Properties

Label 126.10.k.a
Level $126$
Weight $10$
Character orbit 126.k
Analytic conductor $64.895$
Analytic rank $0$
Dimension $48$
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] = [126,10,Mod(17,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.17");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.k (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: \(64.8945153566\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6144 q^{4} - 12720 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6144 q^{4} - 12720 q^{7} + 153792 q^{10} - 1572864 q^{16} - 2546532 q^{19} + 4853376 q^{22} - 6871164 q^{25} + 2380800 q^{28} - 6720912 q^{31} + 1881660 q^{37} + 39370752 q^{40} - 171303864 q^{43} + 54125568 q^{46} + 129836928 q^{49} - 58125312 q^{52} + 180566208 q^{58} - 632749248 q^{61} - 805306368 q^{64} + 721189428 q^{67} + 308075328 q^{70} - 853713108 q^{73} + 1452429384 q^{79} - 1355242752 q^{82} + 368938080 q^{85} + 621232128 q^{88} + 4577539644 q^{91} - 1279193472 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −13.8564 + 8.00000i 0 128.000 221.703i −1351.83 2341.43i 0 −3938.59 + 4984.09i 4096.00i 0 37463.0 + 21629.2i
17.2 −13.8564 + 8.00000i 0 128.000 221.703i 940.085 + 1628.28i 0 −5173.04 + 3686.91i 4096.00i 0 −26052.4 15041.4i
17.3 −13.8564 + 8.00000i 0 128.000 221.703i 788.973 + 1366.54i 0 −3939.49 4983.38i 4096.00i 0 −21864.7 12623.6i
17.4 −13.8564 + 8.00000i 0 128.000 221.703i 718.886 + 1245.15i 0 6327.61 561.268i 4096.00i 0 −19922.3 11502.2i
17.5 −13.8564 + 8.00000i 0 128.000 221.703i −356.977 618.302i 0 3246.94 + 5459.94i 4096.00i 0 9892.83 + 5711.63i
17.6 −13.8564 + 8.00000i 0 128.000 221.703i −349.092 604.645i 0 4657.86 4319.48i 4096.00i 0 9674.32 + 5585.47i
17.7 −13.8564 + 8.00000i 0 128.000 221.703i 262.462 + 454.598i 0 −5311.53 3484.43i 4096.00i 0 −7273.57 4199.40i
17.8 −13.8564 + 8.00000i 0 128.000 221.703i −74.4501 128.951i 0 −4177.48 + 4785.63i 4096.00i 0 2063.22 + 1191.20i
17.9 −13.8564 + 8.00000i 0 128.000 221.703i 554.620 + 960.629i 0 1350.61 + 6207.21i 4096.00i 0 −15370.1 8873.91i
17.10 −13.8564 + 8.00000i 0 128.000 221.703i −558.488 967.329i 0 3171.64 5504.03i 4096.00i 0 15477.3 + 8935.81i
17.11 −13.8564 + 8.00000i 0 128.000 221.703i −999.354 1730.93i 0 −5636.61 2929.55i 4096.00i 0 27694.9 + 15989.7i
17.12 −13.8564 + 8.00000i 0 128.000 221.703i −962.211 1666.60i 0 6242.08 + 1179.01i 4096.00i 0 26665.6 + 15395.4i
17.13 13.8564 8.00000i 0 128.000 221.703i 962.211 + 1666.60i 0 6242.08 + 1179.01i 4096.00i 0 26665.6 + 15395.4i
17.14 13.8564 8.00000i 0 128.000 221.703i 999.354 + 1730.93i 0 −5636.61 2929.55i 4096.00i 0 27694.9 + 15989.7i
17.15 13.8564 8.00000i 0 128.000 221.703i 558.488 + 967.329i 0 3171.64 5504.03i 4096.00i 0 15477.3 + 8935.81i
17.16 13.8564 8.00000i 0 128.000 221.703i −554.620 960.629i 0 1350.61 + 6207.21i 4096.00i 0 −15370.1 8873.91i
17.17 13.8564 8.00000i 0 128.000 221.703i 74.4501 + 128.951i 0 −4177.48 + 4785.63i 4096.00i 0 2063.22 + 1191.20i
17.18 13.8564 8.00000i 0 128.000 221.703i −262.462 454.598i 0 −5311.53 3484.43i 4096.00i 0 −7273.57 4199.40i
17.19 13.8564 8.00000i 0 128.000 221.703i 349.092 + 604.645i 0 4657.86 4319.48i 4096.00i 0 9674.32 + 5585.47i
17.20 13.8564 8.00000i 0 128.000 221.703i 356.977 + 618.302i 0 3246.94 + 5459.94i 4096.00i 0 9892.83 + 5711.63i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.10.k.a 48
3.b odd 2 1 inner 126.10.k.a 48
7.d odd 6 1 inner 126.10.k.a 48
21.g even 6 1 inner 126.10.k.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.10.k.a 48 1.a even 1 1 trivial
126.10.k.a 48 3.b odd 2 1 inner
126.10.k.a 48 7.d odd 6 1 inner
126.10.k.a 48 21.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(126, [\chi])\).