Newform orbit 756.2.be.e

• Label: 756.2.be.e
• Level: $756$
• Weight: $2$
• Character orbit: 756.be
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
107.1	−1.41403	−	0.0228601i		0		1.99895	+	0.0646498i	0.158956	+	0.0917731i		0		1.36747	+	2.26495i	−2.82510	−	0.137113i		0		−0.222670	−	0.133404i
107.2	−1.38037	−	0.307526i		0		1.81086	+	0.849002i	3.20305	+	1.84928i		0		−2.45439	−	0.987919i	−2.23856	−	1.72882i		0		−3.85269	−	3.53771i
107.3	−1.37308	+	0.338582i		0		1.77072	−	0.929803i	−3.53194	−	2.03916i		0		1.92589	−	1.81410i	−2.11654	+	1.87623i		0		5.54007	+	1.60410i
107.4	−1.32269	+	0.500480i		0		1.49904	−	1.32396i	2.34508	+	1.35394i		0		1.75134	+	1.98313i	−1.32015	+	2.50144i		0		−3.77945	−	0.617175i
107.5	−1.28676	−	0.586718i		0		1.31152	+	1.50994i	−1.46375	−	0.845096i		0		−1.83823	+	1.90287i	−0.801716	−	2.71243i		0		1.38767	+	1.94625i
107.6	−1.27826	+	0.605023i		0		1.26789	−	1.54675i	−1.71449	−	0.989859i		0		−2.59406	−	0.520431i	−0.684877	+	2.74426i		0		2.79044	+	0.227993i
107.7	−1.15149	−	0.821011i		0		0.651880	+	1.89078i	−1.46375	−	0.845096i		0		1.83823	−	1.90287i	0.801716	−	2.71243i		0		0.991666	+	2.17488i
107.8	−1.04167	+	0.956516i		0		0.170154	−	1.99275i	0.835158	+	0.482179i		0		2.03771	−	1.68753i	1.72885	+	2.23854i		0		−1.33117	+	0.296571i
107.9	−0.956512	−	1.04167i		0		−0.170171	+	1.99275i	3.20305	+	1.84928i		0		2.45439	+	0.987919i	2.23856	−	1.72882i		0		−1.13740	−	5.10539i
107.10	−0.743813	+	1.20281i		0		−0.893484	−	1.78933i	1.64673	+	0.950741i		0		−0.905127	−	2.48611i	2.81680	+	0.256236i		0		−2.36842	+	1.27353i
107.11	−0.726812	−	1.21315i		0		−0.943489	+	1.76347i	0.158956	+	0.0917731i		0		−1.36747	−	2.26495i	2.82510	−	0.137113i		0		−0.00419589	−	0.259540i
107.12	−0.669754	+	1.24556i		0		−1.10286	−	1.66844i	−1.64673	−	0.950741i		0		0.905127	+	2.48611i	2.81680	−	0.256236i		0		2.28712	−	1.41435i
107.13	−0.393322	−	1.35842i		0		−1.69060	+	1.06859i	−3.53194	−	2.03916i		0		−1.92589	+	1.81410i	2.11654	+	1.87623i		0		−1.38085	+	5.59989i
107.14	−0.307532	+	1.38037i		0		−1.81085	−	0.849017i	−0.835158	−	0.482179i		0		−2.03771	+	1.68753i	1.72885	−	2.23854i		0		0.922423	−	1.00454i
107.15	−0.227919	−	1.39573i		0		−1.89611	+	0.636225i	2.34508	+	1.35394i		0		−1.75134	−	1.98313i	1.32015	+	2.50144i		0		1.35523	−	3.58168i
107.16	−0.115165	−	1.40952i		0		−1.97347	+	0.324653i	−1.71449	−	0.989859i		0		2.59406	+	0.520431i	0.684877	+	2.74426i		0		−1.19777	+	2.53059i
107.17	0.115165	+	1.40952i		0		−1.97347	+	0.324653i	1.71449	+	0.989859i		0		2.59406	+	0.520431i	−0.684877	−	2.74426i		0		−1.19777	+	2.53059i
107.18	0.227919	+	1.39573i		0		−1.89611	+	0.636225i	−2.34508	−	1.35394i		0		−1.75134	−	1.98313i	−1.32015	−	2.50144i		0		1.35523	−	3.58168i
107.19	0.307532	−	1.38037i		0		−1.81085	−	0.849017i	0.835158	+	0.482179i		0		−2.03771	+	1.68753i	−1.72885	+	2.23854i		0		0.922423	−	1.00454i
107.20	0.393322	+	1.35842i		0		−1.69060	+	1.06859i	3.53194	+	2.03916i		0		−1.92589	+	1.81410i	−2.11654	−	1.87623i		0		−1.38085	+	5.59989i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.c	even	3	1	inner
12.b	even	2	1	inner
21.h	odd	6	1	inner
28.g	odd	6	1	inner
84.n	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.be.e	✓	64
3.b	odd	2	1	inner	756.2.be.e	✓	64
4.b	odd	2	1	inner	756.2.be.e	✓	64
7.c	even	3	1	inner	756.2.be.e	✓	64
12.b	even	2	1	inner	756.2.be.e	✓	64
21.h	odd	6	1	inner	756.2.be.e	✓	64
28.g	odd	6	1	inner	756.2.be.e	✓	64
84.n	even	6	1	inner	756.2.be.e	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.be.e	✓	64	1.a	even	1	1	trivial
756.2.be.e	✓	64	3.b	odd	2	1	inner
756.2.be.e	✓	64	4.b	odd	2	1	inner
756.2.be.e	✓	64	7.c	even	3	1	inner
756.2.be.e	✓	64	12.b	even	2	1	inner
756.2.be.e	✓	64	21.h	odd	6	1	inner
756.2.be.e	✓	64	28.g	odd	6	1	inner
756.2.be.e	✓	64	84.n	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

T5^32 - 49*T5^30 + 1478*T5^28 - 28081*T5^26 + 389462*T5^24 - 3870311*T5^22 + 29186549*T5^20 - 164402280*T5^18 + 713820259*T5^16 - 2333719773*T5^14 + 5805327797*T5^12 - 10323476076*T5^10 + 12983505829*T5^8 - 9221411185*T5^6 + 4255293437*T5^4 - 142874436*T5^2 + 4477456

T19^32 - 134*T19^30 + 11525*T19^28 - 580090*T19^26 + 21075775*T19^24 - 533128762*T19^22 + 10047526344*T19^20 - 129552480946*T19^18 + 1173237468205*T19^16 - 5551099766220*T19^14 + 18551989842418*T19^12 - 34223257190146*T19^10 + 43990024036897*T19^8 - 19791946936248*T19^6 + 6679263560112*T19^4 - 138291995520*T19^2 + 2702336256