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Properties

Label 1863.2.c.a
Level $1863$
Weight $2$
Character orbit 1863.c
Analytic conductor $14.876$
Analytic rank $0$
Dimension $12$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1863,2,Mod(1862,1863)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1863, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1863.1862");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1863 = 3^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1863.c (of order \(2\), degree \(1\), 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: \(14.8761298966\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.57352136505929721.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 207)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} - 2) q^{4} + ( - 2 \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{5} - 2) q^{4} + ( - 2 \beta_{4} + \beta_1) q^{8} - \beta_{6} q^{13} + (\beta_{9} - 2 \beta_{5} + 4) q^{16} + (\beta_{11} + \beta_1) q^{23} - 5 q^{25} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{26}+ \cdots + 7 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} + 84 q^{49} + 6 q^{52} - 30 q^{58} - 54 q^{64} - 66 q^{82} + 78 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{9} + 3\nu^{6} - 9\nu^{3} + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{9} + 9\nu^{6} + 21\nu^{3} + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{11} + 6\nu^{10} + 3\nu^{8} - 2\nu^{7} - \nu^{5} + 6\nu^{4} - 60\nu^{2} - 176\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{11} + 12\nu^{10} - 3\nu^{8} - 4\nu^{7} + \nu^{5} + 12\nu^{4} - 192\nu^{2} - 256\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{11} - 4\nu^{10} - \nu^{8} + 3\nu^{5} - 72\nu^{2} + 84\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15\nu^{11} + 18\nu^{10} + 5\nu^{8} - 6\nu^{7} + 9\nu^{5} + 34\nu^{4} + 320\nu^{2} - 384\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{11} + 18\nu^{10} - 3\nu^{8} + 10\nu^{7} + \nu^{5} + 18\nu^{4} - 456\nu^{2} - 400\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\nu^{11} - 36\nu^{10} + \nu^{8} + 12\nu^{7} + 21\nu^{5} - 4\nu^{4} - 440\nu^{2} + 864\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{11} - 4\nu^{10} - 4\nu^{4} - 71\nu^{2} + 96\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -17\nu^{11} - 30\nu^{10} + 3\nu^{8} + 10\nu^{7} - 33\nu^{5} + 18\nu^{4} + 360\nu^{2} + 672\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\nu^{9} - 3\nu^{6} + 9\nu^{3} - 384 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{9} + \beta_{8} + 2\beta_{6} - 4\beta_{5} + 3\beta_{4} - 3\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} + 3\beta_{8} - 3\beta_{7} - 3\beta_{6} - 3\beta_{5} + 5\beta_{4} + \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 3\beta _1 + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{10} - 3\beta_{9} + 3\beta_{8} + 6\beta_{6} + 6\beta_{5} + 13\beta_{4} + 2\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{10} - 5\beta_{9} + 5\beta_{8} + 7\beta_{6} + 10\beta_{5} - 15\beta_{4} - 3\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{11} + 3\beta_{2} + 8\beta _1 + 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{9} + 16\beta_{8} + 18\beta_{7} - 16\beta_{6} - 16\beta_{5} - 24\beta_{4} - 3\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3\beta_{10} + 24\beta_{9} + 12\beta_{8} + 3\beta_{7} + 24\beta_{6} - 39\beta_{5} - 34\beta_{4} + 34\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{11} + \beta _1 + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -6\beta_{10} + 42\beta_{9} + 21\beta_{8} + 6\beta_{7} + 42\beta_{6} - 102\beta_{5} + 67\beta_{4} - 67\beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -59\beta_{9} + 71\beta_{8} - 63\beta_{7} - 71\beta_{6} - 71\beta_{5} + 129\beta_{4} + 33\beta_{3} ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1863\mathbb{Z}\right)^\times\).

\(n\) \(649\) \(1703\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1862.1
1.09400 − 0.896196i
1.41412 − 0.0166831i
−0.692610 − 1.23300i
−1.32313 + 0.499333i
0.229129 − 1.39553i
−0.721506 + 1.21632i
−0.721506 − 1.21632i
0.229129 + 1.39553i
−1.32313 − 0.499333i
−0.692610 + 1.23300i
1.41412 + 0.0166831i
1.09400 + 0.896196i
− 2.79106i 0 −5.79000 0 0 0 10.5781i 0 0
1862.2 − 2.46600i 0 −4.08117 0 0 0 5.13217i 0 0
1862.3 − 2.43264i 0 −3.91772 0 0 0 4.66511i 0 0
1862.4 − 1.79239i 0 −1.21267 0 0 0 − 1.41121i 0 0
1862.5 − 0.998666i 0 1.00267 0 0 0 − 2.99866i 0 0
1862.6 − 0.0333663i 0 1.99889 0 0 0 − 0.133428i 0 0
1862.7 0.0333663i 0 1.99889 0 0 0 0.133428i 0 0
1862.8 0.998666i 0 1.00267 0 0 0 2.99866i 0 0
1862.9 1.79239i 0 −1.21267 0 0 0 1.41121i 0 0
1862.10 2.43264i 0 −3.91772 0 0 0 − 4.66511i 0 0
1862.11 2.46600i 0 −4.08117 0 0 0 − 5.13217i 0 0
1862.12 2.79106i 0 −5.79000 0 0 0 − 10.5781i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1862.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
3.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1863.2.c.a 12
3.b odd 2 1 inner 1863.2.c.a 12
9.c even 3 1 207.2.g.a ✓ 12
9.c even 3 1 621.2.g.a 12
9.d odd 6 1 207.2.g.a ✓ 12
9.d odd 6 1 621.2.g.a 12
23.b odd 2 1 CM 1863.2.c.a 12
69.c even 2 1 inner 1863.2.c.a 12
207.f odd 6 1 207.2.g.a ✓ 12
207.f odd 6 1 621.2.g.a 12
207.g even 6 1 207.2.g.a ✓ 12
207.g even 6 1 621.2.g.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.g.a ✓ 12 9.c even 3 1
207.2.g.a ✓ 12 9.d odd 6 1
207.2.g.a ✓ 12 207.f odd 6 1
207.2.g.a ✓ 12 207.g even 6 1
621.2.g.a 12 9.c even 3 1
621.2.g.a 12 9.d odd 6 1
621.2.g.a 12 207.f odd 6 1
621.2.g.a 12 207.g even 6 1
1863.2.c.a 12 1.a even 1 1 trivial
1863.2.c.a 12 3.b odd 2 1 inner
1863.2.c.a 12 23.b odd 2 1 CM
1863.2.c.a 12 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 24T_{2}^{10} + 216T_{2}^{8} + 889T_{2}^{6} + 1596T_{2}^{4} + 900T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1863, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 24 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 78 T^{4} + \cdots - 1115)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3039868225 \) Copy content Toggle raw display
$31$ \( (T^{6} - 186 T^{4} + \cdots + 28963)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12668628025 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 10306107361 \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{4} + 262 T^{2} + 7225)^{3} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1050758254225 \) Copy content Toggle raw display
$73$ \( (T^{6} - 438 T^{4} + \cdots + 336025)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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