Properties

Label 3960.2.d.c
Level $3960$
Weight $2$
Character orbit 3960.d
Analytic conductor $31.621$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3960,2,Mod(3169,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3960.3169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3960.d (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: \(31.6207592004\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 440)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{5} + \beta q^{7} + q^{11} + 2 \beta q^{17} - 4 q^{19} - 3 \beta q^{23} + ( - 2 \beta - 3) q^{25} + 2 q^{29} + 8 q^{31} + (\beta + 4) q^{35} - 2 \beta q^{37} + 6 q^{41} + 3 \beta q^{43} + \beta q^{47} + 3 q^{49} - 6 \beta q^{53} + ( - \beta + 1) q^{55} + 4 q^{59} + 14 q^{61} + 5 \beta q^{67} - 8 q^{71} - 2 \beta q^{73} + \beta q^{77} + 8 q^{79} + \beta q^{83} + (2 \beta + 8) q^{85} - 10 q^{89} + (4 \beta - 4) q^{95} - 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{11} - 8 q^{19} - 6 q^{25} + 4 q^{29} + 16 q^{31} + 8 q^{35} + 12 q^{41} + 6 q^{49} + 2 q^{55} + 8 q^{59} + 28 q^{61} - 16 q^{71} + 16 q^{79} + 16 q^{85} - 20 q^{89} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times\).

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3169.1
1.00000i
1.00000i
0 0 0 1.00000 2.00000i 0 2.00000i 0 0 0
3169.2 0 0 0 1.00000 + 2.00000i 0 2.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.2.d.c 2
3.b odd 2 1 440.2.b.a 2
5.b even 2 1 inner 3960.2.d.c 2
12.b even 2 1 880.2.b.b 2
15.d odd 2 1 440.2.b.a 2
15.e even 4 1 2200.2.a.c 1
15.e even 4 1 2200.2.a.i 1
60.h even 2 1 880.2.b.b 2
60.l odd 4 1 4400.2.a.h 1
60.l odd 4 1 4400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.a 2 3.b odd 2 1
440.2.b.a 2 15.d odd 2 1
880.2.b.b 2 12.b even 2 1
880.2.b.b 2 60.h even 2 1
2200.2.a.c 1 15.e even 4 1
2200.2.a.i 1 15.e even 4 1
3960.2.d.c 2 1.a even 1 1 trivial
3960.2.d.c 2 5.b even 2 1 inner
4400.2.a.h 1 60.l odd 4 1
4400.2.a.x 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3960, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{29} - 2 \) 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
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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