Newform orbit 123.4.o.a

• Label: 123.4.o.a
• Level: $123$
• Weight: $4$
• Character orbit: 123.o
• Analytic conductor: $7.257$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	123.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.25723493071\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(40\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 12 q^{3} - 40 q^{4} - 76 q^{6} - 32 q^{7} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−0.837999	−	5.29092i	−1.77044	−	4.88524i	−19.6831	+	6.39543i	−3.15927	−	1.60973i	−24.3637	+	13.4611i	−1.55299	+	0.372841i	30.8764	+	60.5983i	−20.7311	+	17.2981i	−5.86947	+	18.0644i
11.2	−0.836276	−	5.28004i	−4.59910	+	2.41832i	−19.5710	+	6.35901i	8.15962	+	4.15754i	16.6150	+	22.2610i	21.9253	−	5.26380i	30.5268	+	59.9123i	15.3034	−	22.2442i	15.1283	−	46.5600i
11.3	−0.835555	−	5.27549i	3.60447	+	3.74270i	−19.5242	+	6.34379i	8.47163	+	4.31651i	16.7328	−	22.1426i	−30.3213	+	7.27950i	30.3811	+	59.6263i	−1.01561	+	26.9809i	15.6932	−	48.2987i
11.4	−0.726677	−	4.58806i	4.74370	−	2.12070i	−12.9137	+	4.19593i	−11.4469	−	5.83249i	−13.1770	−	20.2233i	−5.48634	+	1.31715i	11.7641	+	23.0883i	18.0053	−	20.1199i	−18.4416	+	56.7574i
11.5	−0.694300	−	4.38364i	3.43322	+	3.90039i	−11.1258	+	3.61499i	−5.24873	−	2.67436i	14.7142	−	17.7580i	32.1977	−	7.72998i	7.45192	+	14.6252i	−3.42607	+	26.7817i	−8.07923	+	24.8653i
11.6	−0.690825	−	4.36170i	3.84940	−	3.49029i	−10.9387	+	3.55420i	18.3722	+	9.36109i	−17.8828	−	14.3787i	15.4869	−	3.71808i	7.02025	+	13.7780i	2.63580	−	26.8710i	28.1383	−	86.6007i
11.7	−0.644983	−	4.07226i	−5.19602	+	0.0370747i	−8.55886	+	2.78094i	−13.7235	−	6.99249i	3.50232	+	21.1356i	−7.83158	+	1.88020i	1.87053	+	3.67112i	26.9973	−	0.385282i	−19.6238	+	60.3959i
11.8	−0.582563	−	3.67816i	−3.01671	+	4.23077i	−5.58099	+	1.81338i	9.09546	+	4.63437i	17.3189	+	8.63125i	−18.3975	+	4.41684i	−3.60415	−	7.07353i	−8.79888	−	25.5261i	11.7473	−	36.1543i
11.9	−0.582223	−	3.67601i	−2.94904	−	4.27822i	−5.56560	+	1.80837i	7.57669	+	3.86051i	−14.0098	+	13.3316i	−19.6879	+	4.72665i	−3.62939	−	7.12308i	−9.60632	+	25.2333i	9.77997	−	30.0996i
11.10	−0.581474	−	3.67129i	−0.528850	+	5.16917i	−5.53177	+	1.79738i	−5.62209	−	2.86460i	19.2850	−	1.06418i	2.14736	−	0.515534i	−3.68476	−	7.23174i	−26.4406	−	5.46744i	−7.24765	+	22.3060i
11.11	−0.436277	−	2.75454i	−4.54802	−	2.51306i	0.211278	−	0.0686484i	10.6887	+	5.44616i	−4.93814	+	13.6241i	26.5383	−	6.37129i	−10.4103	−	20.4313i	14.3691	+	22.8589i	10.3385	−	31.8185i
11.12	−0.401056	−	2.53217i	−0.150407	−	5.19398i	1.35741	−	0.441049i	−13.5707	−	6.91461i	−13.0917	+	2.46393i	28.2337	−	6.77831i	−10.9725	−	21.5347i	−26.9548	+	1.56242i	−12.0664	+	37.1364i
11.13	−0.383388	−	2.42062i	5.03463	+	1.28551i	1.89606	−	0.616068i	4.78057	+	2.43582i	1.18152	−	12.6797i	−1.98319	+	0.476121i	−11.1193	−	21.8228i	23.6949	+	12.9442i	4.06338	−	12.5058i
11.14	−0.351172	−	2.21721i	2.23211	−	4.69230i	2.81574	−	0.914891i	0.706842	+	0.360154i	−11.1877	−	3.30127i	−22.9599	+	5.51217i	−11.1704	−	21.9232i	−17.0353	−	20.9475i	0.550315	−	1.69369i
11.15	−0.257875	−	1.62816i	3.33274	+	3.98658i	5.02405	−	1.63241i	−17.0210	−	8.67265i	5.63137	−	6.45428i	−33.3948	+	8.01738i	−9.94047	−	19.5093i	−4.78569	+	26.5725i	−9.73117	+	29.9495i
11.16	−0.235127	−	1.48453i	1.22597	+	5.04946i	5.45990	−	1.77403i	16.9533	+	8.63814i	7.20783	−	3.00725i	5.76772	−	1.38471i	−9.37629	−	18.4020i	−23.9940	+	12.3809i	8.83743	−	27.1988i
11.17	−0.148451	−	0.937284i	−5.16127	+	0.601035i	6.75199	−	2.19385i	−4.71061	−	2.40018i	1.32954	+	4.74836i	2.90236	−	0.696796i	−6.50518	−	12.7671i	26.2775	−	6.20421i	−1.55035	+	4.77149i
11.18	−0.147099	−	0.928746i	5.08366	−	1.07535i	6.76752	−	2.19890i	−1.34227	−	0.683919i	−1.74653	−	4.56325i	13.9329	−	3.34500i	−6.45290	−	12.6645i	24.6872	−	10.9334i	−0.437741	+	1.34723i
11.19	−0.104336	−	0.658755i	−2.64521	+	4.47246i	7.18538	−	2.33467i	−10.5304	−	5.36553i	3.22224	+	1.27590i	13.0903	−	3.14270i	−4.71004	−	9.24398i	−13.0058	−	23.6612i	−2.43586	+	7.49680i
11.20	−0.0667841	−	0.421658i	−5.09569	+	1.01682i	7.43512	−	2.41582i	12.3403	+	6.28769i	0.769063	+	2.08073i	−22.4982	+	5.40133i	−3.06572	−	6.01681i	24.9321	−	10.3628i	1.82712	−	5.62330i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
41.h	odd	40	1	inner
123.o	even	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	123.4.o.a	✓	640
3.b	odd	2	1	inner	123.4.o.a	✓	640
41.h	odd	40	1	inner	123.4.o.a	✓	640
123.o	even	40	1	inner	123.4.o.a	✓	640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
123.4.o.a	✓	640	1.a	even	1	1	trivial
123.4.o.a	✓	640	3.b	odd	2	1	inner
123.4.o.a	✓	640	41.h	odd	40	1	inner
123.4.o.a	✓	640	123.o	even	40	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(123, [\chi])\).