Embedded newform 116.2.l.b.15.10

• Label: 116.2.l.b.15.10
• Level: $116$
• Weight: $2$
• Character: 116.15
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.l (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			15.10
Character	\(\chi\)	\(=\)	116.15
Dual form			116.2.l.b.31.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03719 + 0.961375i) q^{2} +(0.381326 + 0.606877i) q^{3} +(0.151518 + 1.99425i) q^{4} +(-0.388548 - 0.309856i) q^{5} +(-0.187929 + 0.996042i) q^{6} +(-1.45039 - 0.331043i) q^{7} +(-1.76007 + 2.21408i) q^{8} +(1.07876 - 2.24007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03719	+	0.961375i	0.733403	+	0.679794i
							\(3\)	0.381326	+	0.606877i	0.220159	+	0.350380i	0.938482	−	0.345329i	\(-0.112233\pi\)
−0.718323	+	0.695710i	\(0.755090\pi\)
\(5\)	−0.388548	−	0.309856i	−0.173764	−	0.138572i	0.532746	−	0.846275i	\(-0.321160\pi\)
							−0.706510	+	0.707703i	\(0.749731\pi\)
\(7\)	−1.45039	−	0.331043i	−0.548197	−	0.125122i	−0.0605532	−	0.998165i	\(-0.519286\pi\)
							−0.487644	+	0.873043i	\(0.662144\pi\)
\(11\)	1.07892	−	3.08337i	0.325306	−	0.929670i	−0.658836	−	0.752286i	\(-0.728951\pi\)
							0.984142	−	0.177383i	\(-0.0567633\pi\)
\(13\)	0.610177	+	1.26705i	0.169233	+	0.351416i	0.968286	−	0.249845i	\(-0.0803797\pi\)
							−0.799053	+	0.601261i	\(0.794665\pi\)
\(17\)	2.80181	+	2.80181i	0.679538	+	0.679538i	0.959896	−	0.280358i	\(-0.0904531\pi\)
							−0.280358	+	0.959896i	\(0.590453\pi\)
\(19\)	−4.35196	−	2.73452i	−0.998409	−	0.627342i	−0.0694732	−	0.997584i	\(-0.522132\pi\)
							−0.928935	+	0.370242i	\(0.879275\pi\)
\(23\)	0.0143935	−	0.0114784i	0.00300124	−	0.00239341i	−0.621988	−	0.783027i	\(-0.713675\pi\)
							0.624989	+	0.780633i	\(0.285103\pi\)
\(29\)	−3.77159	+	3.84384i	−0.700366	+	0.713784i
							\(31\)	0.262515	+	2.32988i	0.0471491	+	0.418460i	0.995010	+	0.0997753i	\(0.0318124\pi\)
−0.947861	+	0.318684i	\(0.896759\pi\)
\(37\)	−1.69422	+	0.592834i	−0.278528	+	0.0974613i	−0.465929	−	0.884822i	\(-0.654280\pi\)
							0.187400	+	0.982284i	\(0.439994\pi\)
\(41\)	−0.378647	+	0.378647i	−0.0591347	+	0.0591347i	−0.736056	−	0.676921i	\(-0.763314\pi\)
							0.676921	+	0.736056i	\(0.263314\pi\)
\(43\)	−3.80953	−	0.429231i	−0.580948	−	0.0654571i	−0.183400	−	0.983038i	\(-0.558710\pi\)
							−0.397548	+	0.917581i	\(0.630139\pi\)
\(47\)	8.34769	+	2.92098i	1.21764	+	0.426069i	0.861207	−	0.508254i	\(-0.169709\pi\)
							0.356428	+	0.934323i	\(0.383995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.15.10	yes	144
4.3	odd	2	inner	116.2.l.b.15.9	✓	144
29.2	odd	28	inner	116.2.l.b.31.9	yes	144
116.31	even	28	inner	116.2.l.b.31.10	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.15.9	✓	144	4.3	odd	2	inner
116.2.l.b.15.10	yes	144	1.1	even	1	trivial
116.2.l.b.31.9	yes	144	29.2	odd	28	inner
116.2.l.b.31.10	yes	144	116.31	even	28	inner