Newform orbit 138.4.e.c

• Label: 138.4.e.c
• Level: $138$
• Weight: $4$
• Character orbit: 138.e
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	1.30972	+	1.51150i	2.52376	+	1.62192i	−0.569259	+	3.95929i	−4.97931	+	10.9032i	0.853889	+	5.93893i	10.5666	+	3.10264i	−6.73003	+	4.32513i	3.73874	+	8.18669i	−23.0016	+	6.75388i
13.2	1.30972	+	1.51150i	2.52376	+	1.62192i	−0.569259	+	3.95929i	−3.70472	+	8.11221i	0.853889	+	5.93893i	−24.4006	−	7.16466i	−6.73003	+	4.32513i	3.73874	+	8.18669i	−17.1138	+	5.02505i
13.3	1.30972	+	1.51150i	2.52376	+	1.62192i	−0.569259	+	3.95929i	5.61304	−	12.2908i	0.853889	+	5.93893i	18.2993	+	5.37317i	−6.73003	+	4.32513i	3.73874	+	8.18669i	25.9291	−	7.61348i
25.1	−1.68251	−	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	−11.8265	−	3.47259i	−2.49249	+	5.45779i	13.0481	−	15.0583i	1.13852	−	7.91857i	−8.63544	+	2.53559i	16.1434	+	18.6305i
25.2	−1.68251	−	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	4.71256	+	1.38373i	−2.49249	+	5.45779i	3.70690	−	4.27800i	1.13852	−	7.91857i	−8.63544	+	2.53559i	−6.43272	−	7.42375i
25.3	−1.68251	−	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	19.5175	+	5.73085i	−2.49249	+	5.45779i	−20.5565	+	23.7235i	1.13852	−	7.91857i	−8.63544	+	2.53559i	−26.6416	−	30.7461i
31.1	0.284630	−	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	−7.03109	−	8.11432i	5.75696	−	1.69040i	−5.50559	−	3.53823i	−3.32332	+	7.27706i	−5.89375	+	6.80175i	−18.0647	+	11.6095i
31.2	0.284630	−	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	−2.69854	−	3.11428i	5.75696	−	1.69040i	22.2299	+	14.2863i	−3.32332	+	7.27706i	−5.89375	+	6.80175i	−6.93325	+	4.45573i
31.3	0.284630	−	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	13.1550	+	15.1817i	5.75696	−	1.69040i	−15.4591	−	9.93493i	−3.32332	+	7.27706i	−5.89375	+	6.80175i	33.7986	−	21.7210i
49.1	0.284630	+	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	−7.03109	+	8.11432i	5.75696	+	1.69040i	−5.50559	+	3.53823i	−3.32332	−	7.27706i	−5.89375	−	6.80175i	−18.0647	−	11.6095i
49.2	0.284630	+	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	−2.69854	+	3.11428i	5.75696	+	1.69040i	22.2299	−	14.2863i	−3.32332	−	7.27706i	−5.89375	−	6.80175i	−6.93325	−	4.45573i
49.3	0.284630	+	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	13.1550	−	15.1817i	5.75696	+	1.69040i	−15.4591	+	9.93493i	−3.32332	−	7.27706i	−5.89375	−	6.80175i	33.7986	+	21.7210i
55.1	1.91899	−	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	−1.52360	−	10.5969i	−5.04752	−	3.24384i	0.567261	−	1.24213i	5.23889	−	6.04600i	−1.28083	+	8.90839i	−8.89475	−	19.4768i
55.2	1.91899	−	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	1.45978	+	10.1530i	−5.04752	−	3.24384i	−13.4424	+	29.4347i	5.23889	−	6.04600i	−1.28083	+	8.90839i	8.52215	+	18.6609i
55.3	1.91899	−	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	1.75662	+	12.2176i	−5.04752	−	3.24384i	12.8585	−	28.1562i	5.23889	−	6.04600i	−1.28083	+	8.90839i	10.2551	+	22.4555i
73.1	−0.830830	−	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	−16.6025	−	10.6698i	3.92916	+	4.53450i	3.87830	+	26.9742i	7.67594	+	2.25386i	7.57128	−	4.86577i	−5.61729	+	39.0691i
73.2	−0.830830	−	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	−3.38842	−	2.17761i	3.92916	+	4.53450i	−3.30636	−	22.9962i	7.67594	+	2.25386i	7.57128	−	4.86577i	−1.14644	+	7.97366i
73.3	−0.830830	−	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	3.54028	+	2.27520i	3.92916	+	4.53450i	−0.484390	−	3.36901i	7.67594	+	2.25386i	7.57128	−	4.86577i	1.19782	−	8.33101i
85.1	1.30972	−	1.51150i	2.52376	−	1.62192i	−0.569259	−	3.95929i	−4.97931	−	10.9032i	0.853889	−	5.93893i	10.5666	−	3.10264i	−6.73003	−	4.32513i	3.73874	−	8.18669i	−23.0016	−	6.75388i
85.2	1.30972	−	1.51150i	2.52376	−	1.62192i	−0.569259	−	3.95929i	−3.70472	−	8.11221i	0.853889	−	5.93893i	−24.4006	+	7.16466i	−6.73003	−	4.32513i	3.73874	−	8.18669i	−17.1138	−	5.02505i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	138.4.e.c	✓	30
23.c	even	11	1	inner	138.4.e.c	✓	30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.4.e.c	✓	30	1.a	even	1	1	trivial
138.4.e.c	✓	30	23.c	even	11	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^30 + 4*T5^29 + 35*T5^28 + 797*T5^27 + 81670*T5^26 - 589815*T5^25 + 11136436*T5^24 - 319966486*T5^23 + 1800165068*T5^22 + 70972833116*T5^21 + 3670132038238*T5^20 + 91563644296154*T5^19 + 1807208846723463*T5^18 + 29366775824724737*T5^17 + 408658624013719439*T5^16 + 4681691630911220506*T5^15 + 47451894260511337812*T5^14 + 360214646552140254416*T5^13 + 2451297782934002045448*T5^12 + 7198853815244346758980*T5^11 + 12230884172965372914961*T5^10 - 465178892014889525890251*T5^9 - 1524339022925770206799275*T5^8 - 7369391769920466144008172*T5^7 + 94367203119949086180670247*T5^6 + 473351270987283212380501883*T5^5 + 295160703516875176273727665*T5^4 - 10575446706379775024737191801*T5^3 - 8776436682141645540352766045*T5^2 + 115705359382356294057220074739*T5 + 513804341681452470787777189321