Newform orbit 385.2.o.b

• Label: 385.2.o.b
• Level: $385$
• Weight: $2$
• Character orbit: 385.o
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
54.1	−1.30077	+	2.25300i	−0.878341	−	1.52133i	−2.38400	−	4.12920i	−1.24339	+	1.85849i	4.57007			2.30616	+	1.29678i	7.20103			−0.0429672	+	0.0744215i	−2.56981	−	5.21881i
54.2	−1.30077	+	2.25300i	0.878341	+	1.52133i	−2.38400	−	4.12920i	0.987807	−	2.00605i	−4.57007			2.30616	+	1.29678i	7.20103			−0.0429672	+	0.0744215i	3.23472	+	4.83493i
54.3	−1.15617	+	2.00254i	−1.18194	−	2.04718i	−1.67346	−	2.89851i	−2.07840	−	0.824775i	5.46608			−2.64218	+	0.137378i	3.11452			−1.29395	+	2.24119i	4.05463	−	3.20851i
54.4	−1.15617	+	2.00254i	1.18194	+	2.04718i	−1.67346	−	2.89851i	−1.75348	−	1.38756i	−5.46608			−2.64218	+	0.137378i	3.11452			−1.29395	+	2.24119i	4.80597	−	1.90716i
54.5	−1.14254	+	1.97894i	−1.52900	−	2.64830i	−1.61081	−	2.79001i	2.14435	−	0.633864i	6.98779			1.82663	−	1.91400i	2.79153			−3.17567	+	5.50043i	−1.19563	+	4.96776i
54.6	−1.14254	+	1.97894i	1.52900	+	2.64830i	−1.61081	−	2.79001i	0.523230	+	2.17399i	−6.98779			1.82663	−	1.91400i	2.79153			−3.17567	+	5.50043i	−4.90002	−	1.44844i
54.7	−0.901007	+	1.56059i	−0.984554	−	1.70530i	−0.623628	−	1.08016i	1.31996	−	1.80491i	3.54836			−0.686066	+	2.55525i	−1.35646			−0.438692	+	0.759836i	1.62742	+	3.68616i
54.8	−0.901007	+	1.56059i	0.984554	+	1.70530i	−0.623628	−	1.08016i	−0.903114	+	2.04558i	−3.54836			−0.686066	+	2.55525i	−1.35646			−0.438692	+	0.759836i	−2.37860	−	3.25247i
54.9	−0.736827	+	1.27622i	−0.269187	−	0.466245i	−0.0858291	−	0.148660i	−1.48744	+	1.66958i	0.793377			−0.997019	−	2.45070i	−2.69434			1.35508	−	2.34706i	−1.03477	−	3.12850i
54.10	−0.736827	+	1.27622i	0.269187	+	0.466245i	−0.0858291	−	0.148660i	0.702179	−	2.12296i	−0.793377			−0.997019	−	2.45070i	−2.69434			1.35508	−	2.34706i	2.19198	+	2.46039i
54.11	−0.559049	+	0.968301i	−1.09998	−	1.90522i	0.374928	+	0.649395i	1.09488	+	1.94968i	2.45977			0.753941	+	2.53605i	−3.07461			−0.919918	+	1.59334i	−2.49997	−	0.0297884i
54.12	−0.559049	+	0.968301i	1.09998	+	1.90522i	0.374928	+	0.649395i	2.23591	−	0.0266420i	−2.45977			0.753941	+	2.53605i	−3.07461			−0.919918	+	1.59334i	−1.22419	+	2.17993i
54.13	−0.408979	+	0.708373i	−1.44107	−	2.49601i	0.665472	+	1.15263i	−1.82905	+	1.28630i	2.35748			2.08705	−	1.62611i	−2.72457			−2.65339	+	4.59580i	−0.163134	−	1.82172i
54.14	−0.408979	+	0.708373i	1.44107	+	2.49601i	0.665472	+	1.15263i	0.199440	−	2.22716i	−2.35748			2.08705	−	1.62611i	−2.72457			−2.65339	+	4.59580i	1.49609	+	1.05214i
54.15	−0.249992	+	0.433000i	−1.24153	−	2.15040i	0.875008	+	1.51556i	−0.407192	−	2.19868i	1.24149			−2.20302	−	1.46517i	−1.87495			−1.58280	+	2.74149i	1.05382	+	0.373339i
54.16	−0.249992	+	0.433000i	1.24153	+	2.15040i	0.875008	+	1.51556i	−2.10771	+	0.746701i	−1.24149			−2.20302	−	1.46517i	−1.87495			−1.58280	+	2.74149i	0.203590	−	1.09931i
54.17	−0.137269	+	0.237757i	−0.351914	−	0.609533i	0.962314	+	1.66678i	2.12913	−	0.683216i	0.193228			2.52423	−	0.792646i	−1.07746			1.25231	−	2.16907i	−0.129825	+	0.600002i
54.18	−0.137269	+	0.237757i	0.351914	+	0.609533i	0.962314	+	1.66678i	0.472885	+	2.18549i	−0.193228			2.52423	−	0.792646i	−1.07746			1.25231	−	2.16907i	−0.584530	−	0.187569i
54.19	0.137269	−	0.237757i	−0.351914	−	0.609533i	0.962314	+	1.66678i	2.12913	−	0.683216i	−0.193228			−2.52423	+	0.792646i	1.07746			1.25231	−	2.16907i	0.129825	−	0.600002i
54.20	0.137269	−	0.237757i	0.351914	+	0.609533i	0.962314	+	1.66678i	0.472885	+	2.18549i	0.193228			−2.52423	+	0.792646i	1.07746			1.25231	−	2.16907i	0.584530	+	0.187569i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.d	odd	6	1	inner
11.b	odd	2	1	inner
35.i	odd	6	1	inner
55.d	odd	2	1	inner
77.i	even	6	1	inner
385.o	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.o.b	✓	72
5.b	even	2	1	inner	385.2.o.b	✓	72
7.d	odd	6	1	inner	385.2.o.b	✓	72
11.b	odd	2	1	inner	385.2.o.b	✓	72
35.i	odd	6	1	inner	385.2.o.b	✓	72
55.d	odd	2	1	inner	385.2.o.b	✓	72
77.i	even	6	1	inner	385.2.o.b	✓	72
385.o	even	6	1	inner	385.2.o.b	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
385.2.o.b	✓	72	1.a	even	1	1	trivial
385.2.o.b	✓	72	5.b	even	2	1	inner
385.2.o.b	✓	72	7.d	odd	6	1	inner
385.2.o.b	✓	72	11.b	odd	2	1	inner
385.2.o.b	✓	72	35.i	odd	6	1	inner
385.2.o.b	✓	72	55.d	odd	2	1	inner
385.2.o.b	✓	72	77.i	even	6	1	inner
385.2.o.b	✓	72	385.o	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^36 + 25*T2^34 + 372*T2^32 + 3657*T2^30 + 26722*T2^28 + 147177*T2^26 + 631988*T2^24 + 2098050*T2^22 + 5451354*T2^20 + 10838222*T2^18 + 16569017*T2^16 + 18674472*T2^14 + 15652678*T2^12 + 8988030*T2^10 + 3653922*T2^8 + 858468*T2^6 + 138987*T2^4 + 9198*T2^2 + 441