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Properties

Label 630.6.b.b
Level $630$
Weight $6$
Character orbit 630.b
Analytic conductor $101.042$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,6,Mod(251,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.251");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 630.b (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: \(101.041806482\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 384 q^{4} + 600 q^{5} - 196 q^{7} + 6144 q^{16} - 9600 q^{20} + 15000 q^{25} - 3872 q^{26} + 3136 q^{28} - 4900 q^{35} - 33512 q^{37} - 10208 q^{38} + 44968 q^{41} + 8016 q^{43} - 1312 q^{46} + 47240 q^{47}+ \cdots + 257936 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 − 4.00000i 0 −16.0000 25.0000 0 −124.893 + 34.7652i 64.0000i 0 − 100.000i
251.2 − 4.00000i 0 −16.0000 25.0000 0 −121.533 − 45.1314i 64.0000i 0 − 100.000i
251.3 − 4.00000i 0 −16.0000 25.0000 0 −113.004 − 63.5388i 64.0000i 0 − 100.000i
251.4 − 4.00000i 0 −16.0000 25.0000 0 −82.3791 + 100.103i 64.0000i 0 − 100.000i
251.5 − 4.00000i 0 −16.0000 25.0000 0 −43.9610 − 121.961i 64.0000i 0 − 100.000i
251.6 − 4.00000i 0 −16.0000 25.0000 0 −21.7464 − 127.805i 64.0000i 0 − 100.000i
251.7 − 4.00000i 0 −16.0000 25.0000 0 0.790322 + 129.639i 64.0000i 0 − 100.000i
251.8 − 4.00000i 0 −16.0000 25.0000 0 36.2176 − 124.480i 64.0000i 0 − 100.000i
251.9 − 4.00000i 0 −16.0000 25.0000 0 52.2976 + 118.625i 64.0000i 0 − 100.000i
251.10 − 4.00000i 0 −16.0000 25.0000 0 67.1371 + 110.904i 64.0000i 0 − 100.000i
251.11 − 4.00000i 0 −16.0000 25.0000 0 125.323 − 33.1837i 64.0000i 0 − 100.000i
251.12 − 4.00000i 0 −16.0000 25.0000 0 127.751 + 22.0627i 64.0000i 0 − 100.000i
251.13 4.00000i 0 −16.0000 25.0000 0 −124.893 − 34.7652i − 64.0000i 0 100.000i
251.14 4.00000i 0 −16.0000 25.0000 0 −121.533 + 45.1314i − 64.0000i 0 100.000i
251.15 4.00000i 0 −16.0000 25.0000 0 −113.004 + 63.5388i − 64.0000i 0 100.000i
251.16 4.00000i 0 −16.0000 25.0000 0 −82.3791 − 100.103i − 64.0000i 0 100.000i
251.17 4.00000i 0 −16.0000 25.0000 0 −43.9610 + 121.961i − 64.0000i 0 100.000i
251.18 4.00000i 0 −16.0000 25.0000 0 −21.7464 + 127.805i − 64.0000i 0 100.000i
251.19 4.00000i 0 −16.0000 25.0000 0 0.790322 − 129.639i − 64.0000i 0 100.000i
251.20 4.00000i 0 −16.0000 25.0000 0 36.2176 + 124.480i − 64.0000i 0 100.000i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.6.b.b yes 24
3.b odd 2 1 630.6.b.a ✓ 24
7.b odd 2 1 630.6.b.a ✓ 24
21.c even 2 1 inner 630.6.b.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.6.b.a ✓ 24 3.b odd 2 1
630.6.b.a ✓ 24 7.b odd 2 1
630.6.b.b yes 24 1.a even 1 1 trivial
630.6.b.b yes 24 21.c even 2 1 inner