Newform orbit 315.5.c.a

• Label: 315.5.c.a
• Level: $315$
• Weight: $5$
• Character orbit: 315.c
• Analytic conductor: $32.562$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	315.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(32.5615383714\)
Analytic rank:	\(0\)
Dimension:	\(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 200 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
71.1		−	7.63905i		0		−42.3551					11.1803i		0		−18.5203					201.328i		0		85.4072
71.2		−	6.85454i		0		−30.9847					11.1803i		0		18.5203					102.713i		0		76.6361
71.3		−	6.70628i		0		−28.9742				−	11.1803i		0		18.5203					87.0087i		0		−74.9785
71.4		−	6.51398i		0		−26.4320				−	11.1803i		0		−18.5203					67.9538i		0		−72.8286
71.5		−	6.05038i		0		−20.6071					11.1803i		0		18.5203					27.8747i		0		67.6453
71.6		−	5.29757i		0		−12.0643					11.1803i		0		−18.5203				−	20.8497i		0		59.2287
71.7		−	4.58858i		0		−5.05506				−	11.1803i		0		−18.5203				−	50.2217i		0		−51.3019
71.8		−	4.54612i		0		−4.66718					11.1803i		0		18.5203				−	51.5203i		0		50.8271
71.9		−	4.34228i		0		−2.85536					11.1803i		0		−18.5203				−	57.0777i		0		48.5481
71.10		−	3.97875i		0		0.169517				−	11.1803i		0		18.5203				−	64.3345i		0		−44.4838
71.11		−	3.49694i		0		3.77141				−	11.1803i		0		−18.5203				−	69.1394i		0		−39.0970
71.12		−	1.81509i		0		12.7055				−	11.1803i		0		18.5203				−	52.1029i		0		−20.2933
71.13		−	1.69467i		0		13.1281				−	11.1803i		0		18.5203				−	49.3624i		0		−18.9470
71.14		−	1.48549i		0		13.7933					11.1803i		0		−18.5203				−	44.2576i		0		16.6082
71.15		−	1.21589i		0		14.5216					11.1803i		0		18.5203				−	37.1110i		0		13.5941
71.16		−	0.307250i		0		15.9056					11.1803i		0		−18.5203				−	9.80299i		0		3.43516
71.17			0.307250i		0		15.9056				−	11.1803i		0		−18.5203					9.80299i		0		3.43516
71.18			1.21589i		0		14.5216				−	11.1803i		0		18.5203					37.1110i		0		13.5941
71.19			1.48549i		0		13.7933				−	11.1803i		0		−18.5203					44.2576i		0		16.6082
71.20			1.69467i		0		13.1281					11.1803i		0		18.5203					49.3624i		0		−18.9470

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.5.c.a	✓	32
3.b	odd	2	1	inner	315.5.c.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.5.c.a	✓	32	1.a	even	1	1	trivial
315.5.c.a	✓	32	3.b	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(315, [\chi])\).