Embedded newform 240.3.bn.a.91.8

• Label: 240.3.bn.a.91.8
• 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			91.8
Character	\(\chi\)	\(=\)	240.91
Dual form			240.3.bn.a.211.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26185 - 1.55169i) q^{2} +(1.22474 + 1.22474i) q^{3} +(-0.815479 + 3.91599i) q^{4} +(1.58114 + 1.58114i) q^{5} +(0.354982 - 3.44587i) q^{6} -9.52201 q^{7} +(7.10541 - 3.67602i) q^{8} +3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 24 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26185	−	1.55169i	−0.630924	−	0.775845i
							\(3\)	1.22474	+	1.22474i	0.408248	+	0.408248i
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−9.52201			−1.36029			−0.680143	−	0.733079i	\(-0.738083\pi\)
−0.680143	+	0.733079i	\(0.738083\pi\)
\(11\)	−10.0563	+	10.0563i	−0.914207	+	0.914207i	−0.996600	−	0.0823927i	\(-0.973744\pi\)
							0.0823927	+	0.996600i	\(0.473744\pi\)
\(13\)	8.92704	−	8.92704i	0.686695	−	0.686695i	−0.274805	−	0.961500i	\(-0.588613\pi\)
							0.961500	+	0.274805i	\(0.0886132\pi\)
\(17\)	−23.9542			−1.40907			−0.704535	−	0.709670i	\(-0.748844\pi\)
							−0.704535	+	0.709670i	\(0.748844\pi\)
\(19\)	−16.7795	−	16.7795i	−0.883130	−	0.883130i	0.110722	−	0.993851i	\(-0.464684\pi\)
							−0.993851	+	0.110722i	\(0.964684\pi\)
\(23\)	−14.2344			−0.618889			−0.309444	−	0.950918i	\(-0.600143\pi\)
							−0.309444	+	0.950918i	\(0.600143\pi\)
\(29\)	−15.5275	+	15.5275i	−0.535432	+	0.535432i	−0.922184	−	0.386752i	\(-0.873597\pi\)
							0.386752	+	0.922184i	\(0.373597\pi\)
\(31\)			44.5378i			1.43670i	0.695680	+	0.718352i	\(0.255103\pi\)
							−0.695680	+	0.718352i	\(0.744897\pi\)
\(37\)	24.8489	+	24.8489i	0.671593	+	0.671593i	0.958083	−	0.286490i	\(-0.0924886\pi\)
							−0.286490	+	0.958083i	\(0.592489\pi\)
\(41\)			0.878534i			0.0214277i	0.999943	+	0.0107138i	\(0.00341038\pi\)
							−0.999943	+	0.0107138i	\(0.996590\pi\)
\(43\)	33.2851	−	33.2851i	0.774071	−	0.774071i	−0.204744	−	0.978815i	\(-0.565636\pi\)
							0.978815	+	0.204744i	\(0.0656363\pi\)
\(47\)			23.6228i			0.502614i	0.967907	+	0.251307i	\(0.0808603\pi\)
							−0.967907	+	0.251307i	\(0.919140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.bn.a.91.8	✓	64
4.3	odd	2		960.3.bn.a.271.2		64
16.3	odd	4	inner	240.3.bn.a.211.8	yes	64
16.13	even	4		960.3.bn.a.751.2		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.bn.a.91.8	✓	64	1.1	even	1	trivial
240.3.bn.a.211.8	yes	64	16.3	odd	4	inner
960.3.bn.a.271.2		64	4.3	odd	2
960.3.bn.a.751.2		64	16.13	even	4