Embedded newform 240.3.bn.a.91.9

• Label: 240.3.bn.a.91.9
• 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			91.9
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21603 + 1.58785i) q^{2} +(1.22474 + 1.22474i) q^{3} +(-1.04254 - 3.86175i) q^{4} +(1.58114 + 1.58114i) q^{5} +(-3.43404 + 0.455383i) q^{6} -6.89370 q^{7} +(7.39964 + 3.04062i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21603	+	1.58785i	−0.608016	+	0.793925i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−6.89370			−0.984815			−0.492407	−	0.870365i	\(-0.663883\pi\)
−0.492407	+	0.870365i	\(0.663883\pi\)
\(11\)	1.28621	−	1.28621i	0.116928	−	0.116928i	−0.646222	−	0.763150i	\(-0.723652\pi\)
							0.763150	+	0.646222i	\(0.223652\pi\)
\(13\)	−15.5702	+	15.5702i	−1.19771	+	1.19771i	−0.222857	+	0.974851i	\(0.571538\pi\)
							−0.974851	+	0.222857i	\(0.928462\pi\)
\(17\)	−22.3706			−1.31592			−0.657959	−	0.753054i	\(-0.728580\pi\)
							−0.657959	+	0.753054i	\(0.728580\pi\)
\(19\)	17.2511	+	17.2511i	0.907954	+	0.907954i	0.996107	−	0.0881532i	\(-0.0280965\pi\)
							−0.0881532	+	0.996107i	\(0.528097\pi\)
\(23\)	−11.4914			−0.499628			−0.249814	−	0.968294i	\(-0.580369\pi\)
							−0.249814	+	0.968294i	\(0.580369\pi\)
\(29\)	−5.09044	+	5.09044i	−0.175532	+	0.175532i	−0.789405	−	0.613873i	\(-0.789611\pi\)
							0.613873	+	0.789405i	\(0.289611\pi\)
\(31\)			8.17085i			0.263576i	0.991278	+	0.131788i	\(0.0420718\pi\)
							−0.991278	+	0.131788i	\(0.957928\pi\)
\(37\)	−27.9800	−	27.9800i	−0.756215	−	0.756215i	0.219416	−	0.975631i	\(-0.429585\pi\)
							−0.975631	+	0.219416i	\(0.929585\pi\)
\(41\)		−	51.5148i		−	1.25646i	−0.778028	−	0.628229i	\(-0.783780\pi\)
							0.778028	−	0.628229i	\(-0.216220\pi\)
\(43\)	−9.22090	+	9.22090i	−0.214439	+	0.214439i	−0.806150	−	0.591711i	\(-0.798453\pi\)
							0.591711	+	0.806150i	\(0.298453\pi\)
\(47\)			22.5558i			0.479910i	0.970784	+	0.239955i	\(0.0771327\pi\)
							−0.970784	+	0.239955i	\(0.922867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.9	✓	64
4.3	odd	2		960.3.bn.a.271.5		64
16.3	odd	4	inner	240.3.bn.a.211.9	yes	64
16.13	even	4		960.3.bn.a.751.5		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.9	✓	64	1.1	even	1	trivial
240.3.bn.a.211.9	yes	64	16.3	odd	4	inner
960.3.bn.a.271.5		64	4.3	odd	2
960.3.bn.a.751.5		64	16.13	even	4