Embedded newform 240.3.bn.a.91.17

• Label: 240.3.bn.a.91.17
• 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			91.17
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.309897 - 1.97585i) q^{2} +(1.22474 + 1.22474i) q^{3} +(-3.80793 - 1.22462i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(2.79945 - 2.04036i) q^{6} -9.60974 q^{7} +(-3.59972 + 7.14437i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.309897	−	1.97585i	0.154949	−	0.987923i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	−9.60974			−1.37282			−0.686410	−	0.727215i	\(-0.740814\pi\)
−0.686410	+	0.727215i	\(0.740814\pi\)
\(11\)	1.06205	−	1.06205i	0.0965499	−	0.0965499i	−0.657182	−	0.753732i	\(-0.728252\pi\)
							0.753732	+	0.657182i	\(0.228252\pi\)
\(13\)	−11.0267	+	11.0267i	−0.848208	+	0.848208i	−0.989909	−	0.141702i	\(-0.954743\pi\)
							0.141702	+	0.989909i	\(0.454743\pi\)
\(17\)	−2.21516			−0.130304			−0.0651519	−	0.997875i	\(-0.520753\pi\)
							−0.0651519	+	0.997875i	\(0.520753\pi\)
\(19\)	−7.57099	−	7.57099i	−0.398473	−	0.398473i	0.479221	−	0.877694i	\(-0.340919\pi\)
							−0.877694	+	0.479221i	\(0.840919\pi\)
\(23\)	−19.7586			−0.859069			−0.429535	−	0.903050i	\(-0.641322\pi\)
							−0.429535	+	0.903050i	\(0.641322\pi\)
\(29\)	−29.6500	+	29.6500i	−1.02241	+	1.02241i	−0.0226701	+	0.999743i	\(0.507217\pi\)
							−0.999743	+	0.0226701i	\(0.992783\pi\)
\(31\)		−	39.4702i		−	1.27323i	−0.771181	−	0.636616i	\(-0.780334\pi\)
							0.771181	−	0.636616i	\(-0.219666\pi\)
\(37\)	−7.23169	−	7.23169i	−0.195451	−	0.195451i	0.602596	−	0.798047i	\(-0.294133\pi\)
							−0.798047	+	0.602596i	\(0.794133\pi\)
\(41\)		−	29.2296i		−	0.712918i	−0.934311	−	0.356459i	\(-0.883984\pi\)
							0.934311	−	0.356459i	\(-0.116016\pi\)
\(43\)	43.9618	−	43.9618i	1.02237	−	1.02237i	0.0226239	−	0.999744i	\(-0.492798\pi\)
							0.999744	−	0.0226239i	\(-0.00720203\pi\)
\(47\)			63.1115i			1.34280i	0.741096	+	0.671399i	\(0.234306\pi\)
							−0.741096	+	0.671399i	\(0.765694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.17	✓	64
4.3	odd	2		960.3.bn.a.271.3		64
16.3	odd	4	inner	240.3.bn.a.211.17	yes	64
16.13	even	4		960.3.bn.a.751.3		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.17	✓	64	1.1	even	1	trivial
240.3.bn.a.211.17	yes	64	16.3	odd	4	inner
960.3.bn.a.271.3		64	4.3	odd	2
960.3.bn.a.751.3		64	16.13	even	4