Newform orbit 644.2.y.a

• Label: 644.2.y.a
• Level: $644$
• Weight: $2$
• Character orbit: 644.y
• Analytic conductor: $5.142$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.14236589017\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 6 q^{5} - 2 q^{7} + 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1		0		−1.05425	+	3.04607i		0		−2.37758	−	0.951841i		0		−1.23554	+	2.33954i		0		−5.80891	−	4.56818i		0
9.2		0		−0.846905	+	2.44697i		0		−0.759755	−	0.304160i		0		0.160100	−	2.64090i		0		−2.91226	−	2.29023i		0
9.3		0		−0.784486	+	2.26662i		0		2.73026	+	1.09303i		0		−2.14852	−	1.54398i		0		−2.16400	−	1.70179i		0
9.4		0		−0.606056	+	1.75108i		0		2.08712	+	0.835556i		0		0.481145	+	2.60163i		0		−0.340831	−	0.268033i		0
9.5		0		−0.578695	+	1.67203i		0		−2.47193	−	0.989611i		0		2.64323	−	0.115493i		0		−0.102638	−	0.0807154i		0
9.6		0		−0.385621	+	1.11418i		0		3.49946	+	1.40097i		0		2.61624	−	0.394083i		0		1.26547	+	0.995175i		0
9.7		0		−0.105463	+	0.304715i		0		1.18922	+	0.476094i		0		−1.60787	+	2.10113i		0		2.27643	+	1.79020i		0
9.8		0		−0.0146228	+	0.0422497i		0		−3.22138	−	1.28964i		0		−2.02135	−	1.70708i		0		2.35659	+	1.85324i		0
9.9		0		0.00455622	−	0.0131643i		0		−1.13338	−	0.453737i		0		−2.64565	−	0.0227100i		0		2.35801	+	1.85436i		0
9.10		0		0.269670	−	0.779161i		0		−2.63298	−	1.05409i		0		1.66705	+	2.05449i		0		1.82379	+	1.43424i		0
9.11		0		0.452044	−	1.30610i		0		2.95800	+	1.18420i		0		−0.547816	−	2.58842i		0		0.856617	+	0.673651i		0
9.12		0		0.462213	−	1.33548i		0		1.03250	+	0.413351i		0		2.01456	−	1.71509i		0		0.788303	+	0.619929i		0
9.13		0		0.466364	−	1.34747i		0		0.126488	+	0.0506383i		0		1.19869	+	2.35863i		0		0.759980	+	0.597655i		0
9.14		0		0.862910	−	2.49322i		0		0.552798	+	0.221307i		0		−2.49157	−	0.889994i		0		−3.11335	−	2.44837i		0
9.15		0		0.914616	−	2.64261i		0		−2.83463	−	1.13481i		0		1.49208	−	2.18488i		0		−3.78871	−	2.97947i		0
9.16		0		0.943729	−	2.72673i		0		3.75336	+	1.50262i		0		0.590317	+	2.57906i		0		−4.18625	−	3.29210i		0
25.1		0		−3.04369	−	1.21851i		0		−0.281860	−	1.16184i		0		2.53785	−	0.747890i		0		5.60806	+	5.34728i		0
25.2		0		−2.67908	−	1.07254i		0		0.606107	+	2.49841i		0		−2.64323	+	0.115585i		0		3.85591	+	3.67661i		0
25.3		0		−1.91956	−	0.768477i		0		−0.540072	−	2.22621i		0		−1.48945	+	2.18667i		0		0.922965	+	0.880046i		0
25.4		0		−1.80364	−	0.722068i		0		0.578643	+	2.38520i		0		1.99076	+	1.74266i		0		0.560531	+	0.534466i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
23.c	even	11	1	inner
161.m	even	33	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.2.y.a	✓	320
7.c	even	3	1	inner	644.2.y.a	✓	320
23.c	even	11	1	inner	644.2.y.a	✓	320
161.m	even	33	1	inner	644.2.y.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.2.y.a	✓	320	1.a	even	1	1	trivial
644.2.y.a	✓	320	7.c	even	3	1	inner
644.2.y.a	✓	320	23.c	even	11	1	inner
644.2.y.a	✓	320	161.m	even	33	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).