Properties

Label 2184.2.h.b
Level $2184$
Weight $2$
Character orbit 2184.h
Analytic conductor $17.439$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2184,2,Mod(337,2184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2184.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2184.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: \(17.4393278014\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + 3 i q^{5} + i q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 i q^{5} + i q^{7} + q^{9} - 4 i q^{11} + ( - 3 i + 2) q^{13} + 3 i q^{15} + 6 q^{17} + 7 i q^{19} + i q^{21} + q^{23} - 4 q^{25} + q^{27} + q^{29} + 7 i q^{31} - 4 i q^{33} - 3 q^{35} - 12 i q^{37} + ( - 3 i + 2) q^{39} + 6 i q^{41} + 11 q^{43} + 3 i q^{45} - 3 i q^{47} - q^{49} + 6 q^{51} - 3 q^{53} + 12 q^{55} + 7 i q^{57} + 12 i q^{59} - 4 q^{61} + i q^{63} + (6 i + 9) q^{65} - 6 i q^{67} + q^{69} + 12 i q^{71} - 11 i q^{73} - 4 q^{75} + 4 q^{77} - 15 q^{79} + q^{81} + 15 i q^{83} + 18 i q^{85} + q^{87} + 11 i q^{89} + (2 i + 3) q^{91} + 7 i q^{93} - 21 q^{95} + 7 i q^{97} - 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} + 4 q^{13} + 12 q^{17} + 2 q^{23} - 8 q^{25} + 2 q^{27} + 2 q^{29} - 6 q^{35} + 4 q^{39} + 22 q^{43} - 2 q^{49} + 12 q^{51} - 6 q^{53} + 24 q^{55} - 8 q^{61} + 18 q^{65} + 2 q^{69} - 8 q^{75} + 8 q^{77} - 30 q^{79} + 2 q^{81} + 2 q^{87} + 6 q^{91} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2184\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(1639\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 1.00000 0 3.00000i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 3.00000i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2184.2.h.b 2
4.b odd 2 1 4368.2.h.g 2
13.b even 2 1 inner 2184.2.h.b 2
52.b odd 2 1 4368.2.h.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.b 2 1.a even 1 1 trivial
2184.2.h.b 2 13.b even 2 1 inner
4368.2.h.g 2 4.b odd 2 1
4368.2.h.g 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2184, [\chi])\):

\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 49 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 49 \) Copy content Toggle raw display
$37$ \( T^{2} + 144 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T - 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 36 \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 15)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 225 \) Copy content Toggle raw display
$89$ \( T^{2} + 121 \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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