Properties

Label 3330.2.d.h
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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{2} - \beta_1 - 1) q^{10} + (\beta_{3} - \beta_1 + 1) q^{11} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} + \beta_1 + 1) q^{14} + q^{16} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{17} - 3 q^{19} + (\beta_{3} + \beta_{2} + 1) q^{20} + (\beta_{3} - \beta_{2} + \beta_1) q^{22} + \beta_{2} q^{23} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 - 3) q^{26} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{28} + (\beta_{3} - \beta_1 + 6) q^{29} - 2 q^{31} - \beta_{2} q^{32} + (\beta_{3} - \beta_1 - 1) q^{34} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{35}+ \cdots + (2 \beta_{3} + 2 \beta_1) q^{98}+O(q^{100}) \) 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 \( 4 q - 4 q^{4} - 4 q^{5} - 4 q^{10} + 4 q^{11} + 4 q^{14} + 4 q^{16} - 12 q^{19} + 4 q^{20} - 12 q^{26} + 24 q^{29} - 8 q^{31} - 4 q^{34} + 16 q^{35} + 4 q^{40} - 8 q^{41} - 4 q^{44} + 4 q^{46} - 4 q^{50} - 16 q^{55} - 4 q^{56} + 16 q^{59} - 40 q^{61} - 4 q^{64} + 8 q^{70} - 24 q^{71} - 4 q^{74} + 12 q^{76} - 24 q^{79} - 4 q^{80} - 16 q^{85} + 8 q^{86} - 28 q^{89} - 12 q^{91} + 16 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) 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
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.44949i 1.00000i 0 0.224745 + 2.22474i
1999.2 1.00000i 0 −1.00000 0.224745 2.22474i 0 3.44949i 1.00000i 0 −2.22474 0.224745i
1999.3 1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.44949i 1.00000i 0 0.224745 2.22474i
1999.4 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 3.44949i 1.00000i 0 −2.22474 + 0.224745i
\(n\): 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.h 4
3.b odd 2 1 1110.2.d.h 4
5.b even 2 1 inner 3330.2.d.h 4
15.d odd 2 1 1110.2.d.h 4
15.e even 4 1 5550.2.a.bt 2
15.e even 4 1 5550.2.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.h 4 3.b odd 2 1
1110.2.d.h 4 15.d odd 2 1
3330.2.d.h 4 1.a even 1 1 trivial
3330.2.d.h 4 5.b even 2 1 inner
5550.2.a.bt 2 15.e even 4 1
5550.2.a.cc 2 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 \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{4} + 14T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 5 \) Copy content Toggle raw display
\( T_{17}^{4} + 14T_{17}^{2} + 25 \) Copy content Toggle raw display
\( T_{29}^{2} - 12T_{29} + 30 \) 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
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 30T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 44T^{2} + 100 \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 38)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T + 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 194T^{2} + 9025 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T - 18)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T - 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 140T^{2} + 1444 \) Copy content Toggle raw display
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