Embedded newform 240.3.bn.a.91.14

• Label: 240.3.bn.a.91.14
• 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			91.14
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.487538 - 1.93967i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(-3.52461 + 1.89132i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(-1.77849 + 2.97271i) q^{6} -0.906944 q^{7} +(5.38692 + 5.91449i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.487538	−	1.93967i	−0.243769	−	0.969833i
							\(3\)	−1.22474	−	1.22474i	−0.408248	−	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	−0.906944			−0.129563			−0.0647817	−	0.997899i	\(-0.520635\pi\)
−0.0647817	+	0.997899i	\(0.520635\pi\)
\(11\)	−12.8302	+	12.8302i	−1.16638	+	1.16638i	−0.183333	+	0.983051i	\(0.558689\pi\)
							−0.983051	+	0.183333i	\(0.941311\pi\)
\(13\)	4.33113	−	4.33113i	0.333164	−	0.333164i	−0.520623	−	0.853787i	\(-0.674300\pi\)
							0.853787	+	0.520623i	\(0.174300\pi\)
\(17\)	−8.50380			−0.500223			−0.250112	−	0.968217i	\(-0.580467\pi\)
							−0.250112	+	0.968217i	\(0.580467\pi\)
\(19\)	24.2059	+	24.2059i	1.27399	+	1.27399i	0.943974	+	0.330020i	\(0.107056\pi\)
							0.330020	+	0.943974i	\(0.392944\pi\)
\(23\)	18.2047			0.791510			0.395755	−	0.918356i	\(-0.370483\pi\)
							0.395755	+	0.918356i	\(0.370483\pi\)
\(29\)	−25.7394	+	25.7394i	−0.887567	+	0.887567i	−0.994289	−	0.106722i	\(-0.965964\pi\)
							0.106722	+	0.994289i	\(0.465964\pi\)
\(31\)		−	7.75027i		−	0.250009i	−0.992156	−	0.125004i	\(-0.960106\pi\)
							0.992156	−	0.125004i	\(-0.0398945\pi\)
\(37\)	31.9993	+	31.9993i	0.864845	+	0.864845i	0.991896	−	0.127051i	\(-0.0405513\pi\)
							−0.127051	+	0.991896i	\(0.540551\pi\)
\(41\)		−	68.7976i		−	1.67799i	−0.544139	−	0.838995i	\(-0.683143\pi\)
							0.544139	−	0.838995i	\(-0.316857\pi\)
\(43\)	−36.3404	+	36.3404i	−0.845125	+	0.845125i	−0.989520	−	0.144395i	\(-0.953876\pi\)
							0.144395	+	0.989520i	\(0.453876\pi\)
\(47\)			89.4640i			1.90349i	0.306890	+	0.951745i	\(0.400712\pi\)
							−0.306890	+	0.951745i	\(0.599288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.14	✓	64
4.3	odd	2		960.3.bn.a.271.23		64
16.3	odd	4	inner	240.3.bn.a.211.14	yes	64
16.13	even	4		960.3.bn.a.751.23		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.14	✓	64	1.1	even	1	trivial
240.3.bn.a.211.14	yes	64	16.3	odd	4	inner
960.3.bn.a.271.23		64	4.3	odd	2
960.3.bn.a.751.23		64	16.13	even	4