Newform orbit 385.2.u.a

• Label: 385.2.u.a
• Level: $385$
• Weight: $2$
• Character orbit: 385.u
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 28 q^{4} + 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
131.1	−2.37836	−	1.37315i	1.15752	−	0.668297i	2.77106	+	4.79962i	−0.866025	−	0.500000i	−3.67068			0.812731	+	2.51783i		−	9.72771i	−0.606758	+	1.05094i	1.37315	+	2.37836i
131.2	−2.30798	−	1.33251i	−2.59904	+	1.50055i	2.55119	+	4.41878i	0.866025	+	0.500000i	7.99803			−2.40255	−	1.10803i		−	8.26791i	3.00332	−	5.20191i	−1.33251	−	2.30798i
131.3	−2.15906	−	1.24653i	−0.751689	+	0.433988i	2.10768	+	3.65061i	0.866025	+	0.500000i	2.16392			2.43805	+	1.02758i		−	5.52304i	−1.12331	+	1.94563i	−1.24653	−	2.15906i
131.4	−1.99920	−	1.15424i	−1.03932	+	0.600054i	1.66454	+	2.88308i	−0.866025	−	0.500000i	2.77043			0.324545	−	2.62577i		−	3.06818i	−0.779869	+	1.35077i	1.15424	+	1.99920i
131.5	−1.94293	−	1.12175i	1.59571	−	0.921281i	1.51666	+	2.62692i	0.866025	+	0.500000i	−4.13380			1.96415	−	1.77260i		−	2.31824i	0.197519	−	0.342112i	−1.12175	−	1.94293i
131.6	−1.63802	−	0.945714i	2.67129	−	1.54227i	0.788749	+	1.36615i	−0.866025	−	0.500000i	−5.83418			2.61979	−	0.369696i			0.799132i	3.25720	−	5.64163i	0.945714	+	1.63802i
131.7	−1.62822	−	0.940054i	1.55404	−	0.897223i	0.767402	+	1.32918i	0.866025	+	0.500000i	−3.37375			−1.46014	+	2.20635i			0.874620i	0.110018	−	0.190557i	−0.940054	−	1.62822i
131.8	−1.25601	−	0.725160i	0.0469173	−	0.0270877i	0.0517129	+	0.0895694i	0.866025	+	0.500000i	−0.0785716			−1.78723	−	1.95085i			2.75064i	−1.49853	+	2.59553i	−0.725160	−	1.25601i
131.9	−1.09163	−	0.630254i	0.910425	−	0.525634i	−0.205560	−	0.356040i	−0.866025	−	0.500000i	−1.32513			−0.348593	+	2.62269i			3.03924i	−0.947417	+	1.64097i	0.630254	+	1.09163i
131.10	−0.986600	−	0.569614i	−2.80195	+	1.61770i	−0.351080	−	0.608088i	−0.866025	−	0.500000i	3.68587			0.453897	−	2.60653i			3.07838i	3.73394	−	6.46737i	0.569614	+	0.986600i
131.11	−0.928810	−	0.536249i	−0.590583	+	0.340973i	−0.424874	−	0.735903i	−0.866025	−	0.500000i	0.731387			2.31298	+	1.28457i			3.05635i	−1.26747	+	2.19533i	0.536249	+	0.928810i
131.12	−0.738366	−	0.426296i	−2.41904	+	1.39664i	−0.636544	−	1.10253i	0.866025	+	0.500000i	2.38152			−2.05580	+	1.66544i			2.79061i	2.40118	−	4.15897i	−0.426296	−	0.738366i
131.13	−0.737911	−	0.426033i	2.01275	−	1.16206i	−0.636992	−	1.10330i	−0.866025	−	0.500000i	−1.98030			−2.08977	−	1.62260i			2.78965i	1.20077	−	2.07979i	0.426033	+	0.737911i
131.14	−0.205395	−	0.118585i	−1.45411	+	0.839530i	−0.971875	−	1.68334i	−0.866025	−	0.500000i	0.398221			−2.59733	−	0.503865i			0.935337i	−0.0903804	+	0.156543i	0.118585	+	0.205395i
131.15	−0.0852978	−	0.0492467i	−0.752327	+	0.434356i	−0.995150	−	1.72365i	0.866025	+	0.500000i	0.0855624			−1.01809	+	2.44202i			0.393018i	−1.12267	+	1.94452i	−0.0492467	−	0.0852978i
131.16	−0.0679871	−	0.0392524i	2.45941	−	1.41994i	−0.996918	−	1.72671i	0.866025	+	0.500000i	−0.222944			2.62999	+	0.288317i			0.313535i	2.53247	−	4.38636i	−0.0392524	−	0.0679871i
131.17	0.0679871	+	0.0392524i	2.45941	−	1.41994i	−0.996918	−	1.72671i	0.866025	+	0.500000i	0.222944			−2.62999	−	0.288317i		−	0.313535i	2.53247	−	4.38636i	0.0392524	+	0.0679871i
131.18	0.0852978	+	0.0492467i	−0.752327	+	0.434356i	−0.995150	−	1.72365i	0.866025	+	0.500000i	−0.0855624			1.01809	−	2.44202i		−	0.393018i	−1.12267	+	1.94452i	0.0492467	+	0.0852978i
131.19	0.205395	+	0.118585i	−1.45411	+	0.839530i	−0.971875	−	1.68334i	−0.866025	−	0.500000i	−0.398221			2.59733	+	0.503865i		−	0.935337i	−0.0903804	+	0.156543i	−0.118585	−	0.205395i
131.20	0.737911	+	0.426033i	2.01275	−	1.16206i	−0.636992	−	1.10330i	−0.866025	−	0.500000i	1.98030			2.08977	+	1.62260i		−	2.78965i	1.20077	−	2.07979i	−0.426033	−	0.737911i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
11.b	odd	2	1	inner
77.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	385.2.u.a	✓	64
7.d	odd	6	1	inner	385.2.u.a	✓	64
11.b	odd	2	1	inner	385.2.u.a	✓	64
77.i	even	6	1	inner	385.2.u.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
385.2.u.a	✓	64	1.a	even	1	1	trivial
385.2.u.a	✓	64	7.d	odd	6	1	inner
385.2.u.a	✓	64	11.b	odd	2	1	inner
385.2.u.a	✓	64	77.i	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(385, [\chi])\).