Embedded newform 240.3.bn.a.91.2

• Label: 240.3.bn.a.91.2
• 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			91.2
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99652 - 0.117912i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(3.97219 + 0.470829i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(2.30082 + 2.58964i) q^{6} +12.6948 q^{7} +(-7.87505 - 1.40839i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99652	−	0.117912i	−0.998261	−	0.0589562i
							\(3\)	−1.22474	−	1.22474i	−0.408248	−	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	12.6948			1.81354			0.906769	−	0.421628i	\(-0.138541\pi\)
0.906769	+	0.421628i	\(0.138541\pi\)
\(11\)	−8.69027	+	8.69027i	−0.790025	+	0.790025i	−0.981498	−	0.191473i	\(-0.938674\pi\)
							0.191473	+	0.981498i	\(0.438674\pi\)
\(13\)	−1.00134	+	1.00134i	−0.0770265	+	0.0770265i	−0.744570	−	0.667544i	\(-0.767346\pi\)
							0.667544	+	0.744570i	\(0.267346\pi\)
\(17\)	22.4021			1.31777			0.658885	−	0.752244i	\(-0.271028\pi\)
							0.658885	+	0.752244i	\(0.271028\pi\)
\(19\)	−6.21111	−	6.21111i	−0.326900	−	0.326900i	0.524506	−	0.851407i	\(-0.324250\pi\)
							−0.851407	+	0.524506i	\(0.824250\pi\)
\(23\)	23.1440			1.00626			0.503131	−	0.864210i	\(-0.332181\pi\)
							0.503131	+	0.864210i	\(0.332181\pi\)
\(29\)	24.9705	−	24.9705i	0.861053	−	0.861053i	−0.130408	−	0.991460i	\(-0.541629\pi\)
							0.991460	+	0.130408i	\(0.0416286\pi\)
\(31\)		−	42.1800i		−	1.36065i	−0.732912	−	0.680323i	\(-0.761839\pi\)
							0.732912	−	0.680323i	\(-0.238161\pi\)
\(37\)	28.2812	+	28.2812i	0.764356	+	0.764356i	0.977107	−	0.212750i	\(-0.0682421\pi\)
							−0.212750	+	0.977107i	\(0.568242\pi\)
\(41\)			9.87477i			0.240848i	0.992723	+	0.120424i	\(0.0384254\pi\)
							−0.992723	+	0.120424i	\(0.961575\pi\)
\(43\)	28.7490	−	28.7490i	0.668580	−	0.668580i	−0.288807	−	0.957387i	\(-0.593259\pi\)
							0.957387	+	0.288807i	\(0.0932586\pi\)
\(47\)		−	91.2242i		−	1.94094i	−0.241219	−	0.970471i	\(-0.577547\pi\)
							0.241219	−	0.970471i	\(-0.422453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.2	✓	64
4.3	odd	2		960.3.bn.a.271.31		64
16.3	odd	4	inner	240.3.bn.a.211.2	yes	64
16.13	even	4		960.3.bn.a.751.31		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.2	✓	64	1.1	even	1	trivial
240.3.bn.a.211.2	yes	64	16.3	odd	4	inner
960.3.bn.a.271.31		64	4.3	odd	2
960.3.bn.a.751.31		64	16.13	even	4