Newform orbit 138.4.e.b

• Label: 138.4.e.b
• 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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	−3.80011	+	8.32109i	0.853889	+	5.93893i	2.38181	+	0.699363i	6.73003	−	4.32513i	3.73874	+	8.18669i	17.5544	−	5.15444i
13.2	−1.30972	−	1.51150i	−2.52376	−	1.62192i	−0.569259	+	3.95929i	2.02822	−	4.44117i	0.853889	+	5.93893i	−16.2703	−	4.77739i	6.73003	−	4.32513i	3.73874	+	8.18669i	−9.36923	+	2.75105i
13.3	−1.30972	−	1.51150i	−2.52376	−	1.62192i	−0.569259	+	3.95929i	6.04376	−	13.2340i	0.853889	+	5.93893i	29.2734	+	8.59546i	6.73003	−	4.32513i	3.73874	+	8.18669i	−27.9188	+	8.19771i
25.1	1.68251	+	1.08128i	0.426945	+	2.96946i	1.66166	+	3.63853i	−20.3614	−	5.97866i	−2.49249	+	5.45779i	5.13322	−	5.92406i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	−27.7936	−	32.0756i
25.2	1.68251	+	1.08128i	0.426945	+	2.96946i	1.66166	+	3.63853i	2.07213	+	0.608431i	−2.49249	+	5.45779i	−14.7194	+	16.9870i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	2.82848	+	3.26424i
25.3	1.68251	+	1.08128i	0.426945	+	2.96946i	1.66166	+	3.63853i	16.0712	+	4.71893i	−2.49249	+	5.45779i	17.1563	−	19.7994i	−1.13852	+	7.91857i	−8.63544	+	2.53559i	21.9374	+	25.3171i
31.1	−0.284630	+	1.97964i	−1.24625	−	2.72890i	−3.83797	−	1.12693i	−11.4557	−	13.2206i	5.75696	−	1.69040i	16.3172	+	10.4864i	3.32332	−	7.27706i	−5.89375	+	6.80175i	29.4328	−	18.9153i
31.2	−0.284630	+	1.97964i	−1.24625	−	2.72890i	−3.83797	−	1.12693i	1.48009	+	1.70811i	5.75696	−	1.69040i	3.78550	+	2.43279i	3.32332	−	7.27706i	−5.89375	+	6.80175i	−3.80273	+	2.44386i
31.3	−0.284630	+	1.97964i	−1.24625	−	2.72890i	−3.83797	−	1.12693i	11.2596	+	12.9943i	5.75696	−	1.69040i	−30.1361	−	19.3673i	3.32332	−	7.27706i	−5.89375	+	6.80175i	−28.9289	+	18.5915i
49.1	−0.284630	−	1.97964i	−1.24625	+	2.72890i	−3.83797	+	1.12693i	−11.4557	+	13.2206i	5.75696	+	1.69040i	16.3172	−	10.4864i	3.32332	+	7.27706i	−5.89375	−	6.80175i	29.4328	+	18.9153i
49.2	−0.284630	−	1.97964i	−1.24625	+	2.72890i	−3.83797	+	1.12693i	1.48009	−	1.70811i	5.75696	+	1.69040i	3.78550	−	2.43279i	3.32332	+	7.27706i	−5.89375	−	6.80175i	−3.80273	−	2.44386i
49.3	−0.284630	−	1.97964i	−1.24625	+	2.72890i	−3.83797	+	1.12693i	11.2596	−	12.9943i	5.75696	+	1.69040i	−30.1361	+	19.3673i	3.32332	+	7.27706i	−5.89375	−	6.80175i	−28.9289	−	18.5915i
55.1	−1.91899	+	0.563465i	1.96458	+	2.26725i	3.36501	−	2.16256i	−1.40880	−	9.79842i	−5.04752	−	3.24384i	−9.01039	+	19.7300i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	8.22454	+	18.0092i
55.2	−1.91899	+	0.563465i	1.96458	+	2.26725i	3.36501	−	2.16256i	−0.966187	−	6.71998i	−5.04752	−	3.24384i	5.61851	−	12.3028i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	5.64057	+	12.3511i
55.3	−1.91899	+	0.563465i	1.96458	+	2.26725i	3.36501	−	2.16256i	3.11759	+	21.6833i	−5.04752	−	3.24384i	−0.846192	+	1.85290i	−5.23889	+	6.04600i	−1.28083	+	8.90839i	−18.2004	−	39.8533i
73.1	0.830830	+	1.81926i	2.87848	−	0.845198i	−2.61944	+	3.02300i	−11.8567	−	7.61986i	3.92916	+	4.53450i	−2.71532	−	18.8855i	−7.67594	−	2.25386i	7.57128	−	4.86577i	4.01160	−	27.9013i
73.2	0.830830	+	1.81926i	2.87848	−	0.845198i	−2.61944	+	3.02300i	−8.49581	−	5.45993i	3.92916	+	4.53450i	4.18194	+	29.0860i	−7.67594	−	2.25386i	7.57128	−	4.86577i	2.87447	−	19.9924i
73.3	0.830830	+	1.81926i	2.87848	−	0.845198i	−2.61944	+	3.02300i	13.2722	+	8.52953i	3.92916	+	4.53450i	0.849820	+	5.91063i	−7.67594	−	2.25386i	7.57128	−	4.86577i	−4.49052	+	31.2322i
85.1	−1.30972	+	1.51150i	−2.52376	+	1.62192i	−0.569259	−	3.95929i	−3.80011	−	8.32109i	0.853889	−	5.93893i	2.38181	−	0.699363i	6.73003	+	4.32513i	3.73874	−	8.18669i	17.5544	+	5.15444i
85.2	−1.30972	+	1.51150i	−2.52376	+	1.62192i	−0.569259	−	3.95929i	2.02822	+	4.44117i	0.853889	−	5.93893i	−16.2703	+	4.77739i	6.73003	+	4.32513i	3.73874	−	8.18669i	−9.36923	−	2.75105i

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^30 + 6*T5^29 + 65*T5^28 + 6409*T5^27 + 5482*T5^26 - 1016815*T5^25 + 38176278*T5^24 + 432306806*T5^23 - 3451756324*T5^22 - 163702456662*T5^21 + 5703265557290*T5^20 - 9190794675582*T5^19 + 246925166168021*T5^18 - 284792807579417*T5^17 + 204708855646739237*T5^16 - 689048429477561000*T5^15 + 32828448481104139698*T5^14 + 5879146710816920602*T5^13 + 8039465444777039274136*T5^12 + 60921089620793429718032*T5^11 + 1074407353612765250339873*T5^10 + 3404409154744363958003455*T5^9 + 51766587245875316742515857*T5^8 - 135946190209469616369584212*T5^7 + 2295153377974830145959734019*T5^6 - 13502403686497294021324074105*T5^5 + 84493842629376457065888148777*T5^4 - 391741971202251582552550247435*T5^3 + 1101978435402245415521274723015*T5^2 - 1750527293652738457832708716917*T5 + 1256711066253834467096909896369