Embedded newform 240.3.bn.a.91.16

• Label: 240.3.bn.a.91.16
• 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			91.16
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.166064 + 1.99309i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(-3.94485 + 0.661961i) q^{4} +(1.58114 + 1.58114i) q^{5} +(2.23765 - 2.64442i) q^{6} +5.18414 q^{7} +(-1.97445 - 7.75252i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.166064	+	1.99309i	0.0830318	+	0.996547i
							\(3\)	−1.22474	−	1.22474i	−0.408248	−	0.408248i
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	5.18414			0.740592			0.370296	−	0.928914i	\(-0.379256\pi\)
0.370296	+	0.928914i	\(0.379256\pi\)
\(11\)	−13.2861	+	13.2861i	−1.20783	+	1.20783i	−0.236100	+	0.971729i	\(0.575869\pi\)
							−0.971729	+	0.236100i	\(0.924131\pi\)
\(13\)	−12.8084	+	12.8084i	−0.985260	+	0.985260i	−0.999893	−	0.0146333i	\(-0.995342\pi\)
							0.0146333	+	0.999893i	\(0.495342\pi\)
\(17\)	9.46997			0.557057			0.278528	−	0.960428i	\(-0.410153\pi\)
							0.278528	+	0.960428i	\(0.410153\pi\)
\(19\)	−1.40967	−	1.40967i	−0.0741930	−	0.0741930i	0.669037	−	0.743230i	\(-0.266707\pi\)
							−0.743230	+	0.669037i	\(0.766707\pi\)
\(23\)	−44.3571			−1.92857			−0.964284	−	0.264870i	\(-0.914671\pi\)
							−0.964284	+	0.264870i	\(0.914671\pi\)
\(29\)	−21.9180	+	21.9180i	−0.755795	+	0.755795i	−0.975554	−	0.219759i	\(-0.929473\pi\)
							0.219759	+	0.975554i	\(0.429473\pi\)
\(31\)		−	42.9358i		−	1.38502i	−0.721406	−	0.692512i	\(-0.756504\pi\)
							0.721406	−	0.692512i	\(-0.243496\pi\)
\(37\)	22.8352	+	22.8352i	0.617166	+	0.617166i	0.944804	−	0.327637i	\(-0.106252\pi\)
							−0.327637	+	0.944804i	\(0.606252\pi\)
\(41\)			35.0504i			0.854887i	0.904042	+	0.427444i	\(0.140586\pi\)
							−0.904042	+	0.427444i	\(0.859414\pi\)
\(43\)	−10.5624	+	10.5624i	−0.245637	+	0.245637i	−0.819177	−	0.573540i	\(-0.805570\pi\)
							0.573540	+	0.819177i	\(0.305570\pi\)
\(47\)			51.1681i			1.08868i	0.838864	+	0.544342i	\(0.183220\pi\)
							−0.838864	+	0.544342i	\(0.816780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.16	✓	64
4.3	odd	2		960.3.bn.a.271.30		64
16.3	odd	4	inner	240.3.bn.a.211.16	yes	64
16.13	even	4		960.3.bn.a.751.30		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.16	✓	64	1.1	even	1	trivial
240.3.bn.a.211.16	yes	64	16.3	odd	4	inner
960.3.bn.a.271.30		64	4.3	odd	2
960.3.bn.a.751.30		64	16.13	even	4