# 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 $4$

# Related objects

## Newspace parameters

 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

 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.

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

## 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.

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 83.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(126, [\chi])$$.