Properties

Label 6030.2.d.c
Level $6030$
Weight $2$
Character orbit 6030.d
Analytic conductor $48.150$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6030,2,Mod(2411,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("6030.2411");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.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: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + q^{16} + \beta q^{17} + 2 q^{19} + q^{20} - 5 \beta q^{23} + q^{25} + \beta q^{29} - q^{32} - \beta q^{34} - 4 q^{37} - 2 q^{38} - q^{40} - 6 q^{41} - 6 \beta q^{43} + 5 \beta q^{46} - 5 \beta q^{47} + 7 q^{49} - q^{50} - 6 q^{53} - \beta q^{58} + \beta q^{59} + q^{64} + (3 \beta + 7) q^{67} + \beta q^{68} - 5 \beta q^{71} - 4 q^{73} + 4 q^{74} + 2 q^{76} + 12 \beta q^{79} + q^{80} + 6 q^{82} + 7 \beta q^{83} + \beta q^{85} + 6 \beta q^{86} - 11 \beta q^{89} - 5 \beta q^{92} + 5 \beta q^{94} + 2 q^{95} - 12 \beta q^{97} - 7 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} + 2 q^{16} + 4 q^{19} + 2 q^{20} + 2 q^{25} - 2 q^{32} - 8 q^{37} - 4 q^{38} - 2 q^{40} - 12 q^{41} + 14 q^{49} - 2 q^{50} - 12 q^{53} + 2 q^{64} + 14 q^{67} - 8 q^{73} + 8 q^{74} + 4 q^{76} + 2 q^{80} + 12 q^{82} + 4 q^{95} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times\).

\(n\) \(1207\) \(3151\) \(4691\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2411.1
1.41421i
1.41421i
−1.00000 0 1.00000 1.00000 0 0 −1.00000 0 −1.00000
2411.2 −1.00000 0 1.00000 1.00000 0 0 −1.00000 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
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6030.2.d.c 2
3.b odd 2 1 6030.2.d.f yes 2
67.b odd 2 1 6030.2.d.f yes 2
201.d even 2 1 inner 6030.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6030.2.d.c 2 1.a even 1 1 trivial
6030.2.d.c 2 201.d even 2 1 inner
6030.2.d.f yes 2 3.b odd 2 1
6030.2.d.f yes 2 67.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6030, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 50 \) Copy content Toggle raw display
$29$ \( T^{2} + 2 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 72 \) Copy content Toggle raw display
$47$ \( T^{2} + 50 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 67 \) Copy content Toggle raw display
$71$ \( T^{2} + 50 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 288 \) Copy content Toggle raw display
$83$ \( T^{2} + 98 \) Copy content Toggle raw display
$89$ \( T^{2} + 242 \) Copy content Toggle raw display
$97$ \( T^{2} + 288 \) Copy content Toggle raw display
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