Properties

Label 3330.2.h.l
Level $3330$
Weight $2$
Character orbit 3330.h
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3330,2,Mod(2071,3330)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3330.2071");
 
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.h (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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} + \beta_1 q^{5} + q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} + \beta_1 q^{5} + q^{7} - \beta_1 q^{8} - q^{10} - \beta_{3} q^{11} - 2 \beta_{2} q^{13} + \beta_1 q^{14} + q^{16} - 3 \beta_1 q^{17} - 2 \beta_{2} q^{19} - \beta_1 q^{20} - \beta_{2} q^{22} + 6 \beta_1 q^{23} - q^{25} + 2 \beta_{3} q^{26} - q^{28} + 3 \beta_1 q^{29} + 3 \beta_{2} q^{31} + \beta_1 q^{32} + 3 q^{34} + \beta_1 q^{35} + (2 \beta_{2} - 5) q^{37} + 2 \beta_{3} q^{38} + q^{40} - 5 \beta_{3} q^{41} + 3 \beta_{2} q^{43} + \beta_{3} q^{44} - 6 q^{46} - 2 \beta_{3} q^{47} - 6 q^{49} - \beta_1 q^{50} + 2 \beta_{2} q^{52} - 5 \beta_{3} q^{53} - \beta_{2} q^{55} - \beta_1 q^{56} - 3 q^{58} + 12 \beta_1 q^{59} - 7 \beta_{2} q^{61} - 3 \beta_{3} q^{62} - q^{64} + 2 \beta_{3} q^{65} - 2 q^{67} + 3 \beta_1 q^{68} - q^{70} + 2 \beta_{3} q^{71} - 2 q^{73} + ( - 2 \beta_{3} - 5 \beta_1) q^{74} + 2 \beta_{2} q^{76} - \beta_{3} q^{77} - 2 \beta_{2} q^{79} + \beta_1 q^{80} - 5 \beta_{2} q^{82} - 8 \beta_{3} q^{83} + 3 q^{85} - 3 \beta_{3} q^{86} + \beta_{2} q^{88} + 12 \beta_1 q^{89} - 2 \beta_{2} q^{91} - 6 \beta_1 q^{92} - 2 \beta_{2} q^{94} + 2 \beta_{3} q^{95} - \beta_{2} q^{97} - 6 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{7} - 4 q^{10} + 4 q^{16} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 20 q^{37} + 4 q^{40} - 24 q^{46} - 24 q^{49} - 12 q^{58} - 4 q^{64} - 8 q^{67} - 4 q^{70} - 8 q^{73} + 12 q^{85}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2071.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
1.00000i 0 −1.00000 1.00000i 0 1.00000 1.00000i 0 −1.00000
2071.2 1.00000i 0 −1.00000 1.00000i 0 1.00000 1.00000i 0 −1.00000
2071.3 1.00000i 0 −1.00000 1.00000i 0 1.00000 1.00000i 0 −1.00000
2071.4 1.00000i 0 −1.00000 1.00000i 0 1.00000 1.00000i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.b even 2 1 inner
111.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.h.l 4
3.b odd 2 1 inner 3330.2.h.l 4
37.b even 2 1 inner 3330.2.h.l 4
111.d odd 2 1 inner 3330.2.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.h.l 4 1.a even 1 1 trivial
3330.2.h.l 4 3.b odd 2 1 inner
3330.2.h.l 4 37.b even 2 1 inner
3330.2.h.l 4 111.d odd 2 1 inner

Hecke kernels

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

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 12 \) Copy content Toggle raw display
\( T_{41}^{2} - 75 \) 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
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
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