Newform orbit 670.3.b.a

• Label: 670.3.b.a
• Level: $670$
• Weight: $3$
• Character orbit: 670.b
• Analytic conductor: $18.256$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 670 = 2 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	670.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.2561777121\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{4} - 24 q^{6} - 176 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
401.1		−	1.41421i		−	5.70098i	−2.00000				−	2.23607i	−8.06240					12.5457i			2.82843i	−23.5012			−3.16228
401.2		−	1.41421i		−	5.67313i	−2.00000				−	2.23607i	−8.02302				−	4.05202i			2.82843i	−23.1844			−3.16228
401.3		−	1.41421i		−	5.52282i	−2.00000					2.23607i	−7.81045				−	8.84933i			2.82843i	−21.5016			3.16228
401.4		−	1.41421i		−	4.03268i	−2.00000					2.23607i	−5.70307					3.62095i			2.82843i	−7.26249			3.16228
401.5		−	1.41421i		−	3.95896i	−2.00000					2.23607i	−5.59882				−	12.2635i			2.82843i	−6.67340			3.16228
401.6		−	1.41421i		−	3.87122i	−2.00000					2.23607i	−5.47472					6.39484i			2.82843i	−5.98631			3.16228
401.7		−	1.41421i		−	3.47874i	−2.00000				−	2.23607i	−4.91968				−	2.67539i			2.82843i	−3.10161			−3.16228
401.8		−	1.41421i		−	2.59143i	−2.00000					2.23607i	−3.66483					8.49063i			2.82843i	2.28450			3.16228
401.9		−	1.41421i		−	2.45113i	−2.00000				−	2.23607i	−3.46642					0.439059i			2.82843i	2.99196			−3.16228
401.10		−	1.41421i		−	1.96458i	−2.00000				−	2.23607i	−2.77834					12.5241i			2.82843i	5.14042			−3.16228
401.11		−	1.41421i		−	1.38191i	−2.00000				−	2.23607i	−1.95431					0.207632i			2.82843i	7.09034			−3.16228
401.12		−	1.41421i		−	1.29872i	−2.00000					2.23607i	−1.83667				−	7.02143i			2.82843i	7.31332			3.16228
401.13		−	1.41421i		−	0.242536i	−2.00000				−	2.23607i	−0.342997				−	13.5289i			2.82843i	8.94118			−3.16228
401.14		−	1.41421i			0.460833i	−2.00000					2.23607i	0.651716					6.38504i			2.82843i	8.78763			3.16228
401.15		−	1.41421i			1.41262i	−2.00000					2.23607i	1.99774				−	0.971661i			2.82843i	7.00451			3.16228
401.16		−	1.41421i			1.60519i	−2.00000				−	2.23607i	2.27009				−	4.33357i			2.82843i	6.42335			−3.16228
401.17		−	1.41421i			1.64028i	−2.00000					2.23607i	2.31970				−	5.04905i			2.82843i	6.30949			3.16228
401.18		−	1.41421i			2.59628i	−2.00000				−	2.23607i	3.67169					6.95264i			2.82843i	2.25934			−3.16228
401.19		−	1.41421i			3.49503i	−2.00000				−	2.23607i	4.94271				−	10.7812i			2.82843i	−3.21521			−3.16228
401.20		−	1.41421i			3.53928i	−2.00000				−	2.23607i	5.00529					6.36334i			2.82843i	−3.52648			−3.16228

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	670.3.b.a	✓	48
67.b	odd	2	1	inner	670.3.b.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
670.3.b.a	✓	48	1.a	even	1	1	trivial
670.3.b.a	✓	48	67.b	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(670, [\chi])\).