} - 15 q^{81} - 21 q^{82} - 30 q^{83} - 54 q^{86} - 48 q^{87} - 3 q^{88} - 33 q^{89} + 21 q^{90} - 18 q^{91} - 45 q^{92} - 15 q^{93} - 12 q^{94} - 6 q^{96} + 9 q^{97} - 6 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 9x^{7} - x^{6} + 24x^{5} + 3x^{4} - 22x^{3} + 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - \nu^{6} - 7\nu^{5} + 6\nu^{4} + 11\nu^{3} - 9\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{8} + \nu^{7} + 7\nu^{6} - 6\nu^{5} - 10\nu^{4} + 9\nu^{3} - 3\nu^{2} - 5\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{8} - \nu^{7} - 15\nu^{6} + 4\nu^{5} + 27\nu^{4} - 8\nu^{2} - 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 2\nu^{7} - 7\nu^{6} + 14\nu^{5} + 12\nu^{4} - 25\nu^{3} - 5\nu^{2} + 10\nu - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{8} - \nu^{7} - 16\nu^{6} + 5\nu^{5} + 34\nu^{4} - 5\nu^{3} - 19\nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\nu^{7} + 2\nu^{6} + 15\nu^{5} - 12\nu^{4} - 28\nu^{3} + 16\nu^{2} + 11\nu - 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 3\nu^{7} - 6\nu^{6} + 21\nu^{5} + 6\nu^{4} - 36\nu^{3} + 3\nu^{2} + 13\nu - 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{5} - \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{8} - \beta_{6} + 7\beta_{5} + \beta_{4} + \beta_{3} - 5\beta_{2} + 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{8} + 7\beta_{7} + 6\beta_{6} + 2\beta_{5} + 6\beta_{3} - 6\beta_{2} + 13\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{8} + 2\beta_{7} - 7\beta_{6} + 40\beta_{5} + 8\beta_{4} + 8\beta_{3} - 25\beta_{2} + 16\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -52\beta_{8} + 40\beta_{7} + 30\beta_{6} + 21\beta_{5} + 2\beta_{4} + 33\beta_{3} - 34\beta_{2} + 65\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 188 \beta_{8} + 21 \beta_{7} - 36 \beta_{6} + 216 \beta_{5} + 48 \beta_{4} + 51 \beta_{3} + \cdots + 191 \) 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
−2.14017
−1.31684
0.190305
0.537083
2.35697
−1.36957
−0.752334
0.891991
1.60257
−2.72117 −0.162275 5.40478 3.69794 0.441578 1.00000 −9.26499 −2.97367 −10.0627
1.2 −2.17089 −1.16346 2.71276 −2.32890 2.52574 1.00000 −1.54732 −1.64636 5.05578
1.3 −1.56855 1.82495 0.460353 −1.05042 −2.86253 1.00000 2.41501 0.330436 1.64764
1.4 −1.49267 2.85261 0.228066 −0.0203680 −4.25801 1.00000 2.64491 5.13740 0.0304027
1.5 −1.27531 −0.797374 −0.373586 −3.45084 1.01690 1.00000 3.02706 −2.36419 4.40089
1.6 −0.167857 1.37224 −1.97182 2.67737 −0.230341 1.00000 0.666699 −1.11696 −0.449415
1.7 0.131471 −2.03645 −1.98272 0.469880 −0.267734 1.00000 −0.523612 1.14712 0.0617756
1.8 1.32255 −2.10696 −0.250857 1.12078 −2.78656 1.00000 −2.97687 1.43926 1.48228
1.9 1.94243 0.216712 1.77302 −1.11543 0.420947 1.00000 −0.440885 −2.95304 −2.16664
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(17\) \(-1\)

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 2023.2.a.k ✓ 9
17.b even 2 1 2023.2.a.l yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2023.2.a.k ✓ 9 1.a even 1 1 trivial
2023.2.a.l yes 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2023))\):

\( T_{2}^{9} + 6T_{2}^{8} + 6T_{2}^{7} - 26T_{2}^{6} - 57T_{2}^{5} + 76T_{2}^{3} + 48T_{2}^{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{9} - 12T_{3}^{7} - 4T_{3}^{6} + 42T_{3}^{5} + 21T_{3}^{4} - 44T_{3}^{3} - 27T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 6 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{9} - 12 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} - 21 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 3 T^{8} + \cdots - 163 \) Copy content Toggle raw display
$13$ \( T^{9} + 18 T^{8} + \cdots - 53 \) Copy content Toggle raw display
$17$ \( T^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + 12 T^{8} + \cdots + 5417 \) Copy content Toggle raw display
$23$ \( T^{9} + 21 T^{8} + \cdots - 742211 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots - 182989 \) Copy content Toggle raw display
$31$ \( T^{9} - 9 T^{8} + \cdots - 2291921 \) Copy content Toggle raw display
$37$ \( T^{9} + 12 T^{8} + \cdots - 568297 \) Copy content Toggle raw display
$41$ \( T^{9} + 3 T^{8} + \cdots - 4339 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} + \cdots - 866521 \) Copy content Toggle raw display
$47$ \( T^{9} + 24 T^{8} + \cdots + 329111 \) Copy content Toggle raw display
$53$ \( T^{9} + 15 T^{8} + \cdots - 4331539 \) Copy content Toggle raw display
$59$ \( T^{9} - 6 T^{8} + \cdots + 8927909 \) Copy content Toggle raw display
$61$ \( T^{9} - 39 T^{8} + \cdots - 49286087 \) Copy content Toggle raw display
$67$ \( T^{9} + 18 T^{8} + \cdots - 68707 \) Copy content Toggle raw display
$71$ \( T^{9} + 39 T^{8} + \cdots + 21079873 \) Copy content Toggle raw display
$73$ \( T^{9} - 15 T^{8} + \cdots + 13693409 \) Copy content Toggle raw display
$79$ \( T^{9} - 12 T^{8} + \cdots - 4419343 \) Copy content Toggle raw display
$83$ \( T^{9} + 30 T^{8} + \cdots + 127529117 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 1680246289 \) Copy content Toggle raw display
$97$ \( T^{9} - 9 T^{8} + \cdots - 1097513 \) Copy content Toggle raw display
show more
show less