Properties

Label 1881.2.h.a
Level $1881$
Weight $2$
Character orbit 1881.h
Analytic conductor $15.020$
Analytic rank $1$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1881,2,Mod(208,1881)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1881, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1881.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.h (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: \(15.0198606202\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-19})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{4} - q^{5} + (2 \beta - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - q^{5} + (2 \beta - 1) q^{7} + ( - \beta - 2) q^{11} + 4 q^{16} + (2 \beta - 1) q^{17} + (2 \beta - 1) q^{19} + 2 q^{20} + 4 q^{23} - 4 q^{25} + ( - 4 \beta + 2) q^{28} + ( - 2 \beta + 1) q^{35} + ( - 6 \beta + 3) q^{43} + (2 \beta + 4) q^{44} - 13 q^{47} - 12 q^{49} + (\beta + 2) q^{55} + ( - 2 \beta + 1) q^{61} - 8 q^{64} + ( - 4 \beta + 2) q^{68} + ( - 6 \beta + 3) q^{73} + ( - 4 \beta + 2) q^{76} + ( - 5 \beta + 12) q^{77} - 4 q^{80} + (4 \beta - 2) q^{83} + ( - 2 \beta + 1) q^{85} - 8 q^{92} + ( - 2 \beta + 1) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 2 q^{5} - 5 q^{11} + 8 q^{16} + 4 q^{20} + 8 q^{23} - 8 q^{25} + 10 q^{44} - 26 q^{47} - 24 q^{49} + 5 q^{55} - 16 q^{64} + 19 q^{77} - 8 q^{80} - 16 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1881\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(496\) \(1046\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
0.500000 2.17945i
0.500000 + 2.17945i
0 0 −2.00000 −1.00000 0 4.35890i 0 0 0
208.2 0 0 −2.00000 −1.00000 0 4.35890i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
11.b odd 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1881.2.h.a 2
3.b odd 2 1 209.2.d.a 2
11.b odd 2 1 inner 1881.2.h.a 2
19.b odd 2 1 CM 1881.2.h.a 2
33.d even 2 1 209.2.d.a 2
57.d even 2 1 209.2.d.a 2
209.d even 2 1 inner 1881.2.h.a 2
627.b odd 2 1 209.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.d.a 2 3.b odd 2 1
209.2.d.a 2 33.d even 2 1
209.2.d.a 2 57.d even 2 1
209.2.d.a 2 627.b odd 2 1
1881.2.h.a 2 1.a even 1 1 trivial
1881.2.h.a 2 11.b odd 2 1 inner
1881.2.h.a 2 19.b odd 2 1 CM
1881.2.h.a 2 209.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1881, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 19 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 11 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19 \) Copy content Toggle raw display
$19$ \( T^{2} + 19 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 171 \) Copy content Toggle raw display
$47$ \( (T + 13)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 19 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 171 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 76 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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