Embedded newform 116.2.l.b.31.4

• Label: 116.2.l.b.31.4
• Level: $116$
• Weight: $2$
• Character: 116.31
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• 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			31.4
Character	\(\chi\)	\(=\)	116.31
Dual form			116.2.l.b.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.444450 + 1.34256i) q^{2} +(-1.33661 + 2.12720i) q^{3} +(-1.60493 - 1.19340i) q^{4} +(-0.462574 + 0.368891i) q^{5} +(-2.26183 - 2.73990i) q^{6} +(-1.17855 + 0.268996i) q^{7} +(2.31552 - 1.62430i) q^{8} +(-1.43680 - 2.98354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.444450	+	1.34256i	−0.314274	+	0.949332i
							\(3\)	−1.33661	+	2.12720i	−0.771690	+	1.22814i	0.197132	+	0.980377i	\(0.436837\pi\)
−0.968821	+	0.247760i	\(0.920306\pi\)
\(5\)	−0.462574	+	0.368891i	−0.206869	+	0.164973i	−0.721444	−	0.692473i	\(-0.756521\pi\)
							0.514575	+	0.857446i	\(0.327950\pi\)
\(7\)	−1.17855	+	0.268996i	−0.445449	+	0.101671i	−0.439361	−	0.898311i	\(-0.644795\pi\)
							−0.00608794	+	0.999981i	\(0.501938\pi\)
\(11\)	−0.0746709	−	0.213397i	−0.0225141	−	0.0643416i	0.932078	−	0.362258i	\(-0.117994\pi\)
							−0.954592	+	0.297916i	\(0.903708\pi\)
\(13\)	−2.16393	+	4.49345i	−0.600167	+	1.24626i	0.350650	+	0.936507i	\(0.385961\pi\)
							−0.950817	+	0.309753i	\(0.899754\pi\)
\(17\)	1.53510	−	1.53510i	0.372316	−	0.372316i	−0.496004	−	0.868320i	\(-0.665200\pi\)
							0.868320	+	0.496004i	\(0.165200\pi\)
\(19\)	−4.81834	+	3.02756i	−1.10540	+	0.694571i	−0.955754	−	0.294168i	\(-0.904957\pi\)
							−0.149650	+	0.988739i	\(0.547815\pi\)
\(23\)	6.30724	+	5.02986i	1.31515	+	1.04880i	0.994837	+	0.101486i	\(0.0323599\pi\)
							0.320314	+	0.947311i	\(0.396212\pi\)
\(29\)	4.97931	+	2.05098i	0.924634	+	0.380857i
							\(31\)	0.762184	−	6.76457i	0.136892	−	1.21495i	−0.717961	−	0.696084i	\(-0.754924\pi\)
0.854853	−	0.518870i	\(-0.173647\pi\)
\(37\)	7.20883	+	2.52248i	1.18512	+	0.414693i	0.849645	−	0.527355i	\(-0.176816\pi\)
							0.335479	+	0.942048i	\(0.391102\pi\)
\(41\)	−5.76718	−	5.76718i	−0.900682	−	0.900682i	0.0948129	−	0.995495i	\(-0.469775\pi\)
							−0.995495	+	0.0948129i	\(0.969775\pi\)
\(43\)	6.38114	−	0.718982i	0.973114	−	0.109644i	0.388932	−	0.921266i	\(-0.372844\pi\)
							0.584182	+	0.811623i	\(0.301415\pi\)
\(47\)	7.38621	−	2.58455i	1.07739	−	0.376995i	0.267488	−	0.963561i	\(-0.413806\pi\)
							0.809901	+	0.586566i	\(0.199521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.31.4	yes	144
4.3	odd	2	inner	116.2.l.b.31.6	yes	144
29.15	odd	28	inner	116.2.l.b.15.6	yes	144
116.15	even	28	inner	116.2.l.b.15.4	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.15.4	✓	144	116.15	even	28	inner
116.2.l.b.15.6	yes	144	29.15	odd	28	inner
116.2.l.b.31.4	yes	144	1.1	even	1	trivial
116.2.l.b.31.6	yes	144	4.3	odd	2	inner