Properties

Label 6300.2.k.f
Level $6300$
Weight $2$
Character orbit 6300.k
Analytic conductor $50.306$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(6049,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6300.6049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.k (of order \(2\), degree \(1\), not 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: \(50.3057532734\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{7} - 2 q^{11} - 4 i q^{13} - 2 i q^{17} + 2 q^{19} + 4 i q^{23} - 2 q^{29} - 6 q^{31} - 6 i q^{37} - 6 q^{41} + 4 i q^{43} - q^{49} + 8 i q^{53} - 10 q^{61} - 12 i q^{67} + 14 q^{71} - 4 i q^{73} + 2 i q^{77} + 8 q^{79} + 12 i q^{83} - 14 q^{89} - 4 q^{91} + 8 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{11} + 4 q^{19} - 4 q^{29} - 12 q^{31} - 12 q^{41} - 2 q^{49} - 20 q^{61} + 28 q^{71} + 16 q^{79} - 28 q^{89} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
6049.2 0 0 0 0 0 1.00000i 0 0 0
\(n\): 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 6300.2.k.f 2
3.b odd 2 1 2100.2.k.h 2
5.b even 2 1 inner 6300.2.k.f 2
5.c odd 4 1 1260.2.a.a 1
5.c odd 4 1 6300.2.a.u 1
15.d odd 2 1 2100.2.k.h 2
15.e even 4 1 420.2.a.d 1
15.e even 4 1 2100.2.a.h 1
20.e even 4 1 5040.2.a.r 1
35.f even 4 1 8820.2.a.s 1
60.l odd 4 1 1680.2.a.i 1
60.l odd 4 1 8400.2.a.bq 1
105.k odd 4 1 2940.2.a.c 1
105.w odd 12 2 2940.2.q.l 2
105.x even 12 2 2940.2.q.b 2
120.q odd 4 1 6720.2.a.bs 1
120.w even 4 1 6720.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.d 1 15.e even 4 1
1260.2.a.a 1 5.c odd 4 1
1680.2.a.i 1 60.l odd 4 1
2100.2.a.h 1 15.e even 4 1
2100.2.k.h 2 3.b odd 2 1
2100.2.k.h 2 15.d odd 2 1
2940.2.a.c 1 105.k odd 4 1
2940.2.q.b 2 105.x even 12 2
2940.2.q.l 2 105.w odd 12 2
5040.2.a.r 1 20.e even 4 1
6300.2.a.u 1 5.c odd 4 1
6300.2.k.f 2 1.a even 1 1 trivial
6300.2.k.f 2 5.b even 2 1 inner
6720.2.a.e 1 120.w even 4 1
6720.2.a.bs 1 120.q odd 4 1
8400.2.a.bq 1 60.l odd 4 1
8820.2.a.s 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\):

\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 64 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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