Newform orbit 756.2.ca.a

• Label: 756.2.ca.a
• Level: $756$
• Weight: $2$
• Character orbit: 756.ca
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.ca (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
173.1		0		−1.73005	−	0.0832183i		0		0.849177	−	0.712544i		0		−0.774464	+	2.52986i		0		2.98615	+	0.287944i		0
173.2		0		−1.69481	−	0.357227i		0		1.90051	−	1.59472i		0		1.42097	−	2.23178i		0		2.74478	+	1.21086i		0
173.3		0		−1.38241	−	1.04352i		0		−2.01034	+	1.68687i		0		−2.51700	−	0.815311i		0		0.822127	+	2.88515i		0
173.4		0		−1.34169	−	1.09539i		0		−2.77975	+	2.33249i		0		2.63835	−	0.197791i		0		0.600250	+	2.93934i		0
173.5		0		−1.31760	+	1.12424i		0		−2.86428	+	2.40342i		0		−1.48545	+	2.18939i		0		0.472151	−	2.96261i		0
173.6		0		−1.31609	+	1.12602i		0		−0.481690	+	0.404186i		0		2.64564	+	0.0243961i		0		0.464163	−	2.96387i		0
173.7		0		−1.16896	+	1.27810i		0		0.898527	−	0.753954i		0		−1.31798	−	2.29411i		0		−0.267080	−	2.98809i		0
173.8		0		−1.04679	−	1.37994i		0		0.700470	−	0.587764i		0		−0.673988	+	2.55846i		0		−0.808445	+	2.88902i		0
173.9		0		−0.586799	−	1.62962i		0		3.01033	−	2.52597i		0		−2.34045	−	1.23381i		0		−2.31133	+	1.91252i		0
173.10		0		−0.568097	+	1.63624i		0		3.32029	−	2.78605i		0		1.31877	+	2.29366i		0		−2.35453	−	1.85908i		0
173.11		0		−0.385495	−	1.68861i		0		−1.16198	+	0.975020i		0		0.818663	−	2.51591i		0		−2.70279	+	1.30190i		0
173.12		0		−0.267657	+	1.71125i		0		0.977821	−	0.820489i		0		−2.63440	−	0.244842i		0		−2.85672	−	0.916052i		0
173.13		0		0.0527582	+	1.73125i		0		−3.03928	+	2.55026i		0		0.308602	−	2.62769i		0		−2.99443	+	0.182675i		0
173.14		0		0.280100	−	1.70925i		0		−0.516309	+	0.433235i		0		1.59408	+	2.11161i		0		−2.84309	−	0.957524i		0
173.15		0		0.294282	+	1.70687i		0		−0.796960	+	0.668729i		0		0.995586	+	2.45129i		0		−2.82680	+	1.00460i		0
173.16		0		0.785443	−	1.54372i		0		1.98604	−	1.66648i		0		2.56407	−	0.652322i		0		−1.76616	−	2.42501i		0
173.17		0		0.839673	−	1.51491i		0		−1.81938	+	1.52664i		0		−2.37423	+	1.16749i		0		−1.58990	−	2.54406i		0
173.18		0		1.12890	+	1.31362i		0		0.424429	−	0.356139i		0		−2.49967	+	0.866968i		0		−0.451185	+	2.96588i		0
173.19		0		1.37133	+	1.05805i		0		−1.39267	+	1.16859i		0		0.270511	−	2.63189i		0		0.761074	+	2.90186i		0
173.20		0		1.47752	+	0.903842i		0		1.37147	−	1.15080i		0		2.42417	−	1.05990i		0		1.36614	+	2.67089i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.bd	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.ca.a	✓	144
7.d	odd	6	1		756.2.ck.a	yes	144
27.f	odd	18	1		756.2.ck.a	yes	144
189.bd	even	18	1	inner	756.2.ca.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.ca.a	✓	144	1.a	even	1	1	trivial
756.2.ca.a	✓	144	189.bd	even	18	1	inner
756.2.ck.a	yes	144	7.d	odd	6	1
756.2.ck.a	yes	144	27.f	odd	18	1

Hecke kernels

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