↑

Properties

Label 1872.3.i.h
Level $1872$
Weight $3$
Character orbit 1872.i
Analytic conductor $51.008$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1872,3,Mod(415,1872)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1872.415"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1872.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,12,0,0,0,-24,0,22,0,0,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(51.0083054882\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 624)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} + 6 q^{7} - 12 q^{11} + ( - 2 \beta + 11) q^{13} - 14 q^{17} - 18 q^{19} + 2 \beta q^{23} + 13 q^{25} - 38 q^{29} - 30 q^{31} - 6 \beta q^{35} + 14 \beta q^{37} - 9 \beta q^{41} + 2 \beta q^{43} + \cdots - 42 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{7} - 24 q^{11} + 22 q^{13} - 28 q^{17} - 36 q^{19} + 26 q^{25} - 76 q^{29} - 60 q^{31} + 48 q^{47} - 26 q^{49} - 20 q^{53} - 144 q^{59} + 124 q^{61} - 48 q^{65} - 156 q^{67} + 216 q^{71} - 144 q^{77}+ \cdots + 132 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 − 3.46410i 0 6.00000 0 0 0
415.2 0 0 0 3.46410i 0 6.00000 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
52.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.3.i.h 2
3.b odd 2 1 624.3.i.b yes 2
4.b odd 2 1 1872.3.i.b 2
12.b even 2 1 624.3.i.a ✓ 2
13.b even 2 1 1872.3.i.b 2
39.d odd 2 1 624.3.i.a ✓ 2
52.b odd 2 1 inner 1872.3.i.h 2
156.h even 2 1 624.3.i.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.3.i.a ✓ 2 12.b even 2 1
624.3.i.a ✓ 2 39.d odd 2 1
624.3.i.b yes 2 3.b odd 2 1
624.3.i.b yes 2 156.h even 2 1
1872.3.i.b 2 4.b odd 2 1
1872.3.i.b 2 13.b even 2 1
1872.3.i.h 2 1.a even 1 1 trivial
1872.3.i.h 2 52.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7} - 6 \) Copy content Toggle raw display
\( T_{17} + 14 \) 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
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( (T - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 22T + 169 \) Copy content Toggle raw display
$17$ \( (T + 14)^{2} \) Copy content Toggle raw display
$19$ \( (T + 18)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 48 \) Copy content Toggle raw display
$29$ \( (T + 38)^{2} \) Copy content Toggle raw display
$31$ \( (T + 30)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2352 \) Copy content Toggle raw display
$41$ \( T^{2} + 972 \) Copy content Toggle raw display
$43$ \( T^{2} + 48 \) Copy content Toggle raw display
$47$ \( (T - 24)^{2} \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( (T + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T - 62)^{2} \) Copy content Toggle raw display
$67$ \( (T + 78)^{2} \) Copy content Toggle raw display
$71$ \( (T - 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( T^{2} + 13872 \) Copy content Toggle raw display
$83$ \( (T + 72)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10092 \) Copy content Toggle raw display
$97$ \( T^{2} + 21168 \) Copy content Toggle raw display
show more
show less