Newform orbit 138.4.e.a

• Label: 138.4.e.a
• 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} - 9 q^{3} - 12 q^{4} - 20 q^{5} - 18 q^{6} - 10 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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	−2.73603	+	5.99106i	−0.853889	−	5.93893i	−4.46365	−	1.31065i	6.73003	−	4.32513i	3.73874	+	8.18669i	12.6389	−	3.71112i
13.2	−1.30972	−	1.51150i	2.52376	+	1.62192i	−0.569259	+	3.95929i	0.452038	−	0.989824i	−0.853889	−	5.93893i	6.64594	+	1.95142i	6.73003	−	4.32513i	3.73874	+	8.18669i	−2.08816	+	0.613140i
13.3	−1.30972	−	1.51150i	2.52376	+	1.62192i	−0.569259	+	3.95929i	7.75778	−	16.9872i	−0.853889	−	5.93893i	−16.8919	−	4.95990i	6.73003	−	4.32513i	3.73874	+	8.18669i	−35.8366	+	10.5226i
25.1	1.68251	+	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	−10.4597	−	3.07126i	2.49249	−	5.45779i	11.4417	−	13.2044i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	−14.2777	−	16.4773i
25.2	1.68251	+	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	−10.3127	−	3.02807i	2.49249	−	5.45779i	−14.8488	+	17.1364i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	−14.0769	−	16.2456i
25.3	1.68251	+	1.08128i	−0.426945	−	2.96946i	1.66166	+	3.63853i	15.5804	+	4.57481i	2.49249	−	5.45779i	−6.98605	+	8.06233i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	21.2674	+	24.5439i
31.1	−0.284630	+	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	−13.9794	−	16.1331i	−5.75696	+	1.69040i	18.5923	+	11.9485i	3.32332	−	7.27706i	−5.89375	+	6.80175i	35.9168	−	23.0823i
31.2	−0.284630	+	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	−1.58076	−	1.82430i	−5.75696	+	1.69040i	−19.6617	−	12.6358i	3.32332	−	7.27706i	−5.89375	+	6.80175i	4.06139	−	2.61010i
31.3	−0.284630	+	1.97964i	1.24625	+	2.72890i	−3.83797	−	1.12693i	3.37498	+	3.89494i	−5.75696	+	1.69040i	15.9890	+	10.2755i	3.32332	−	7.27706i	−5.89375	+	6.80175i	−8.67120	+	5.57264i
49.1	−0.284630	−	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	−13.9794	+	16.1331i	−5.75696	−	1.69040i	18.5923	−	11.9485i	3.32332	+	7.27706i	−5.89375	−	6.80175i	35.9168	+	23.0823i
49.2	−0.284630	−	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	−1.58076	+	1.82430i	−5.75696	−	1.69040i	−19.6617	+	12.6358i	3.32332	+	7.27706i	−5.89375	−	6.80175i	4.06139	+	2.61010i
49.3	−0.284630	−	1.97964i	1.24625	−	2.72890i	−3.83797	+	1.12693i	3.37498	−	3.89494i	−5.75696	−	1.69040i	15.9890	−	10.2755i	3.32332	+	7.27706i	−5.89375	−	6.80175i	−8.67120	−	5.57264i
55.1	−1.91899	+	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	−2.21950	−	15.4369i	5.04752	+	3.24384i	4.21769	−	9.23545i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	12.9574	+	28.3727i
55.2	−1.91899	+	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	−0.141588	−	0.984766i	5.04752	+	3.24384i	−6.60477	+	14.4624i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	0.826586	+	1.80997i
55.3	−1.91899	+	0.563465i	−1.96458	−	2.26725i	3.36501	−	2.16256i	1.68648	+	11.7297i	5.04752	+	3.24384i	8.69247	−	19.0339i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	−9.84560	−	21.5588i
73.1	0.830830	+	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	−10.0675	−	6.46998i	−3.92916	−	4.53450i	2.57211	+	17.8894i	−7.67594	−	2.25386i	7.57128	−	4.86577i	3.40623	−	23.6909i
73.2	0.830830	+	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	0.656703	+	0.422037i	−3.92916	−	4.53450i	−4.63598	−	32.2439i	−7.67594	−	2.25386i	7.57128	−	4.86577i	−0.222189	+	1.54536i
73.3	0.830830	+	1.81926i	−2.87848	+	0.845198i	−2.61944	+	3.02300i	11.9888	+	7.70474i	−3.92916	−	4.53450i	0.941694	+	6.54963i	−7.67594	−	2.25386i	7.57128	−	4.86577i	−4.05629	+	28.2122i
85.1	−1.30972	+	1.51150i	2.52376	−	1.62192i	−0.569259	−	3.95929i	−2.73603	−	5.99106i	−0.853889	+	5.93893i	−4.46365	+	1.31065i	6.73003	+	4.32513i	3.73874	−	8.18669i	12.6389	+	3.71112i
85.2	−1.30972	+	1.51150i	2.52376	−	1.62192i	−0.569259	−	3.95929i	0.452038	+	0.989824i	−0.853889	+	5.93893i	6.64594	−	1.95142i	6.73003	+	4.32513i	3.73874	−	8.18669i	−2.08816	−	0.613140i

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.a	✓	30
23.c	even	11	1	inner	138.4.e.a	✓	30

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^30 + 20*T5^29 + 333*T5^28 + 1351*T5^27 + 64930*T5^26 - 217335*T5^25 - 19997534*T5^24 - 71804068*T5^23 + 1906786500*T5^22 + 108295846524*T5^21 + 2442466437490*T5^20 + 10969021274834*T5^19 - 98156844432163*T5^18 + 754804065669827*T5^17 + 37664579803444009*T5^16 + 853637499463796120*T5^15 + 16547199535055386564*T5^14 + 177031292598656025236*T5^13 + 1047231172724713121974*T5^12 + 4436170055329721231454*T5^11 + 19543810313116389021609*T5^10 + 117990249939583511057035*T5^9 + 810698898231135082085507*T5^8 + 1063210845513491646652510*T5^7 + 1754889314516135889274221*T5^6 - 1790105640197102598519247*T5^5 + 6657009335335824022465911*T5^4 - 7019883299399089569009359*T5^3 + 7601016814659062567705827*T5^2 - 5338684698919489632979995*T5 + 2740393092436069619045641