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Properties

Label 6300.2.ch.f
Level $6300$
Weight $2$
Character orbit 6300.ch
Analytic conductor $50.306$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(1601,6300)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6300.1601");
 
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.ch (of order \(6\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1 0 0 0 0 0 −2.48577 + 0.906070i 0 0 0
1601.2 0 0 0 0 0 −2.48577 + 0.906070i 0 0 0
1601.3 0 0 0 0 0 −1.43620 − 2.22201i 0 0 0
1601.4 0 0 0 0 0 −1.43620 − 2.22201i 0 0 0
1601.5 0 0 0 0 0 −1.29877 + 2.30504i 0 0 0
1601.6 0 0 0 0 0 −1.29877 + 2.30504i 0 0 0
1601.7 0 0 0 0 0 −1.25359 − 2.32992i 0 0 0
1601.8 0 0 0 0 0 −1.25359 − 2.32992i 0 0 0
1601.9 0 0 0 0 0 1.25359 + 2.32992i 0 0 0
1601.10 0 0 0 0 0 1.25359 + 2.32992i 0 0 0
1601.11 0 0 0 0 0 1.29877 − 2.30504i 0 0 0
1601.12 0 0 0 0 0 1.29877 − 2.30504i 0 0 0
1601.13 0 0 0 0 0 1.43620 + 2.22201i 0 0 0
1601.14 0 0 0 0 0 1.43620 + 2.22201i 0 0 0
1601.15 0 0 0 0 0 2.48577 − 0.906070i 0 0 0
1601.16 0 0 0 0 0 2.48577 − 0.906070i 0 0 0
4301.1 0 0 0 0 0 −2.48577 − 0.906070i 0 0 0
4301.2 0 0 0 0 0 −2.48577 − 0.906070i 0 0 0
4301.3 0 0 0 0 0 −1.43620 + 2.22201i 0 0 0
4301.4 0 0 0 0 0 −1.43620 + 2.22201i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1601.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 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.ch.f 32
3.b odd 2 1 inner 6300.2.ch.f 32
5.b even 2 1 inner 6300.2.ch.f 32
5.c odd 4 2 1260.2.dc.a ✓ 32
7.d odd 6 1 inner 6300.2.ch.f 32
15.d odd 2 1 inner 6300.2.ch.f 32
15.e even 4 2 1260.2.dc.a ✓ 32
21.g even 6 1 inner 6300.2.ch.f 32
35.i odd 6 1 inner 6300.2.ch.f 32
35.k even 12 2 1260.2.dc.a ✓ 32
35.k even 12 2 8820.2.f.a 32
35.l odd 12 2 8820.2.f.a 32
105.p even 6 1 inner 6300.2.ch.f 32
105.w odd 12 2 1260.2.dc.a ✓ 32
105.w odd 12 2 8820.2.f.a 32
105.x even 12 2 8820.2.f.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.dc.a ✓ 32 5.c odd 4 2
1260.2.dc.a ✓ 32 15.e even 4 2
1260.2.dc.a ✓ 32 35.k even 12 2
1260.2.dc.a ✓ 32 105.w odd 12 2
6300.2.ch.f 32 1.a even 1 1 trivial
6300.2.ch.f 32 3.b odd 2 1 inner
6300.2.ch.f 32 5.b even 2 1 inner
6300.2.ch.f 32 7.d odd 6 1 inner
6300.2.ch.f 32 15.d odd 2 1 inner
6300.2.ch.f 32 21.g even 6 1 inner
6300.2.ch.f 32 35.i odd 6 1 inner
6300.2.ch.f 32 105.p even 6 1 inner
8820.2.f.a 32 35.k even 12 2
8820.2.f.a 32 35.l odd 12 2
8820.2.f.a 32 105.w odd 12 2
8820.2.f.a 32 105.x even 12 2

Hecke kernels

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

\( T_{11}^{16} - 56 T_{11}^{14} + 2296 T_{11}^{12} - 43004 T_{11}^{10} + 591808 T_{11}^{8} - 1607312 T_{11}^{6} + \cdots + 614656 \) Copy content Toggle raw display
\( T_{37}^{16} + 232 T_{37}^{14} + 37812 T_{37}^{12} + 3065044 T_{37}^{10} + 180918823 T_{37}^{8} + \cdots + 9116621361 \) Copy content Toggle raw display