Embedded newform 116.2.l.b.15.11

• Label: 116.2.l.b.15.11
• 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			15.11
Character	\(\chi\)	\(=\)	116.15
Dual form			116.2.l.b.31.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29785 - 0.561759i) q^{2} +(1.55227 + 2.47043i) q^{3} +(1.36885 - 1.45816i) q^{4} +(-2.53559 - 2.02207i) q^{5} +(3.40241 + 2.33426i) q^{6} +(-2.96001 - 0.675603i) q^{7} +(0.957438 - 2.66145i) q^{8} +(-2.39182 + 4.96667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29785	−	0.561759i	0.917722	−	0.397224i
							\(3\)	1.55227	+	2.47043i	0.896206	+	1.42630i	0.904563	+	0.426339i	\(0.140197\pi\)
−0.00835729	+	0.999965i	\(0.502660\pi\)
\(5\)	−2.53559	−	2.02207i	−1.13395	−	0.904295i	−0.137671	−	0.990478i	\(-0.543962\pi\)
							−0.996279	+	0.0861827i	\(0.972533\pi\)
\(7\)	−2.96001	−	0.675603i	−1.11878	−	0.255354i	−0.377142	−	0.926155i	\(-0.623093\pi\)
							−0.741637	+	0.670801i	\(0.765950\pi\)
\(11\)	−0.356315	+	1.01829i	−0.107433	+	0.307026i	−0.985032	−	0.172369i	\(-0.944858\pi\)
							0.877600	+	0.479394i	\(0.159144\pi\)
\(13\)	1.93837	+	4.02507i	0.537607	+	1.11635i	0.976041	+	0.217589i	\(0.0698191\pi\)
							−0.438434	+	0.898764i	\(0.644467\pi\)
\(17\)	−1.15274	−	1.15274i	−0.279579	−	0.279579i	0.553362	−	0.832941i	\(-0.313345\pi\)
							−0.832941	+	0.553362i	\(0.813345\pi\)
\(19\)	−0.711839	−	0.447278i	−0.163307	−	0.102613i	0.447899	−	0.894084i	\(-0.352172\pi\)
							−0.611206	+	0.791472i	\(0.709315\pi\)
\(23\)	4.45472	−	3.55252i	0.928874	−	0.740752i	−0.0371215	−	0.999311i	\(-0.511819\pi\)
							0.965996	+	0.258558i	\(0.0832474\pi\)
\(29\)	4.63270	−	2.74556i	0.860271	−	0.509837i
							\(31\)	0.542220	+	4.81233i	0.0973855	+	0.864321i	0.943512	+	0.331338i	\(0.107500\pi\)
−0.846126	+	0.532982i	\(0.821071\pi\)
\(37\)	−4.22908	+	1.47982i	−0.695257	+	0.243281i	−0.654692	−	0.755896i	\(-0.727202\pi\)
							−0.0405651	+	0.999177i	\(0.512916\pi\)
\(41\)	−3.70601	+	3.70601i	−0.578782	+	0.578782i	−0.934568	−	0.355786i	\(-0.884213\pi\)
							0.355786	+	0.934568i	\(0.384213\pi\)
\(43\)	8.74943	+	0.985824i	1.33428	+	0.150337i	0.750118	−	0.661304i	\(-0.229997\pi\)
							0.584157	+	0.811641i	\(0.301425\pi\)
\(47\)	−6.77839	−	2.37186i	−0.988730	−	0.345972i	−0.213011	−	0.977050i	\(-0.568327\pi\)
							−0.775719	+	0.631078i	\(0.782613\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.15.11	✓	144
4.3	odd	2	inner	116.2.l.b.15.12	yes	144
29.2	odd	28	inner	116.2.l.b.31.12	yes	144
116.31	even	28	inner	116.2.l.b.31.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.15.11	✓	144	1.1	even	1	trivial
116.2.l.b.15.12	yes	144	4.3	odd	2	inner
116.2.l.b.31.11	yes	144	116.31	even	28	inner
116.2.l.b.31.12	yes	144	29.2	odd	28	inner