Newform orbit 124.4.p.a

• Label: 124.4.p.a
• Level: $124$
• Weight: $4$
• Character orbit: 124.p
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.82362	−	0.164797i	−0.134736	+	1.28193i	7.94568	+	0.930647i	−0.0229181	+	0.0396953i	0.591703	−	3.59749i	1.94572	+	9.15391i	−22.2822	−	3.93722i	24.7848	+	5.26817i	0.0712537	−	0.108308i
3.2	−2.81278	+	0.297076i	−0.949976	+	9.03842i	7.82349	−	1.67122i	6.63863	−	11.4984i	−0.0130205	−	25.7053i	−2.08859	−	9.82603i	−21.5093	+	7.02495i	−54.3805	−	11.5589i	−15.2571	+	34.3148i
3.3	−2.80456	−	0.366691i	0.244891	−	2.32998i	7.73108	+	2.05681i	−8.79653	+	15.2360i	−1.54119	+	6.44476i	−4.62413	−	21.7548i	−20.9280	−	8.60336i	21.0412	+	4.47243i	30.2573	−	39.5047i
3.4	−2.80221	+	0.384244i	0.781206	−	7.43268i	7.70471	−	2.15346i	−1.74753	+	3.02682i	0.666864	+	21.1281i	6.94622	+	32.6794i	−20.7627	+	8.99494i	−28.2244	−	5.99928i	3.73391	−	9.15324i
3.5	−2.70584	+	0.823655i	0.572364	−	5.44568i	6.64318	−	4.45737i	9.37792	−	16.2430i	2.93664	+	15.2066i	−6.65861	−	31.3263i	−14.3041	+	17.5326i	−2.91787	−	0.620213i	−11.9965	+	51.6753i
3.6	−2.62585	+	1.05116i	−0.804966	+	7.65874i	5.79014	−	5.52035i	−9.96666	+	17.2628i	−5.93681	−	20.9568i	3.05831	+	14.3882i	−9.40128	+	20.5819i	−31.5984	−	6.71644i	8.02507	−	55.8059i
3.7	−2.50811	−	1.30742i	0.215453	−	2.04990i	4.58128	+	6.55834i	10.3394	−	17.9083i	−3.22047	+	4.85970i	3.46899	+	16.3203i	−2.91584	−	22.4388i	22.2543	+	4.73030i	−49.3462	+	31.3982i
3.8	−2.46001	−	1.39582i	−0.479513	+	4.56226i	4.10335	+	6.86750i	0.676684	−	1.17205i	7.54773	−	10.5539i	−5.59922	−	26.3423i	−0.508472	−	22.6217i	5.82568	+	1.23829i	−3.30063	+	1.93873i
3.9	−2.41499	−	1.47235i	0.915780	−	8.71306i	3.66440	+	7.11141i	0.646337	−	1.11949i	−15.0402	+	19.6937i	−1.72916	−	8.13508i	1.62096	−	22.5693i	−48.6688	−	10.3449i	−3.20918	+	1.75193i
3.10	−2.30351	+	1.64129i	−0.315000	+	2.99703i	2.61231	−	7.56147i	0.474237	−	0.821402i	−4.19340	−	7.42069i	−1.56190	−	7.34815i	6.39311	+	21.7055i	17.5270	+	3.72549i	0.255753	+	2.67047i
3.11	−2.27279	+	1.68358i	0.315000	−	2.99703i	2.33112	−	7.65284i	0.474237	−	0.821402i	4.32981	+	7.34193i	1.56190	+	7.34815i	7.58603	+	21.3179i	17.5270	+	3.72549i	0.305057	+	2.66529i
3.12	−2.25611	−	1.70586i	−0.718973	+	6.84057i	2.18011	+	7.69722i	−2.50613	+	4.34075i	13.2911	−	14.2066i	4.92655	+	23.1776i	8.21177	−	21.0848i	−19.8664	−	4.22274i	13.0588	−	5.51813i
3.13	−1.81114	+	2.17250i	0.804966	−	7.65874i	−1.43954	−	7.86942i	−9.96666	+	17.2628i	15.1807	+	15.6198i	−3.05831	−	14.3882i	19.7036	+	11.1252i	−31.5984	−	6.71644i	−19.4524	−	52.9179i
3.14	−1.61949	+	2.31889i	−0.572364	+	5.44568i	−2.75448	−	7.51085i	9.37792	−	16.2430i	−11.7010	−	10.1465i	6.65861	+	31.3263i	21.8777	+	5.77646i	−2.91787	−	0.620213i	22.4783	+	48.0518i
3.15	−1.48223	−	2.40894i	0.430178	−	4.09287i	−3.60602	+	7.14119i	−4.68227	+	8.10993i	−10.4971	+	5.03029i	1.43654	+	6.75841i	22.5477	−	1.89817i	9.84342	+	2.09228i	26.4765	−	0.741420i
3.16	−1.23137	+	2.54632i	−0.781206	+	7.43268i	−4.96747	−	6.27090i	−1.74753	+	3.02682i	−17.9640	−	11.1416i	−6.94622	−	32.6794i	22.0845	−	4.92697i	−28.2244	−	5.99928i	−5.55538	−	8.17690i
3.17	−1.20752	−	2.55771i	−0.00192290	+	0.0182952i	−5.08377	+	6.17699i	6.27992	−	10.8771i	0.0491158	−	0.0171736i	−1.11632	−	5.25185i	21.9377	+	5.54396i	26.4097	+	5.61355i	−35.4037	−	2.92782i
3.18	−1.15173	+	2.58331i	0.949976	−	9.03842i	−5.34702	−	5.95058i	6.63863	−	11.4984i	22.2549	+	12.8639i	2.08859	+	9.82603i	21.5306	−	6.95955i	−54.3805	−	11.5589i	22.0582	+	30.3928i
3.19	−1.00949	−	2.64215i	−0.650852	+	6.19245i	−5.96188	+	5.33442i	−8.39771	+	14.5453i	17.0184	−	4.53154i	−3.41539	−	16.0681i	20.1127	+	10.3671i	−11.5128	−	2.44712i	46.9081	+	7.50475i
3.20	−0.737328	−	2.73063i	−0.901200	+	8.57435i	−6.91270	+	4.02674i	8.05240	−	13.9472i	24.0779	−	3.86126i	−1.43706	−	6.76081i	16.0925	+	15.9070i	−46.2973	−	9.84079i	−44.0218	−	11.7045i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
31.h	odd	30	1	inner
124.p	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	124.4.p.a	✓	368
4.b	odd	2	1	inner	124.4.p.a	✓	368
31.h	odd	30	1	inner	124.4.p.a	✓	368
124.p	even	30	1	inner	124.4.p.a	✓	368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
124.4.p.a	✓	368	1.a	even	1	1	trivial
124.4.p.a	✓	368	4.b	odd	2	1	inner
124.4.p.a	✓	368	31.h	odd	30	1	inner
124.4.p.a	✓	368	124.p	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(124, [\chi])\).