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Properties

Label 3960.2.d.e
Level $3960$
Weight $2$
Character orbit 3960.d
Analytic conductor $31.621$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1320)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{5} + \beta_{2} q^{7} + q^{11} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} + 6 q^{11} + 4 q^{19} - 2 q^{25} - 24 q^{29} - 8 q^{31} - 12 q^{35} - 8 q^{41} + 18 q^{49} + 2 q^{55} - 24 q^{59} - 12 q^{61} + 20 q^{65} + 48 q^{71} + 12 q^{79} - 44 q^{85} - 36 q^{89} + 8 q^{91} + 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 2\nu^{2} - 4\nu - 76 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 9\nu^{4} + 16\nu^{3} + 4\nu^{2} + 38\nu - 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} - 12\nu^{4} + 6\nu^{3} + 36\nu^{2} + 112\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} - 2\nu^{3} - 2\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} - 2\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - \beta_{3} - 5\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{5} + 3\beta_{4} + \beta_{3} - 8\beta_{2} - 4\beta _1 - 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.403032 + 0.403032i
0.403032 − 0.403032i
1.45161 − 1.45161i
1.45161 + 1.45161i
−0.854638 − 0.854638i
−0.854638 + 0.854638i
0 0 0 −1.48119 − 1.67513i 0 0.806063i 0 0 0
3169.2 0 0 0 −1.48119 + 1.67513i 0 − 0.806063i 0 0 0
3169.3 0 0 0 0.311108 − 2.21432i 0 − 2.90321i 0 0 0
3169.4 0 0 0 0.311108 + 2.21432i 0 2.90321i 0 0 0
3169.5 0 0 0 2.17009 − 0.539189i 0 − 1.70928i 0 0 0
3169.6 0 0 0 2.17009 + 0.539189i 0 1.70928i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3169.6
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.e 6
3.b odd 2 1 1320.2.d.a ✓ 6
5.b even 2 1 inner 3960.2.d.e 6
12.b even 2 1 2640.2.d.g 6
15.d odd 2 1 1320.2.d.a ✓ 6
15.e even 4 1 6600.2.a.bp 3
15.e even 4 1 6600.2.a.bt 3
60.h even 2 1 2640.2.d.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.d.a ✓ 6 3.b odd 2 1
1320.2.d.a ✓ 6 15.d odd 2 1
2640.2.d.g 6 12.b even 2 1
2640.2.d.g 6 60.h even 2 1
3960.2.d.e 6 1.a even 1 1 trivial
3960.2.d.e 6 5.b even 2 1 inner
6600.2.a.bp 3 15.e even 4 1
6600.2.a.bt 3 15.e even 4 1

Hecke kernels

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

\( T_{7}^{6} + 12T_{7}^{4} + 32T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{29}^{3} + 12T_{29}^{2} + 32T_{29} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 24 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{6} + 68 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 20 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} + 12 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 80 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 48 T + 80)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{3} + 12 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 192 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( (T^{3} - 24 T^{2} + \cdots - 400)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 104 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( (T^{3} - 6 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 328 T^{4} + \cdots + 300304 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + \cdots - 488)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 240 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
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