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Properties

Label 6300.2.dd.d
Level $6300$
Weight $2$
Character orbit 6300.dd
Analytic conductor $50.306$
Analytic rank $0$
Dimension $40$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6300,2,Mod(1349,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{31} - 8 q^{49} + 12 q^{61} - 4 q^{79} - 80 q^{91}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −2.56421 − 0.651797i 0 0 0
1349.2 0 0 0 0 0 −2.56421 − 0.651797i 0 0 0
1349.3 0 0 0 0 0 −2.14507 − 1.54876i 0 0 0
1349.4 0 0 0 0 0 −2.14507 − 1.54876i 0 0 0
1349.5 0 0 0 0 0 −2.05705 + 1.66390i 0 0 0
1349.6 0 0 0 0 0 −2.05705 + 1.66390i 0 0 0
1349.7 0 0 0 0 0 −1.17299 + 2.37152i 0 0 0
1349.8 0 0 0 0 0 −1.17299 + 2.37152i 0 0 0
1349.9 0 0 0 0 0 −0.464924 − 2.60458i 0 0 0
1349.10 0 0 0 0 0 −0.464924 − 2.60458i 0 0 0
1349.11 0 0 0 0 0 0.464924 + 2.60458i 0 0 0
1349.12 0 0 0 0 0 0.464924 + 2.60458i 0 0 0
1349.13 0 0 0 0 0 1.17299 − 2.37152i 0 0 0
1349.14 0 0 0 0 0 1.17299 − 2.37152i 0 0 0
1349.15 0 0 0 0 0 2.05705 − 1.66390i 0 0 0
1349.16 0 0 0 0 0 2.05705 − 1.66390i 0 0 0
1349.17 0 0 0 0 0 2.14507 + 1.54876i 0 0 0
1349.18 0 0 0 0 0 2.14507 + 1.54876i 0 0 0
1349.19 0 0 0 0 0 2.56421 + 0.651797i 0 0 0
1349.20 0 0 0 0 0 2.56421 + 0.651797i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.20
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.dd.d 40
3.b odd 2 1 inner 6300.2.dd.d 40
5.b even 2 1 inner 6300.2.dd.d 40
5.c odd 4 1 6300.2.ch.d ✓ 20
5.c odd 4 1 6300.2.ch.e yes 20
7.d odd 6 1 inner 6300.2.dd.d 40
15.d odd 2 1 inner 6300.2.dd.d 40
15.e even 4 1 6300.2.ch.d ✓ 20
15.e even 4 1 6300.2.ch.e yes 20
21.g even 6 1 inner 6300.2.dd.d 40
35.i odd 6 1 inner 6300.2.dd.d 40
35.k even 12 1 6300.2.ch.d ✓ 20
35.k even 12 1 6300.2.ch.e yes 20
105.p even 6 1 inner 6300.2.dd.d 40
105.w odd 12 1 6300.2.ch.d ✓ 20
105.w odd 12 1 6300.2.ch.e yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6300.2.ch.d ✓ 20 5.c odd 4 1
6300.2.ch.d ✓ 20 15.e even 4 1
6300.2.ch.d ✓ 20 35.k even 12 1
6300.2.ch.d ✓ 20 105.w odd 12 1
6300.2.ch.e yes 20 5.c odd 4 1
6300.2.ch.e yes 20 15.e even 4 1
6300.2.ch.e yes 20 35.k even 12 1
6300.2.ch.e yes 20 105.w odd 12 1
6300.2.dd.d 40 1.a even 1 1 trivial
6300.2.dd.d 40 3.b odd 2 1 inner
6300.2.dd.d 40 5.b even 2 1 inner
6300.2.dd.d 40 7.d odd 6 1 inner
6300.2.dd.d 40 15.d odd 2 1 inner
6300.2.dd.d 40 21.g even 6 1 inner
6300.2.dd.d 40 35.i odd 6 1 inner
6300.2.dd.d 40 105.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{20} - 44 T_{11}^{18} + 1408 T_{11}^{16} - 20576 T_{11}^{14} + 219724 T_{11}^{12} - 645992 T_{11}^{10} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\). Copy content Toggle raw display