Properties

Label 1840.2.m.d
Level $1840$
Weight $2$
Character orbit 1840.m
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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: \(\Q(\sqrt{6}, \sqrt{-14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{2} q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{2} q^{5} - 2 q^{9} + ( - 2 \beta_{2} + 2 \beta_1) q^{11} + \beta_{3} q^{13} + \beta_{2} q^{15} + ( - \beta_{2} + \beta_1) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + (\beta_{2} + \beta_1 - 3) q^{23} + ( - \beta_{3} - 2) q^{25} - 5 q^{27} - 3 q^{29} - \beta_{3} q^{31} + ( - 2 \beta_{2} + 2 \beta_1) q^{33} + (2 \beta_{2} - 2 \beta_1) q^{37} + \beta_{3} q^{39} - 3 q^{41} + (3 \beta_{2} + 3 \beta_1) q^{43} - 2 \beta_{2} q^{45} - 9 q^{47} + 7 q^{49} + ( - \beta_{2} + \beta_1) q^{51} + ( - 2 \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{3} - 6) q^{55} + ( - \beta_{2} + \beta_1) q^{57} + 2 \beta_{3} q^{59} + (3 \beta_{2} + 3 \beta_1) q^{61} + (2 \beta_{2} - 5 \beta_1) q^{65} + (\beta_{2} + \beta_1 - 3) q^{69} + 3 \beta_{3} q^{71} + \beta_{3} q^{73} + ( - \beta_{3} - 2) q^{75} + ( - 3 \beta_{2} + 3 \beta_1) q^{79} + q^{81} + ( - \beta_{2} - \beta_1) q^{83} + (\beta_{3} - 3) q^{85} - 3 q^{87} + ( - \beta_{2} - \beta_1) q^{89} - \beta_{3} q^{93} + (\beta_{3} - 3) q^{95} + (4 \beta_{2} - 4 \beta_1) q^{97} + (4 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{9} - 12 q^{23} - 8 q^{25} - 20 q^{27} - 12 q^{29} - 12 q^{41} - 36 q^{47} + 28 q^{49} - 24 q^{55} - 12 q^{69} - 8 q^{75} + 4 q^{81} - 12 q^{85} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 5 \) 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
\(\nu^{3}\)\(=\) \( 5\beta_{2} - 4\beta_1 \) 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
1.22474 1.87083i
1.22474 + 1.87083i
−1.22474 1.87083i
−1.22474 + 1.87083i
0 1.00000 0 −1.22474 1.87083i 0 0 0 −2.00000 0
1839.2 0 1.00000 0 −1.22474 + 1.87083i 0 0 0 −2.00000 0
1839.3 0 1.00000 0 1.22474 1.87083i 0 0 0 −2.00000 0
1839.4 0 1.00000 0 1.22474 + 1.87083i 0 0 0 −2.00000 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 inner
23.b 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.d yes 4
4.b odd 2 1 1840.2.m.a 4
5.b even 2 1 1840.2.m.a 4
20.d odd 2 1 inner 1840.2.m.d yes 4
23.b odd 2 1 inner 1840.2.m.d yes 4
92.b even 2 1 1840.2.m.a 4
115.c odd 2 1 1840.2.m.a 4
460.g even 2 1 inner 1840.2.m.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.a 4 4.b odd 2 1
1840.2.m.a 4 5.b even 2 1
1840.2.m.a 4 92.b even 2 1
1840.2.m.a 4 115.c odd 2 1
1840.2.m.d yes 4 1.a even 1 1 trivial
1840.2.m.d yes 4 20.d odd 2 1 inner
1840.2.m.d yes 4 23.b odd 2 1 inner
1840.2.m.d yes 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 21)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 23)^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 21)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 126)^{2} \) Copy content Toggle raw display
$47$ \( (T + 9)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 126)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 189)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 21)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 14)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
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