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Properties

Label 126.4.m.a
Level $126$
Weight $4$
Character orbit 126.m
Analytic conductor $7.434$
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] = [126,4,Mod(41,126)]
 
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([5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.m (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: \(7.43424066072\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{4} - 12 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{4} - 12 q^{7} - 36 q^{9} + 24 q^{11} - 132 q^{14} - 120 q^{15} - 384 q^{16} + 120 q^{18} + 180 q^{21} + 348 q^{23} - 600 q^{25} - 96 q^{28} - 84 q^{29} + 192 q^{30} + 96 q^{36} - 672 q^{37} + 1368 q^{39} + 1128 q^{42} + 84 q^{43} - 1008 q^{46} - 42 q^{49} + 456 q^{50} + 2016 q^{51} - 528 q^{56} + 732 q^{57} + 504 q^{58} - 1008 q^{60} - 774 q^{63} - 3072 q^{64} - 6972 q^{65} + 1176 q^{67} + 216 q^{70} - 384 q^{72} + 2520 q^{74} + 1500 q^{77} + 2832 q^{78} + 348 q^{79} + 2268 q^{81} - 1080 q^{84} + 720 q^{85} + 1200 q^{86} + 180 q^{91} + 1392 q^{92} + 5232 q^{93} - 5892 q^{95} + 972 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −1.73205 + 1.00000i −5.10721 − 0.957281i 2.00000 − 3.46410i 0.0529940 − 0.0917884i 9.80323 − 3.44915i 18.4211 + 1.91364i 8.00000i 25.1672 + 9.77807i 0.211976i
41.2 −1.73205 + 1.00000i −5.04167 + 1.25759i 2.00000 − 3.46410i 0.984584 − 1.70535i 7.47485 − 7.21988i −17.4955 − 6.07527i 8.00000i 23.8370 − 12.6807i 3.93834i
41.3 −1.73205 + 1.00000i −3.28373 + 4.02705i 2.00000 − 3.46410i −10.0781 + 17.4558i 1.66054 − 10.2588i 3.08758 + 18.2611i 8.00000i −5.43421 − 26.4475i − 40.3124i
41.4 −1.73205 + 1.00000i −2.30109 − 4.65886i 2.00000 − 3.46410i −2.59824 + 4.50028i 8.64447 + 5.76829i −3.66839 + 18.1533i 8.00000i −16.4100 + 21.4409i − 10.3929i
41.5 −1.73205 + 1.00000i −2.25730 − 4.68024i 2.00000 − 3.46410i 7.47541 − 12.9478i 8.58999 + 5.84911i −6.77961 − 17.2348i 8.00000i −16.8092 + 21.1294i 29.9016i
41.6 −1.73205 + 1.00000i −0.111457 + 5.19496i 2.00000 − 3.46410i 8.24952 − 14.2886i −5.00191 − 9.10939i −17.4194 + 6.29008i 8.00000i −26.9752 − 1.15803i 32.9981i
41.7 −1.73205 + 1.00000i 0.111457 − 5.19496i 2.00000 − 3.46410i −8.24952 + 14.2886i 5.00191 + 9.10939i 14.1571 − 11.9406i 8.00000i −26.9752 − 1.15803i − 32.9981i
41.8 −1.73205 + 1.00000i 2.25730 + 4.68024i 2.00000 − 3.46410i −7.47541 + 12.9478i −8.58999 − 5.84911i −11.5359 − 14.4887i 8.00000i −16.8092 + 21.1294i − 29.9016i
41.9 −1.73205 + 1.00000i 2.30109 + 4.65886i 2.00000 − 3.46410i 2.59824 − 4.50028i −8.64447 − 5.76829i 17.5554 + 5.89974i 8.00000i −16.4100 + 21.4409i 10.3929i
41.10 −1.73205 + 1.00000i 3.28373 − 4.02705i 2.00000 − 3.46410i 10.0781 − 17.4558i −1.66054 + 10.2588i 14.2708 + 11.8045i 8.00000i −5.43421 − 26.4475i 40.3124i
41.11 −1.73205 + 1.00000i 5.04167 − 1.25759i 2.00000 − 3.46410i −0.984584 + 1.70535i −7.47485 + 7.21988i 3.48639 − 18.1891i 8.00000i 23.8370 − 12.6807i − 3.93834i
41.12 −1.73205 + 1.00000i 5.10721 + 0.957281i 2.00000 − 3.46410i −0.0529940 + 0.0917884i −9.80323 + 3.44915i −7.55330 + 16.9100i 8.00000i 25.1672 + 9.77807i − 0.211976i
41.13 1.73205 − 1.00000i −5.19582 + 0.0586938i 2.00000 − 3.46410i −2.95803 + 5.12346i −8.94073 + 5.29748i 13.8716 + 12.2711i − 8.00000i 26.9931 − 0.609925i 11.8321i
41.14 1.73205 − 1.00000i −4.77115 − 2.05818i 2.00000 − 3.46410i 9.07575 − 15.7197i −10.3221 + 1.20629i −16.5045 + 8.40239i − 8.00000i 18.5278 + 19.6398i − 36.3030i
41.15 1.73205 − 1.00000i −4.36858 + 2.81345i 2.00000 − 3.46410i −5.62817 + 9.74828i −4.75315 + 9.24162i 2.97811 − 18.2792i − 8.00000i 11.1690 − 24.5816i 22.5127i
41.16 1.73205 − 1.00000i −2.86447 + 4.33530i 2.00000 − 3.46410i 4.49442 − 7.78456i −0.626101 + 10.3734i −13.8476 − 12.2981i − 8.00000i −10.5897 − 24.8366i − 17.9777i
41.17 1.73205 − 1.00000i −2.71645 − 4.42955i 2.00000 − 3.46410i −7.20413 + 12.4779i −9.13458 − 4.95575i −17.9832 + 4.42778i − 8.00000i −12.2418 + 24.0653i 28.8165i
41.18 1.73205 − 1.00000i −0.618827 − 5.15917i 2.00000 − 3.46410i 4.67738 − 8.10146i −6.23101 − 8.31712i 18.4619 − 1.46876i − 8.00000i −26.2341 + 6.38527i − 18.7095i
41.19 1.73205 − 1.00000i 0.618827 + 5.15917i 2.00000 − 3.46410i −4.67738 + 8.10146i 6.23101 + 8.31712i −10.5029 + 15.2541i − 8.00000i −26.2341 + 6.38527i 18.7095i
41.20 1.73205 − 1.00000i 2.71645 + 4.42955i 2.00000 − 3.46410i 7.20413 − 12.4779i 9.13458 + 4.95575i 12.8262 − 13.3600i − 8.00000i −12.2418 + 24.0653i − 28.8165i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.m.a ✓ 48
3.b odd 2 1 378.4.m.a 48
7.b odd 2 1 inner 126.4.m.a ✓ 48
9.c even 3 1 378.4.m.a 48
9.c even 3 1 1134.4.d.b 48
9.d odd 6 1 inner 126.4.m.a ✓ 48
9.d odd 6 1 1134.4.d.b 48
21.c even 2 1 378.4.m.a 48
63.l odd 6 1 378.4.m.a 48
63.l odd 6 1 1134.4.d.b 48
63.o even 6 1 inner 126.4.m.a ✓ 48
63.o even 6 1 1134.4.d.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.m.a ✓ 48 1.a even 1 1 trivial
126.4.m.a ✓ 48 7.b odd 2 1 inner
126.4.m.a ✓ 48 9.d odd 6 1 inner
126.4.m.a ✓ 48 63.o even 6 1 inner
378.4.m.a 48 3.b odd 2 1
378.4.m.a 48 9.c even 3 1
378.4.m.a 48 21.c even 2 1
378.4.m.a 48 63.l odd 6 1
1134.4.d.b 48 9.c even 3 1
1134.4.d.b 48 9.d odd 6 1
1134.4.d.b 48 63.l odd 6 1
1134.4.d.b 48 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(126, [\chi])\).