Properties

Label 3330.2.d.l
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
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] = [3330,2,Mod(1999,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3330.1999");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + (2 \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + (2 \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} + \beta_1 q^{8} + ( - \beta_1 + 2) q^{10} + \beta_{3} q^{11} + 4 \beta_1 q^{13} + ( - 2 \beta_{3} + 1) q^{14} + q^{16} + ( - 2 \beta_{2} + 3 \beta_1) q^{17} - 4 \beta_{3} q^{19} + ( - 2 \beta_1 - 1) q^{20} - \beta_{2} q^{22} + (2 \beta_{2} + 2 \beta_1) q^{23} + (4 \beta_1 - 3) q^{25} + 4 q^{26} + (2 \beta_{2} - \beta_1) q^{28} - 3 q^{29} + ( - \beta_{3} - 4) q^{31} - \beta_1 q^{32} + ( - 2 \beta_{3} + 3) q^{34} + (4 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{35} - \beta_1 q^{37} + 4 \beta_{2} q^{38} + (\beta_1 - 2) q^{40} + ( - 3 \beta_{3} - 4) q^{41} + (3 \beta_{2} - 2 \beta_1) q^{43} - \beta_{3} q^{44} + (2 \beta_{3} + 2) q^{46} + ( - 2 \beta_{2} - 8 \beta_1) q^{47} + (4 \beta_{3} - 6) q^{49} + (3 \beta_1 + 4) q^{50} - 4 \beta_1 q^{52} + (\beta_{2} - 2 \beta_1) q^{53} + (\beta_{3} + 2 \beta_{2}) q^{55} + (2 \beta_{3} - 1) q^{56} + 3 \beta_1 q^{58} + ( - 2 \beta_{3} - 6) q^{59} - 3 \beta_{3} q^{61} + (\beta_{2} + 4 \beta_1) q^{62} - q^{64} + (4 \beta_1 - 8) q^{65} + ( - 2 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{2} - 3 \beta_1) q^{68} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 1) q^{70}+ \cdots + ( - 4 \beta_{2} + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} + 8 q^{10} + 4 q^{14} + 4 q^{16} - 4 q^{20} - 12 q^{25} + 16 q^{26} - 12 q^{29} - 16 q^{31} + 12 q^{34} - 8 q^{35} - 8 q^{40} - 16 q^{41} + 8 q^{46} - 24 q^{49} + 16 q^{50} - 4 q^{56} - 24 q^{59} - 4 q^{64} - 32 q^{65} + 4 q^{70} + 24 q^{71} - 4 q^{74} + 32 q^{79} + 4 q^{80} - 24 q^{85} - 8 q^{86} - 24 q^{89} - 16 q^{91} - 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.00000i 0 −1.00000 1.00000 + 2.00000i 0 2.46410i 1.00000i 0 2.00000 1.00000i
1999.2 1.00000i 0 −1.00000 1.00000 + 2.00000i 0 4.46410i 1.00000i 0 2.00000 1.00000i
1999.3 1.00000i 0 −1.00000 1.00000 2.00000i 0 4.46410i 1.00000i 0 2.00000 + 1.00000i
1999.4 1.00000i 0 −1.00000 1.00000 2.00000i 0 2.46410i 1.00000i 0 2.00000 + 1.00000i
\(n\): 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 3330.2.d.l yes 4
3.b odd 2 1 3330.2.d.i 4
5.b even 2 1 inner 3330.2.d.l yes 4
15.d odd 2 1 3330.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.d.i 4 3.b odd 2 1
3330.2.d.i 4 15.d odd 2 1
3330.2.d.l yes 4 1.a even 1 1 trivial
3330.2.d.l yes 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{4} + 26T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{17}^{4} + 42T_{17}^{2} + 9 \) Copy content Toggle raw display
\( T_{29} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 3)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 13)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8 T - 11)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 62T^{2} + 529 \) Copy content Toggle raw display
$47$ \( T^{4} + 152T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 494 T^{2} + 57121 \) Copy content Toggle raw display
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