Properties

Label 2128.2.a.k
Level $2128$
Weight $2$
Character orbit 2128.a
Self dual yes
Analytic conductor $16.992$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2128,2,Mod(1,2128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2128.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2128.a (trivial)

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: yes
Analytic conductor: \(16.9921655501\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 3 q^{5} - q^{7} + (\beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 3 q^{5} - q^{7} + (\beta + 2) q^{9} + (\beta + 1) q^{11} - q^{13} + 3 \beta q^{15} + (\beta + 1) q^{17} - q^{19} - \beta q^{21} + ( - 2 \beta + 1) q^{23} + 4 q^{25} + 5 q^{27} + (\beta + 1) q^{29} + ( - 3 \beta + 1) q^{31} + (2 \beta + 5) q^{33} - 3 q^{35} + 5 q^{37} - \beta q^{39} + ( - \beta - 1) q^{41} - 2 q^{43} + (3 \beta + 6) q^{45} + ( - 2 \beta - 5) q^{47} + q^{49} + (2 \beta + 5) q^{51} - 3 \beta q^{53} + (3 \beta + 3) q^{55} - \beta q^{57} + ( - 4 \beta - 1) q^{59} - q^{61} + ( - \beta - 2) q^{63} - 3 q^{65} + (3 \beta + 1) q^{67} + ( - \beta - 10) q^{69} + (4 \beta + 1) q^{71} + (3 \beta + 8) q^{73} + 4 \beta q^{75} + ( - \beta - 1) q^{77} + 10 q^{79} + (2 \beta - 6) q^{81} + ( - 3 \beta - 6) q^{83} + (3 \beta + 3) q^{85} + (2 \beta + 5) q^{87} + (2 \beta + 2) q^{89} + q^{91} + ( - 2 \beta - 15) q^{93} - 3 q^{95} - 7 q^{97} + (4 \beta + 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} - 2 q^{13} + 3 q^{15} + 3 q^{17} - 2 q^{19} - q^{21} + 8 q^{25} + 10 q^{27} + 3 q^{29} - q^{31} + 12 q^{33} - 6 q^{35} + 10 q^{37} - q^{39} - 3 q^{41} - 4 q^{43} + 15 q^{45} - 12 q^{47} + 2 q^{49} + 12 q^{51} - 3 q^{53} + 9 q^{55} - q^{57} - 6 q^{59} - 2 q^{61} - 5 q^{63} - 6 q^{65} + 5 q^{67} - 21 q^{69} + 6 q^{71} + 19 q^{73} + 4 q^{75} - 3 q^{77} + 20 q^{79} - 10 q^{81} - 15 q^{83} + 9 q^{85} + 12 q^{87} + 6 q^{89} + 2 q^{91} - 32 q^{93} - 6 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.79129
2.79129
0 −1.79129 0 3.00000 0 −1.00000 0 0.208712 0
1.2 0 2.79129 0 3.00000 0 −1.00000 0 4.79129 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2128.2.a.k 2
4.b odd 2 1 532.2.a.c 2
8.b even 2 1 8512.2.a.m 2
8.d odd 2 1 8512.2.a.t 2
12.b even 2 1 4788.2.a.g 2
28.d even 2 1 3724.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.c 2 4.b odd 2 1
2128.2.a.k 2 1.a even 1 1 trivial
3724.2.a.e 2 28.d even 2 1
4788.2.a.g 2 12.b even 2 1
8512.2.a.m 2 8.b even 2 1
8512.2.a.t 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2128))\):

\( T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 21 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 47 \) Copy content Toggle raw display
$37$ \( (T - 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 45 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 75 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 5T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 75 \) Copy content Toggle raw display
$73$ \( T^{2} - 19T + 43 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 15T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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