Embedded newform 240.2.bb.a.173.9

• Label: 240.2.bb.a.173.9
• Level: $240$
• Weight: $2$
• Character: 240.173
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.bb (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			173.9
Character	\(\chi\)	\(=\)	240.173
Dual form			240.2.bb.a.197.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18598 - 0.770357i) q^{2} +(-1.09486 - 1.34211i) q^{3} +(0.813100 + 1.82726i) q^{4} +(1.71343 - 1.43671i) q^{5} +(0.264580 + 2.43516i) q^{6} +(3.11495 + 3.11495i) q^{7} +(0.443321 - 2.79347i) q^{8} +(-0.602544 + 2.93887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18598	−	0.770357i	−0.838615	−	0.544725i
							\(3\)	−1.09486	−	1.34211i	−0.632120	−	0.774870i
\(5\)	1.71343	−	1.43671i	0.766270	−	0.642518i
							\(7\)	3.11495	+	3.11495i	1.17734	+	1.17734i	0.980420	+	0.196919i	\(0.0630938\pi\)
0.196919	+	0.980420i	\(0.436906\pi\)
\(11\)	3.15836	+	3.15836i	0.952283	+	0.952283i	0.998912	−	0.0466296i	\(-0.0148480\pi\)
							−0.0466296	+	0.998912i	\(0.514848\pi\)
\(13\)		−	4.72071i		−	1.30929i	−0.755936	−	0.654645i	\(-0.772818\pi\)
							0.755936	−	0.654645i	\(-0.227182\pi\)
\(17\)	−1.27146	−	1.27146i	−0.308376	−	0.308376i	0.535904	−	0.844279i	\(-0.319971\pi\)
							−0.844279	+	0.535904i	\(0.819971\pi\)
\(19\)	1.33059	−	1.33059i	0.305259	−	0.305259i	−0.537808	−	0.843067i	\(-0.680747\pi\)
							0.843067	+	0.537808i	\(0.180747\pi\)
\(23\)	1.09034	+	1.09034i	0.227352	+	0.227352i	0.811585	−	0.584234i	\(-0.198605\pi\)
							−0.584234	+	0.811585i	\(0.698605\pi\)
\(29\)	−1.67641	−	1.67641i	−0.311302	−	0.311302i	0.534112	−	0.845414i	\(-0.320646\pi\)
							−0.845414	+	0.534112i	\(0.820646\pi\)
\(31\)	−3.42372			−0.614918			−0.307459	−	0.951561i	\(-0.599479\pi\)
							−0.307459	+	0.951561i	\(0.599479\pi\)
\(37\)			1.11057i			0.182576i	0.995825	+	0.0912880i	\(0.0290984\pi\)
							−0.995825	+	0.0912880i	\(0.970902\pi\)
\(41\)	0.0298672			0.00466447			0.00233223	−	0.999997i	\(-0.499258\pi\)
							0.00233223	+	0.999997i	\(0.499258\pi\)
\(43\)	−5.45244			−0.831490			−0.415745	−	0.909481i	\(-0.636479\pi\)
							−0.415745	+	0.909481i	\(0.636479\pi\)
\(47\)	0.608051	+	0.608051i	0.0886933	+	0.0886933i	0.750061	−	0.661368i	\(-0.230024\pi\)
							−0.661368	+	0.750061i	\(0.730024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.2.bb.a.173.9	✓	88
3.2	odd	2	inner	240.2.bb.a.173.36	yes	88
4.3	odd	2		960.2.bb.a.593.32		88
5.2	odd	4		240.2.bf.a.77.31	yes	88
12.11	even	2		960.2.bb.a.593.13		88
15.2	even	4		240.2.bf.a.77.14	yes	88
16.5	even	4		240.2.bf.a.53.14	yes	88
16.11	odd	4		960.2.bf.a.113.33		88
20.7	even	4		960.2.bf.a.17.33		88
48.5	odd	4		240.2.bf.a.53.31	yes	88
48.11	even	4		960.2.bf.a.113.34		88
60.47	odd	4		960.2.bf.a.17.34		88
80.27	even	4		960.2.bb.a.497.13		88
80.37	odd	4	inner	240.2.bb.a.197.36	yes	88
240.107	odd	4		960.2.bb.a.497.32		88
240.197	even	4	inner	240.2.bb.a.197.9	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.bb.a.173.9	✓	88	1.1	even	1	trivial
240.2.bb.a.173.36	yes	88	3.2	odd	2	inner
240.2.bb.a.197.9	yes	88	240.197	even	4	inner
240.2.bb.a.197.36	yes	88	80.37	odd	4	inner
240.2.bf.a.53.14	yes	88	16.5	even	4
240.2.bf.a.53.31	yes	88	48.5	odd	4
240.2.bf.a.77.14	yes	88	15.2	even	4
240.2.bf.a.77.31	yes	88	5.2	odd	4
960.2.bb.a.497.13		88	80.27	even	4
960.2.bb.a.497.32		88	240.107	odd	4
960.2.bb.a.593.13		88	12.11	even	2
960.2.bb.a.593.32		88	4.3	odd	2
960.2.bf.a.17.33		88	20.7	even	4
960.2.bf.a.17.34		88	60.47	odd	4
960.2.bf.a.113.33		88	16.11	odd	4
960.2.bf.a.113.34		88	48.11	even	4