Embedded newform 240.3.bn.a.91.11

• Label: 240.3.bn.a.91.11
• Level: $240$
• Weight: $3$
• Character: 240.91
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.bn (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53952634465\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			91.11
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.979211 + 1.74389i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(-2.08229 - 3.41527i) q^{4} +(1.58114 + 1.58114i) q^{5} +(3.33510 - 0.936535i) q^{6} -6.97297 q^{7} +(7.99485 - 0.287019i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.979211	+	1.74389i	−0.489605	+	0.871944i
							\(3\)	−1.22474	−	1.22474i	−0.408248	−	0.408248i
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−6.97297			−0.996139			−0.498069	−	0.867137i	\(-0.665958\pi\)
−0.498069	+	0.867137i	\(0.665958\pi\)
\(11\)	−3.36527	+	3.36527i	−0.305934	+	0.305934i	−0.843330	−	0.537396i	\(-0.819408\pi\)
							0.537396	+	0.843330i	\(0.319408\pi\)
\(13\)	13.6766	−	13.6766i	1.05204	−	1.05204i	0.0534743	−	0.998569i	\(-0.482970\pi\)
							0.998569	−	0.0534743i	\(-0.0170295\pi\)
\(17\)	23.4860			1.38153			0.690765	−	0.723079i	\(-0.257274\pi\)
							0.690765	+	0.723079i	\(0.257274\pi\)
\(19\)	9.12012	+	9.12012i	0.480006	+	0.480006i	0.905134	−	0.425127i	\(-0.139771\pi\)
							−0.425127	+	0.905134i	\(0.639771\pi\)
\(23\)	27.6644			1.20280			0.601399	−	0.798949i	\(-0.294610\pi\)
							0.601399	+	0.798949i	\(0.294610\pi\)
\(29\)	12.3702	−	12.3702i	0.426557	−	0.426557i	−0.460897	−	0.887454i	\(-0.652472\pi\)
							0.887454	+	0.460897i	\(0.152472\pi\)
\(31\)			51.9511i			1.67584i	0.545792	+	0.837921i	\(0.316229\pi\)
							−0.545792	+	0.837921i	\(0.683771\pi\)
\(37\)	22.5553	+	22.5553i	0.609604	+	0.609604i	0.942842	−	0.333239i	\(-0.108142\pi\)
							−0.333239	+	0.942842i	\(0.608142\pi\)
\(41\)		−	36.1760i		−	0.882341i	−0.897423	−	0.441171i	\(-0.854563\pi\)
							0.897423	−	0.441171i	\(-0.145437\pi\)
\(43\)	9.54629	−	9.54629i	0.222007	−	0.222007i	−0.587336	−	0.809343i	\(-0.699823\pi\)
							0.809343	+	0.587336i	\(0.199823\pi\)
\(47\)		−	58.9486i		−	1.25422i	−0.778929	−	0.627112i	\(-0.784237\pi\)
							0.778929	−	0.627112i	\(-0.215763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.11	✓	64
4.3	odd	2		960.3.bn.a.271.21		64
16.3	odd	4	inner	240.3.bn.a.211.11	yes	64
16.13	even	4		960.3.bn.a.751.21		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.11	✓	64	1.1	even	1	trivial
240.3.bn.a.211.11	yes	64	16.3	odd	4	inner
960.3.bn.a.271.21		64	4.3	odd	2
960.3.bn.a.751.21		64	16.13	even	4