Embedded newform 240.3.bn.a.91.13

• Label: 240.3.bn.a.91.13
• 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			91.13
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.712831 + 1.86865i) q^{2} +(1.22474 + 1.22474i) q^{3} +(-2.98374 - 2.66407i) q^{4} +(-1.58114 - 1.58114i) q^{5} +(-3.16166 + 1.41559i) q^{6} +10.5937 q^{7} +(7.10514 - 3.67655i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.712831	+	1.86865i	−0.356416	+	0.934327i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	−1.58114	−	1.58114i	−0.316228	−	0.316228i
							\(7\)	10.5937			1.51338			0.756690	−	0.653773i	\(-0.226815\pi\)
0.756690	+	0.653773i	\(0.226815\pi\)
\(11\)	2.72452	−	2.72452i	0.247684	−	0.247684i	−0.572336	−	0.820020i	\(-0.693963\pi\)
							0.820020	+	0.572336i	\(0.193963\pi\)
\(13\)	3.54573	−	3.54573i	0.272749	−	0.272749i	−0.557457	−	0.830206i	\(-0.688223\pi\)
							0.830206	+	0.557457i	\(0.188223\pi\)
\(17\)	1.85808			0.109299			0.0546495	−	0.998506i	\(-0.482596\pi\)
							0.0546495	+	0.998506i	\(0.482596\pi\)
\(19\)	25.8736	+	25.8736i	1.36177	+	1.36177i	0.871667	+	0.490099i	\(0.163039\pi\)
							0.490099	+	0.871667i	\(0.336961\pi\)
\(23\)	−6.37845			−0.277324			−0.138662	−	0.990340i	\(-0.544280\pi\)
							−0.138662	+	0.990340i	\(0.544280\pi\)
\(29\)	−17.0713	+	17.0713i	−0.588665	+	0.588665i	−0.937270	−	0.348604i	\(-0.886656\pi\)
							0.348604	+	0.937270i	\(0.386656\pi\)
\(31\)		−	32.4982i		−	1.04833i	−0.851617	−	0.524165i	\(-0.824378\pi\)
							0.851617	−	0.524165i	\(-0.175622\pi\)
\(37\)	33.1832	+	33.1832i	0.896844	+	0.896844i	0.995156	−	0.0983120i	\(-0.0313443\pi\)
							−0.0983120	+	0.995156i	\(0.531344\pi\)
\(41\)		−	14.8307i		−	0.361725i	−0.983508	−	0.180862i	\(-0.942111\pi\)
							0.983508	−	0.180862i	\(-0.0578888\pi\)
\(43\)	−6.19522	+	6.19522i	−0.144075	+	0.144075i	−0.775465	−	0.631390i	\(-0.782485\pi\)
							0.631390	+	0.775465i	\(0.282485\pi\)
\(47\)		−	25.1723i		−	0.535581i	−0.963477	−	0.267791i	\(-0.913706\pi\)
							0.963477	−	0.267791i	\(-0.0862935\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.13	✓	64
4.3	odd	2		960.3.bn.a.271.15		64
16.3	odd	4	inner	240.3.bn.a.211.13	yes	64
16.13	even	4		960.3.bn.a.751.15		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.13	✓	64	1.1	even	1	trivial
240.3.bn.a.211.13	yes	64	16.3	odd	4	inner
960.3.bn.a.271.15		64	4.3	odd	2
960.3.bn.a.751.15		64	16.13	even	4