Embedded newform 296.2.q.a.101.7

• Label: 296.2.q.a.101.7
• Level: $296$
• Weight: $2$
• Character: 296.101
• Analytic conductor: $2.364$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	296.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.36357189983\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.7
Character	\(\chi\)	\(=\)	296.101
Dual form			296.2.q.a.85.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26355 + 0.635167i) q^{2} +(-2.54245 + 1.46788i) q^{3} +(1.19312 - 1.60513i) q^{4} +(-2.01329 - 3.48712i) q^{5} +(2.28016 - 3.46962i) q^{6} +(0.477860 + 0.827678i) q^{7} +(-0.488046 + 2.78600i) q^{8} +(2.80936 - 4.86595i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} - 2 q^{4} + 2 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/296\mathbb{Z}\right)^\times\).

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26355	+	0.635167i	−0.893466	+	0.449131i
							\(3\)	−2.54245	+	1.46788i	−1.46788	+	0.847482i	−0.999353	−	0.0359656i	\(-0.988549\pi\)
−0.468529	+	0.883448i	\(0.655216\pi\)
\(5\)	−2.01329	−	3.48712i	−0.900370	−	1.55949i	−0.827015	−	0.562180i	\(-0.809963\pi\)
							−0.0733552	−	0.997306i	\(-0.523371\pi\)
\(7\)	0.477860	+	0.827678i	0.180614	+	0.312833i	0.942090	−	0.335361i	\(-0.108858\pi\)
							−0.761476	+	0.648193i	\(0.775525\pi\)
\(11\)			0.266146i			0.0802461i	0.999195	+	0.0401230i	\(0.0127750\pi\)
							−0.999195	+	0.0401230i	\(0.987225\pi\)
\(13\)	2.19590	+	3.80341i	0.609032	+	1.05488i	0.991400	+	0.130865i	\(0.0417754\pi\)
							−0.382368	+	0.924010i	\(0.624891\pi\)
\(17\)	−4.25701	−	2.45778i	−1.03248	−	0.596100i	−0.114783	−	0.993391i	\(-0.536617\pi\)
							−0.917693	+	0.397291i	\(0.869950\pi\)
\(19\)	0.537084	+	0.930256i	0.123215	+	0.213415i	0.921034	−	0.389482i	\(-0.127346\pi\)
							−0.797819	+	0.602898i	\(0.794013\pi\)
\(23\)			4.38312i			0.913943i	0.889481	+	0.456972i	\(0.151066\pi\)
							−0.889481	+	0.456972i	\(0.848934\pi\)
\(29\)	9.06214			1.68280			0.841399	−	0.540415i	\(-0.181733\pi\)
							0.841399	+	0.540415i	\(0.181733\pi\)
\(31\)			3.47979i			0.624989i	0.949920	+	0.312494i	\(0.101165\pi\)
							−0.949920	+	0.312494i	\(0.898835\pi\)
\(37\)	−0.468706	+	6.06468i	−0.0770548	+	0.997027i
							\(41\)	2.60116	+	4.50534i	0.406233	+	0.703616i	0.994464	−	0.105077i	\(-0.0335088\pi\)
−0.588231	+	0.808693i	\(0.700175\pi\)
\(43\)	2.65469			0.404836			0.202418	−	0.979299i	\(-0.435120\pi\)
							0.202418	+	0.979299i	\(0.435120\pi\)
\(47\)	9.88608			1.44203			0.721017	−	0.692918i	\(-0.243675\pi\)
							0.721017	+	0.692918i	\(0.243675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	296.2.q.a.101.7	yes	72
4.3	odd	2		1184.2.y.a.1137.34		72
8.3	odd	2		1184.2.y.a.1137.3		72
8.5	even	2	inner	296.2.q.a.101.20	yes	72
37.11	even	6	inner	296.2.q.a.85.20	yes	72
148.11	odd	6		1184.2.y.a.529.3		72
296.11	odd	6		1184.2.y.a.529.34		72
296.85	even	6	inner	296.2.q.a.85.7	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.q.a.85.7	✓	72	296.85	even	6	inner
296.2.q.a.85.20	yes	72	37.11	even	6	inner
296.2.q.a.101.7	yes	72	1.1	even	1	trivial
296.2.q.a.101.20	yes	72	8.5	even	2	inner
1184.2.y.a.529.3		72	148.11	odd	6
1184.2.y.a.529.34		72	296.11	odd	6
1184.2.y.a.1137.3		72	8.3	odd	2
1184.2.y.a.1137.34		72	4.3	odd	2