Embedded newform 116.2.l.b.11.11

• Label: 116.2.l.b.11.11
• Level: $116$
• Weight: $2$
• Character: 116.11
• 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			11.11
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13737 + 0.840464i) q^{2} +(0.198741 - 0.0223927i) q^{3} +(0.587240 + 1.91184i) q^{4} +(1.17216 - 2.43402i) q^{5} +(0.244863 + 0.141566i) q^{6} +(-0.635177 - 0.506537i) q^{7} +(-0.938925 + 2.66804i) q^{8} +(-2.88579 + 0.658662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13737	+	0.840464i	0.804245	+	0.594298i
							\(3\)	0.198741	−	0.0223927i	0.114743	−	0.0129284i	−0.0544062	−	0.998519i	\(-0.517327\pi\)
0.169149	+	0.985590i	\(0.445898\pi\)
\(5\)	1.17216	−	2.43402i	0.524207	−	1.08853i	−0.455895	−	0.890034i	\(-0.650681\pi\)
							0.980102	−	0.198494i	\(-0.0636051\pi\)
\(7\)	−0.635177	−	0.506537i	−0.240074	−	0.191453i	0.496060	−	0.868288i	\(-0.334780\pi\)
							−0.736135	+	0.676835i	\(0.763351\pi\)
\(11\)	−3.17740	+	1.99649i	−0.958022	+	0.601965i	−0.917722	−	0.397223i	\(-0.869974\pi\)
							−0.0402996	+	0.999188i	\(0.512831\pi\)
\(13\)	1.70771	+	0.389773i	0.473633	+	0.108104i	0.452673	−	0.891677i	\(-0.350471\pi\)
							0.0209601	+	0.999780i	\(0.493328\pi\)
\(17\)	0.662411	−	0.662411i	0.160658	−	0.160658i	−0.622200	−	0.782858i	\(-0.713761\pi\)
							0.782858	+	0.622200i	\(0.213761\pi\)
\(19\)	0.725538	−	6.43933i	0.166450	−	1.47728i	−0.581510	−	0.813539i	\(-0.697538\pi\)
							0.747960	−	0.663744i	\(-0.231033\pi\)
\(23\)	−2.06557	−	4.28920i	−0.430701	−	0.894360i	−0.997511	−	0.0705114i	\(-0.977537\pi\)
							0.566810	−	0.823849i	\(-0.308177\pi\)
\(29\)	3.86646	+	3.74839i	0.717984	+	0.696059i
							\(31\)	2.22131	+	6.34814i	0.398959	+	1.14016i	0.951759	+	0.306845i	\(0.0992734\pi\)
−0.552801	+	0.833314i	\(0.686441\pi\)
\(37\)	−1.57097	+	2.50018i	−0.258265	+	0.411027i	−0.950538	−	0.310610i	\(-0.899467\pi\)
							0.692272	+	0.721636i	\(0.256610\pi\)
\(41\)	−3.80228	−	3.80228i	−0.593817	−	0.593817i	0.344843	−	0.938660i	\(-0.387932\pi\)
							−0.938660	+	0.344843i	\(0.887932\pi\)
\(43\)	11.2470	+	3.93549i	1.71515	+	0.600156i	0.995550	−	0.0942354i	\(-0.0300406\pi\)
							0.719597	+	0.694391i	\(0.244326\pi\)
\(47\)	0.750767	+	1.19484i	0.109511	+	0.174285i	0.896983	−	0.442064i	\(-0.145754\pi\)
							−0.787473	+	0.616350i	\(0.788611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.11.11	yes	144
4.3	odd	2	inner	116.2.l.b.11.8	✓	144
29.8	odd	28	inner	116.2.l.b.95.8	yes	144
116.95	even	28	inner	116.2.l.b.95.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.8	✓	144	4.3	odd	2	inner
116.2.l.b.11.11	yes	144	1.1	even	1	trivial
116.2.l.b.95.8	yes	144	29.8	odd	28	inner
116.2.l.b.95.11	yes	144	116.95	even	28	inner