Embedded newform 116.2.l.b.19.12

• Label: 116.2.l.b.19.12
• Level: $116$
• Weight: $2$
• Character: 116.19
• 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			19.12
Character	\(\chi\)	\(=\)	116.19
Dual form			116.2.l.b.55.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39663 - 0.222345i) q^{2} +(-1.60021 - 0.559939i) q^{3} +(1.90113 - 0.621065i) q^{4} +(1.69266 + 0.386339i) q^{5} +(-2.35940 - 0.426226i) q^{6} +(-0.260564 - 0.541066i) q^{7} +(2.51707 - 1.29010i) q^{8} +(-0.0983443 - 0.0784270i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39663	−	0.222345i	0.987563	−	0.157222i
							\(3\)	−1.60021	−	0.559939i	−0.923883	−	0.323281i	−0.173920	−	0.984760i	\(-0.555643\pi\)
−0.749964	+	0.661479i	\(0.769929\pi\)
\(5\)	1.69266	+	0.386339i	0.756982	+	0.172776i	0.583559	−	0.812071i	\(-0.301660\pi\)
							0.173424	+	0.984847i	\(0.444517\pi\)
\(7\)	−0.260564	−	0.541066i	−0.0984839	−	0.204504i	0.845906	−	0.533332i	\(-0.179060\pi\)
							−0.944390	+	0.328829i	\(0.893346\pi\)
\(11\)	−0.116172	+	1.03105i	−0.0350272	+	0.310875i	0.963967	+	0.266024i	\(0.0857099\pi\)
							−0.998994	+	0.0448511i	\(0.985719\pi\)
\(13\)	−3.87172	+	3.08759i	−1.07382	+	0.856344i	−0.990131	−	0.140148i	\(-0.955242\pi\)
							−0.0836906	+	0.996492i	\(0.526671\pi\)
\(17\)	−1.06372	+	1.06372i	−0.257990	+	0.257990i	−0.824236	−	0.566246i	\(-0.808395\pi\)
							0.566246	+	0.824236i	\(0.308395\pi\)
\(19\)	1.36842	+	3.91071i	0.313937	+	0.897180i	0.987440	+	0.157995i	\(0.0505031\pi\)
							−0.673503	+	0.739184i	\(0.735211\pi\)
\(23\)	−7.10568	+	1.62182i	−1.48164	+	0.338174i	−0.885474	−	0.464690i	\(-0.846166\pi\)
							−0.596163	+	0.802864i	\(0.703309\pi\)
\(29\)	1.22082	−	5.24496i	0.226701	−	0.973964i
							\(31\)	3.05397	−	1.91894i	0.548510	−	0.344651i	−0.229096	−	0.973404i	\(-0.573577\pi\)
0.777606	+	0.628752i	\(0.216434\pi\)
\(37\)	5.84292	−	0.658339i	0.960570	−	0.108230i	0.382267	−	0.924052i	\(-0.375144\pi\)
							0.578304	+	0.815822i	\(0.303715\pi\)
\(41\)	−3.60638	−	3.60638i	−0.563222	−	0.563222i	0.366999	−	0.930221i	\(-0.380385\pi\)
							−0.930221	+	0.366999i	\(0.880385\pi\)
\(43\)	4.74262	−	7.54784i	0.723243	−	1.15104i	−0.259443	−	0.965758i	\(-0.583539\pi\)
							0.982686	−	0.185277i	\(-0.0593182\pi\)
\(47\)	0.738546	+	0.0832142i	0.107728	+	0.0121380i	0.165664	−	0.986182i	\(-0.447023\pi\)
							−0.0579358	+	0.998320i	\(0.518452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.19.12	yes	144
4.3	odd	2	inner	116.2.l.b.19.4	✓	144
29.26	odd	28	inner	116.2.l.b.55.4	yes	144
116.55	even	28	inner	116.2.l.b.55.12	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.19.4	✓	144	4.3	odd	2	inner
116.2.l.b.19.12	yes	144	1.1	even	1	trivial
116.2.l.b.55.4	yes	144	29.26	odd	28	inner
116.2.l.b.55.12	yes	144	116.55	even	28	inner