Embedded newform 240.3.bn.a.91.15

• Label: 240.3.bn.a.91.15
• 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			91.15
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.319664 - 1.97429i) q^{2} +(1.22474 + 1.22474i) q^{3} +(-3.79563 + 1.26222i) q^{4} +(1.58114 + 1.58114i) q^{5} +(2.02649 - 2.80951i) q^{6} +6.96024 q^{7} +(3.70531 + 7.09018i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.319664	−	1.97429i	−0.159832	−	0.987144i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	6.96024			0.994320			0.497160	−	0.867659i	\(-0.334376\pi\)
0.497160	+	0.867659i	\(0.334376\pi\)
\(11\)	−9.97993	+	9.97993i	−0.907266	+	0.907266i	−0.996051	−	0.0887844i	\(-0.971702\pi\)
							0.0887844	+	0.996051i	\(0.471702\pi\)
\(13\)	−16.4708	+	16.4708i	−1.26698	+	1.26698i	−0.319347	+	0.947638i	\(0.603464\pi\)
							−0.947638	+	0.319347i	\(0.896536\pi\)
\(17\)	29.2345			1.71968			0.859838	−	0.510568i	\(-0.170565\pi\)
							0.859838	+	0.510568i	\(0.170565\pi\)
\(19\)	2.95035	+	2.95035i	0.155282	+	0.155282i	0.780472	−	0.625191i	\(-0.214979\pi\)
							−0.625191	+	0.780472i	\(0.714979\pi\)
\(23\)	23.2333			1.01014			0.505072	−	0.863077i	\(-0.331466\pi\)
							0.505072	+	0.863077i	\(0.331466\pi\)
\(29\)	29.3698	−	29.3698i	1.01275	−	1.01275i	0.0128332	−	0.999918i	\(-0.495915\pi\)
							0.999918	−	0.0128332i	\(-0.00408505\pi\)
\(31\)			13.0342i			0.420458i	0.977652	+	0.210229i	\(0.0674209\pi\)
							−0.977652	+	0.210229i	\(0.932579\pi\)
\(37\)	−23.8138	−	23.8138i	−0.643616	−	0.643616i	0.307826	−	0.951443i	\(-0.400398\pi\)
							−0.951443	+	0.307826i	\(0.900398\pi\)
\(41\)			8.18200i			0.199561i	0.995009	+	0.0997804i	\(0.0318140\pi\)
							−0.995009	+	0.0997804i	\(0.968186\pi\)
\(43\)	−8.04095	+	8.04095i	−0.186999	+	0.186999i	−0.794397	−	0.607399i	\(-0.792213\pi\)
							0.607399	+	0.794397i	\(0.292213\pi\)
\(47\)			43.3282i			0.921878i	0.887432	+	0.460939i	\(0.152487\pi\)
							−0.887432	+	0.460939i	\(0.847513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.15	✓	64
4.3	odd	2		960.3.bn.a.271.13		64
16.3	odd	4	inner	240.3.bn.a.211.15	yes	64
16.13	even	4		960.3.bn.a.751.13		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.15	✓	64	1.1	even	1	trivial
240.3.bn.a.211.15	yes	64	16.3	odd	4	inner
960.3.bn.a.271.13		64	4.3	odd	2
960.3.bn.a.751.13		64	16.13	even	4