Newform orbit 381.3.h.a

• Label: 381.3.h.a
• Level: $381$
• Weight: $3$
• Character orbit: 381.h
• Analytic conductor: $10.381$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	381.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3814980721\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 2 q^{2} - 63 q^{3} + 86 q^{4} - 3 q^{6} + 30 q^{7} + 28 q^{8} + 63 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
235.1	−3.93967			−1.50000	−	0.866025i	11.5210					0.245375i	5.90951	+	3.41186i	4.02042	+	2.32119i	−29.6304			1.50000	+	2.59808i		−	0.966698i
235.2	−3.27862			−1.50000	−	0.866025i	6.74932				−	2.44544i	4.91792	+	2.83936i	−1.66076	−	0.958843i	−9.01396			1.50000	+	2.59808i			8.01764i
235.3	−3.06107			−1.50000	−	0.866025i	5.37013					7.06047i	4.59160	+	2.65096i	−8.16126	−	4.71191i	−4.19406			1.50000	+	2.59808i		−	21.6126i
235.4	−3.06047			−1.50000	−	0.866025i	5.36645					4.88605i	4.59070	+	2.65044i	8.45105	+	4.87922i	−4.18199			1.50000	+	2.59808i		−	14.9536i
235.5	−2.69015			−1.50000	−	0.866025i	3.23693				−	5.83417i	4.03523	+	2.32974i	−1.51557	−	0.875012i	2.05277			1.50000	+	2.59808i			15.6948i
235.6	−1.80793			−1.50000	−	0.866025i	−0.731399				−	3.62152i	2.71189	+	1.56571i	6.52168	+	3.76529i	8.55403			1.50000	+	2.59808i			6.54744i
235.7	−1.42604			−1.50000	−	0.866025i	−1.96641					7.03999i	2.13906	+	1.23499i	7.13697	+	4.12053i	8.50834			1.50000	+	2.59808i		−	10.0393i
235.8	−1.38915			−1.50000	−	0.866025i	−2.07025				−	2.98064i	2.08373	+	1.20304i	−3.19454	−	1.84437i	8.43251			1.50000	+	2.59808i			4.14057i
235.9	−1.22733			−1.50000	−	0.866025i	−2.49365					5.12021i	1.84100	+	1.06290i	−5.55317	−	3.20612i	7.96988			1.50000	+	2.59808i		−	6.28421i
235.10	−0.394097			−1.50000	−	0.866025i	−3.84469				−	7.24268i	0.591146	+	0.341298i	10.1127	+	5.83854i	3.09157			1.50000	+	2.59808i			2.85432i
235.11	0.352710			−1.50000	−	0.866025i	−3.87560					0.563566i	−0.529064	−	0.305455i	−1.07479	−	0.620532i	−2.77780			1.50000	+	2.59808i			0.198775i
235.12	0.613322			−1.50000	−	0.866025i	−3.62384					4.76175i	−0.919984	−	0.531153i	−6.21008	−	3.58539i	−4.67587			1.50000	+	2.59808i			2.92049i
235.13	0.642642			−1.50000	−	0.866025i	−3.58701				−	9.34072i	−0.963964	−	0.556545i	−5.63829	−	3.25527i	−4.87573			1.50000	+	2.59808i		−	6.00274i
235.14	1.07563			−1.50000	−	0.866025i	−2.84302					5.19427i	−1.61344	−	0.931521i	6.24536	+	3.60576i	−7.36055			1.50000	+	2.59808i			5.58710i
235.15	2.04912			−1.50000	−	0.866025i	0.198891				−	0.581057i	−3.07368	−	1.77459i	−2.37943	−	1.37376i	−7.78893			1.50000	+	2.59808i		−	1.19066i
235.16	2.21493			−1.50000	−	0.866025i	0.905904				−	0.500159i	−3.32239	−	1.91818i	9.06847	+	5.23569i	−6.85320			1.50000	+	2.59808i		−	1.10781i
235.17	2.42993			−1.50000	−	0.866025i	1.90458				−	5.08168i	−3.64490	−	2.10438i	0.432202	+	0.249532i	−5.09174			1.50000	+	2.59808i		−	12.3481i
235.18	3.07277			−1.50000	−	0.866025i	5.44189					6.37145i	−4.60915	−	2.66109i	5.32719	+	3.07565i	4.43060			1.50000	+	2.59808i			19.5780i
235.19	3.18197			−1.50000	−	0.866025i	6.12491				−	1.68960i	−4.77295	−	2.75566i	−11.5695	−	6.67968i	6.76139			1.50000	+	2.59808i		−	5.37625i
235.20	3.72212			−1.50000	−	0.866025i	9.85416				−	5.60105i	−5.58318	−	3.22345i	7.67665	+	4.43212i	21.7899			1.50000	+	2.59808i		−	20.8478i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
127.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	381.3.h.a	✓	42
127.d	odd	6	1	inner	381.3.h.a	✓	42

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
381.3.h.a	✓	42	1.a	even	1	1	trivial
381.3.h.a	✓	42	127.d	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^21 - T2^20 - 63*T2^19 + 56*T2^18 + 1666*T2^17 - 1293*T2^16 - 24113*T2^15 + 16007*T2^14 + 208789*T2^13 - 115557*T2^12 - 1110942*T2^11 + 497400*T2^10 + 3596618*T2^9 - 1277258*T2^8 - 6779737*T2^7 + 2017239*T2^6 + 6789644*T2^5 - 2095224*T2^4 - 3050361*T2^3 + 1149586*T2^2 + 338994*T2 - 132651