Embedded newform 240.3.bn.a.91.18

• Label: 240.3.bn.a.91.18
• 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			91.18
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.328478 + 1.97284i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(-3.78420 + 1.29607i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(2.01393 - 2.81853i) q^{6} +4.74581 q^{7} +(-3.79997 - 7.03990i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.328478	+	1.97284i	0.164239	+	0.986421i
							\(3\)	−1.22474	−	1.22474i	−0.408248	−	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	4.74581			0.677973			0.338987	−	0.940791i	\(-0.389916\pi\)
0.338987	+	0.940791i	\(0.389916\pi\)
\(11\)	7.48992	−	7.48992i	0.680902	−	0.680902i	−0.279302	−	0.960203i	\(-0.590103\pi\)
							0.960203	+	0.279302i	\(0.0901029\pi\)
\(13\)	3.55128	−	3.55128i	0.273175	−	0.273175i	−0.557202	−	0.830377i	\(-0.688125\pi\)
							0.830377	+	0.557202i	\(0.188125\pi\)
\(17\)	15.9415			0.937735			0.468868	−	0.883268i	\(-0.344662\pi\)
							0.468868	+	0.883268i	\(0.344662\pi\)
\(19\)	6.28848	+	6.28848i	0.330973	+	0.330973i	0.852956	−	0.521983i	\(-0.174808\pi\)
							−0.521983	+	0.852956i	\(0.674808\pi\)
\(23\)	14.7521			0.641395			0.320697	−	0.947182i	\(-0.396083\pi\)
							0.320697	+	0.947182i	\(0.396083\pi\)
\(29\)	33.7383	−	33.7383i	1.16339	−	1.16339i	0.179663	−	0.983728i	\(-0.442499\pi\)
							0.983728	−	0.179663i	\(-0.0575007\pi\)
\(31\)			18.2479i			0.588641i	0.955707	+	0.294320i	\(0.0950933\pi\)
							−0.955707	+	0.294320i	\(0.904907\pi\)
\(37\)	−20.9285	−	20.9285i	−0.565635	−	0.565635i	0.365267	−	0.930903i	\(-0.380978\pi\)
							−0.930903	+	0.365267i	\(0.880978\pi\)
\(41\)		−	57.7112i		−	1.40759i	−0.710403	−	0.703795i	\(-0.751487\pi\)
							0.710403	−	0.703795i	\(-0.248513\pi\)
\(43\)	−12.0735	+	12.0735i	−0.280778	+	0.280778i	−0.833419	−	0.552641i	\(-0.813620\pi\)
							0.552641	+	0.833419i	\(0.313620\pi\)
\(47\)			46.0219i			0.979190i	0.871950	+	0.489595i	\(0.162855\pi\)
							−0.871950	+	0.489595i	\(0.837145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.18	✓	64
4.3	odd	2		960.3.bn.a.271.28		64
16.3	odd	4	inner	240.3.bn.a.211.18	yes	64
16.13	even	4		960.3.bn.a.751.28		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.18	✓	64	1.1	even	1	trivial
240.3.bn.a.211.18	yes	64	16.3	odd	4	inner
960.3.bn.a.271.28		64	4.3	odd	2
960.3.bn.a.751.28		64	16.13	even	4