Properties

Label 6300.2.dd.b
Level $6300$
Weight $2$
Character orbit 6300.dd
Analytic conductor $50.306$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(1349,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6300.1349");
 
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.dd (of order \(6\), degree \(2\), 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: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{11} + 12 q^{19} - 12 q^{31} + 16 q^{41} - 44 q^{49} - 28 q^{79} - 40 q^{89} + 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1349.1 0 0 0 0 0 −2.63285 0.260926i 0 0 0
1349.2 0 0 0 0 0 −2.08318 + 1.63107i 0 0 0
1349.3 0 0 0 0 0 −1.98249 1.75207i 0 0 0
1349.4 0 0 0 0 0 −0.390758 2.61674i 0 0 0
1349.5 0 0 0 0 0 −0.380350 + 2.61827i 0 0 0
1349.6 0 0 0 0 0 −0.0290059 2.64559i 0 0 0
1349.7 0 0 0 0 0 0.0290059 + 2.64559i 0 0 0
1349.8 0 0 0 0 0 0.380350 2.61827i 0 0 0
1349.9 0 0 0 0 0 0.390758 + 2.61674i 0 0 0
1349.10 0 0 0 0 0 1.98249 + 1.75207i 0 0 0
1349.11 0 0 0 0 0 2.08318 1.63107i 0 0 0
1349.12 0 0 0 0 0 2.63285 + 0.260926i 0 0 0
4049.1 0 0 0 0 0 −2.63285 + 0.260926i 0 0 0
4049.2 0 0 0 0 0 −2.08318 1.63107i 0 0 0
4049.3 0 0 0 0 0 −1.98249 + 1.75207i 0 0 0
4049.4 0 0 0 0 0 −0.390758 + 2.61674i 0 0 0
4049.5 0 0 0 0 0 −0.380350 2.61827i 0 0 0
4049.6 0 0 0 0 0 −0.0290059 + 2.64559i 0 0 0
4049.7 0 0 0 0 0 0.0290059 2.64559i 0 0 0
4049.8 0 0 0 0 0 0.380350 + 2.61827i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.dd.b 24
3.b odd 2 1 6300.2.dd.c 24
5.b even 2 1 inner 6300.2.dd.b 24
5.c odd 4 1 1260.2.cg.b yes 12
5.c odd 4 1 6300.2.ch.b 12
7.d odd 6 1 6300.2.dd.c 24
15.d odd 2 1 6300.2.dd.c 24
15.e even 4 1 1260.2.cg.a 12
15.e even 4 1 6300.2.ch.c 12
21.g even 6 1 inner 6300.2.dd.b 24
35.i odd 6 1 6300.2.dd.c 24
35.k even 12 1 1260.2.cg.a 12
35.k even 12 1 6300.2.ch.c 12
35.k even 12 1 8820.2.d.b 12
35.l odd 12 1 8820.2.d.a 12
105.p even 6 1 inner 6300.2.dd.b 24
105.w odd 12 1 1260.2.cg.b yes 12
105.w odd 12 1 6300.2.ch.b 12
105.w odd 12 1 8820.2.d.a 12
105.x even 12 1 8820.2.d.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cg.a 12 15.e even 4 1
1260.2.cg.a 12 35.k even 12 1
1260.2.cg.b yes 12 5.c odd 4 1
1260.2.cg.b yes 12 105.w odd 12 1
6300.2.ch.b 12 5.c odd 4 1
6300.2.ch.b 12 105.w odd 12 1
6300.2.ch.c 12 15.e even 4 1
6300.2.ch.c 12 35.k even 12 1
6300.2.dd.b 24 1.a even 1 1 trivial
6300.2.dd.b 24 5.b even 2 1 inner
6300.2.dd.b 24 21.g even 6 1 inner
6300.2.dd.b 24 105.p even 6 1 inner
6300.2.dd.c 24 3.b odd 2 1
6300.2.dd.c 24 7.d odd 6 1
6300.2.dd.c 24 15.d odd 2 1
6300.2.dd.c 24 35.i odd 6 1
8820.2.d.a 12 35.l odd 12 1
8820.2.d.a 12 105.w odd 12 1
8820.2.d.b 12 35.k even 12 1
8820.2.d.b 12 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 12 T_{11}^{11} + 38 T_{11}^{10} - 120 T_{11}^{9} - 636 T_{11}^{8} + 1320 T_{11}^{7} + 10748 T_{11}^{6} + 10056 T_{11}^{5} - 37340 T_{11}^{4} - 57600 T_{11}^{3} + 131040 T_{11}^{2} + 375840 T_{11} + 272484 \) acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\). Copy content Toggle raw display