Properties

Label 1840.2.m.c
Level $1840$
Weight $2$
Character orbit 1840.m
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2 \beta_1) q^{3} + \beta_{3} q^{5} + 3 \beta_{2} q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2 \beta_1) q^{3} + \beta_{3} q^{5} + 3 \beta_{2} q^{7} + 7 q^{9} + 5 \beta_{2} q^{15} + 6 \beta_{3} q^{21} + (\beta_{2} - 3 \beta_1) q^{23} - 5 q^{25} + ( - 4 \beta_{2} + 8 \beta_1) q^{27} - 6 q^{29} + (3 \beta_{2} - 6 \beta_1) q^{35} + 12 q^{41} - 9 \beta_{2} q^{43} + 7 \beta_{3} q^{45} + ( - 3 \beta_{2} + 6 \beta_1) q^{47} - 11 q^{49} - 6 \beta_{3} q^{61} + 21 \beta_{2} q^{63} + 3 \beta_{2} q^{67} + ( - \beta_{3} - 15) q^{69} + (5 \beta_{2} - 10 \beta_1) q^{75} + 19 q^{81} + 11 \beta_{2} q^{83} + (6 \beta_{2} - 12 \beta_1) q^{87} - 8 \beta_{3} q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{9} - 20 q^{25} - 24 q^{29} + 48 q^{41} - 44 q^{49} - 60 q^{69} + 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1839.1
−1.58114 + 0.707107i
−1.58114 0.707107i
1.58114 0.707107i
1.58114 + 0.707107i
0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.2 0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.3 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.4 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.m.c 4
4.b odd 2 1 inner 1840.2.m.c 4
5.b even 2 1 inner 1840.2.m.c 4
20.d odd 2 1 CM 1840.2.m.c 4
23.b odd 2 1 inner 1840.2.m.c 4
92.b even 2 1 inner 1840.2.m.c 4
115.c odd 2 1 inner 1840.2.m.c 4
460.g even 2 1 inner 1840.2.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.c 4 1.a even 1 1 trivial
1840.2.m.c 4 4.b odd 2 1 inner
1840.2.m.c 4 5.b even 2 1 inner
1840.2.m.c 4 20.d odd 2 1 CM
1840.2.m.c 4 23.b odd 2 1 inner
1840.2.m.c 4 92.b even 2 1 inner
1840.2.m.c 4 115.c odd 2 1 inner
1840.2.m.c 4 460.g 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{2} - 10 \) Copy content Toggle raw display
\( T_{7}^{2} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 44T^{2} + 529 \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 242)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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