Newform orbit 756.2.bb.a

• Label: 756.2.bb.a
• Level: $756$
• Weight: $2$
• Character orbit: 756.bb
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 252)
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{4} + 6 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
611.1	−1.41415	+	0.0135638i		0		1.99963	−	0.0383626i	2.92354	−	1.68790i		0		0.479205	+	2.60199i	−2.82726	+	0.0813731i		0		−4.11142	+	2.42660i
611.2	−1.41013	+	0.107328i		0		1.97696	−	0.302694i	0.996165	−	0.575136i		0		−2.47311	−	0.940076i	−2.75529	+	0.639023i		0		−1.34300	+	0.917936i
611.3	−1.40608	−	0.151434i		0		1.95414	+	0.425857i	0.941391	−	0.543513i		0		2.38059	−	1.15446i	−2.68319	−	0.894712i		0		−1.40598	+	0.621665i
611.4	−1.35995	−	0.387995i		0		1.69892	+	1.05531i	−3.19234	+	1.84310i		0		−2.29354	+	1.31898i	−1.90099	−	2.09434i		0		5.05653	−	1.26790i
611.5	−1.30394	−	0.547479i		0		1.40053	+	1.42776i	1.40549	−	0.811459i		0		−1.53742	−	2.15322i	−1.04455	−	2.62848i		0		−2.27693	+	0.288621i
611.6	−1.28108	+	0.599020i		0		1.28235	−	1.53479i	−0.426828	+	0.246429i		0		−1.23286	−	2.34095i	−0.723425	+	2.73435i		0		0.399186	−	0.571375i
611.7	−1.26657	+	0.629119i		0		1.20842	−	1.59365i	−2.83919	+	1.63921i		0		−0.591598	+	2.57876i	−0.527954	+	2.77872i		0		2.56478	−	3.86236i
611.8	−1.24253	+	0.675360i		0		1.08778	−	1.67831i	1.44577	−	0.834716i		0		0.369619	+	2.61981i	−0.218133	+	2.82000i		0		−1.23268	+	2.01358i
611.9	−1.12310	−	0.859445i		0		0.522707	+	1.93049i	−2.12868	+	1.22899i		0		2.32866	−	1.25593i	1.07210	−	2.61737i		0		3.44698	+	0.449201i
611.10	−1.08038	−	0.912567i		0		0.334444	+	1.97184i	−0.934343	+	0.539443i		0		−0.423599	+	2.61162i	1.43811	−	2.43554i		0		1.50172	+	0.269846i
611.11	−1.07292	−	0.921328i		0		0.302311	+	1.97702i	2.48793	−	1.43641i		0		−2.33229	+	1.24916i	1.49713	−	2.39971i		0		−3.99275	−	0.751051i
611.12	−0.966497	+	1.03242i		0		−0.131768	−	1.99565i	1.42201	−	0.820999i		0		2.57988	−	0.586703i	2.18770	+	1.79275i		0		−0.526757	+	2.26160i
611.13	−0.803010	−	1.16412i		0		−0.710349	+	1.86960i	1.33842	−	0.772734i		0		2.59837	+	0.498458i	2.74686	−	0.674378i		0		−1.97432	−	0.937562i
611.14	−0.801193	+	1.16537i		0		−0.716179	−	1.86737i	−2.71842	+	1.56948i		0		−2.39714	−	1.11970i	2.74998	+	0.661514i		0		0.348953	−	4.42542i
611.15	−0.700426	+	1.22858i		0		−1.01881	−	1.72105i	−1.88207	+	1.08661i		0		2.38085	+	1.15394i	2.82805	−	0.0462147i		0		−0.0167398	−	3.07336i
611.16	−0.692480	−	1.23307i		0		−1.04094	+	1.70776i	−1.47575	+	0.852024i		0		0.540398	−	2.58997i	2.82662	+	0.100974i		0		2.07253	+	1.22970i
611.17	−0.614155	+	1.27390i		0		−1.24563	−	1.56474i	3.69227	−	2.13173i		0		−1.08153	−	2.41460i	2.75833	−	0.625806i		0		0.447980	+	6.01278i
611.18	−0.301845	+	1.38163i		0		−1.81778	−	0.834072i	−0.434289	+	0.250737i		0		0.412779	−	2.61335i	1.70106	−	2.25973i		0		−0.215337	−	0.675709i
611.19	−0.298214	−	1.38241i		0		−1.82214	+	0.824510i	−2.82413	+	1.63051i		0		−2.14986	−	1.54211i	1.68320	+	2.27307i		0		3.09624	+	3.41787i
611.20	−0.237258	+	1.39417i		0		−1.88742	−	0.661555i	0.722678	−	0.417239i		0		−2.04221	+	1.68208i	1.37012	−	2.47442i		0		0.410240	+	1.10653i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
63.j	odd	6	1	inner
252.bb	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.bb.a		88
3.b	odd	2	1		252.2.bb.a	yes	88
4.b	odd	2	1	inner	756.2.bb.a		88
7.c	even	3	1		756.2.o.a		88
9.c	even	3	1		252.2.o.a	✓	88
9.d	odd	6	1		756.2.o.a		88
12.b	even	2	1		252.2.bb.a	yes	88
21.h	odd	6	1		252.2.o.a	✓	88
28.g	odd	6	1		756.2.o.a		88
36.f	odd	6	1		252.2.o.a	✓	88
36.h	even	6	1		756.2.o.a		88
63.h	even	3	1		252.2.bb.a	yes	88
63.j	odd	6	1	inner	756.2.bb.a		88
84.n	even	6	1		252.2.o.a	✓	88
252.u	odd	6	1		252.2.bb.a	yes	88
252.bb	even	6	1	inner	756.2.bb.a		88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.o.a	✓	88	9.c	even	3	1
252.2.o.a	✓	88	21.h	odd	6	1
252.2.o.a	✓	88	36.f	odd	6	1
252.2.o.a	✓	88	84.n	even	6	1
252.2.bb.a	yes	88	3.b	odd	2	1
252.2.bb.a	yes	88	12.b	even	2	1
252.2.bb.a	yes	88	63.h	even	3	1
252.2.bb.a	yes	88	252.u	odd	6	1
756.2.o.a		88	7.c	even	3	1
756.2.o.a		88	9.d	odd	6	1
756.2.o.a		88	28.g	odd	6	1
756.2.o.a		88	36.h	even	6	1
756.2.bb.a		88	1.a	even	1	1	trivial
756.2.bb.a		88	4.b	odd	2	1	inner
756.2.bb.a		88	63.j	odd	6	1	inner
756.2.bb.a		88	252.bb	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).