Embedded newform 240.3.bn.a.91.1

• Label: 240.3.bn.a.91.1
• 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			91.1
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99836 - 0.0809611i) q^{2} +(1.22474 + 1.22474i) q^{3} +(3.98689 + 0.323579i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(-2.34833 - 2.54664i) q^{6} -2.50432 q^{7} +(-7.94105 - 0.969411i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99836	−	0.0809611i	−0.999180	−	0.0404806i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	−2.50432			−0.357760			−0.178880	−	0.983871i	\(-0.557247\pi\)
−0.178880	+	0.983871i	\(0.557247\pi\)
\(11\)	−9.23989	+	9.23989i	−0.839990	+	0.839990i	−0.988857	−	0.148867i	\(-0.952437\pi\)
							0.148867	+	0.988857i	\(0.452437\pi\)
\(13\)	−3.33745	+	3.33745i	−0.256727	+	0.256727i	−0.823722	−	0.566995i	\(-0.808106\pi\)
							0.566995	+	0.823722i	\(0.308106\pi\)
\(17\)	−2.84118			−0.167128			−0.0835640	−	0.996502i	\(-0.526630\pi\)
							−0.0835640	+	0.996502i	\(0.526630\pi\)
\(19\)	8.26427	+	8.26427i	0.434961	+	0.434961i	0.890312	−	0.455351i	\(-0.150486\pi\)
							−0.455351	+	0.890312i	\(0.650486\pi\)
\(23\)	−37.3138			−1.62234			−0.811170	−	0.584811i	\(-0.801169\pi\)
							−0.811170	+	0.584811i	\(0.801169\pi\)
\(29\)	−8.66072	+	8.66072i	−0.298646	+	0.298646i	−0.840483	−	0.541838i	\(-0.817729\pi\)
							0.541838	+	0.840483i	\(0.317729\pi\)
\(31\)			17.7707i			0.573248i	0.958043	+	0.286624i	\(0.0925331\pi\)
							−0.958043	+	0.286624i	\(0.907467\pi\)
\(37\)	−5.98826	−	5.98826i	−0.161845	−	0.161845i	0.621539	−	0.783383i	\(-0.286508\pi\)
							−0.783383	+	0.621539i	\(0.786508\pi\)
\(41\)			34.8206i			0.849283i	0.905362	+	0.424642i	\(0.139600\pi\)
							−0.905362	+	0.424642i	\(0.860400\pi\)
\(43\)	−53.9801	+	53.9801i	−1.25535	+	1.25535i	−0.302063	+	0.953288i	\(0.597675\pi\)
							−0.953288	+	0.302063i	\(0.902325\pi\)
\(47\)		−	11.6617i		−	0.248120i	−0.992275	−	0.124060i	\(-0.960408\pi\)
							0.992275	−	0.124060i	\(-0.0395916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.1	✓	64
4.3	odd	2		960.3.bn.a.271.6		64
16.3	odd	4	inner	240.3.bn.a.211.1	yes	64
16.13	even	4		960.3.bn.a.751.6		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.1	✓	64	1.1	even	1	trivial
240.3.bn.a.211.1	yes	64	16.3	odd	4	inner
960.3.bn.a.271.6		64	4.3	odd	2
960.3.bn.a.751.6		64	16.13	even	4