Embedded newform 115.4.g.a.16.5

• Label: 115.4.g.a.16.5
• Level: $115$
• Weight: $4$
• Character: 115.16
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $110$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(110\)
Relative dimension:	\(11\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			16.5
Character	\(\chi\)	\(=\)	115.16
Dual form			115.4.g.a.36.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.861960 + 0.994755i) q^{2} +(-0.892278 + 0.573432i) q^{3} +(0.891957 + 6.20369i) q^{4} +(2.07708 + 4.54816i) q^{5} +(0.198683 - 1.38187i) q^{6} +(31.2385 - 9.17246i) q^{7} +(-15.7984 - 10.1530i) q^{8} +(-10.7489 + 23.5367i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.861960	+	0.994755i	−0.304749	+	0.351699i	−0.887381	−	0.461037i	\(-0.847477\pi\)
							0.582632	+	0.812736i	\(0.302023\pi\)
\(3\)	−0.892278	+	0.573432i	−0.171719	+	0.110357i	−0.623677	−	0.781682i	\(-0.714362\pi\)
							0.451958	+	0.892039i	\(0.350726\pi\)
\(5\)	2.07708	+	4.54816i	0.185779	+	0.406800i
							\(7\)	31.2385	−	9.17246i	1.68672	−	0.495266i	0.709007	−	0.705202i	\(-0.249144\pi\)
0.977715	+	0.209936i	\(0.0673254\pi\)
\(11\)	12.0872	+	13.9493i	0.331310	+	0.382352i	0.896825	−	0.442386i	\(-0.145868\pi\)
							−0.565515	+	0.824738i	\(0.691322\pi\)
\(13\)	−46.8944	−	13.7694i	−1.00047	−	0.293766i	−0.259824	−	0.965656i	\(-0.583665\pi\)
							−0.740651	+	0.671890i	\(0.765483\pi\)
\(17\)	−16.3103	+	113.440i	−0.232695	+	1.61843i	0.453667	+	0.891171i	\(0.350115\pi\)
							−0.686363	+	0.727260i	\(0.740794\pi\)
\(19\)	−0.754174	−	5.24540i	−0.00910629	−	0.0633356i	0.984762	−	0.173910i	\(-0.0556401\pi\)
							−0.993868	+	0.110574i	\(0.964731\pi\)
\(23\)	91.9473	+	60.9319i	0.833579	+	0.552400i
							\(29\)	−8.90313	+	61.9226i	−0.0570093	+	0.396508i	0.941259	+	0.337685i	\(0.109644\pi\)
−0.998268	+	0.0588232i	\(0.981265\pi\)
\(31\)	−253.699	−	163.043i	−1.46986	−	0.944624i	−0.998017	−	0.0629427i	\(-0.979951\pi\)
							−0.471846	−	0.881681i	\(-0.656412\pi\)
\(37\)	−89.3887	+	195.734i	−0.397173	+	0.869688i	0.600376	+	0.799718i	\(0.295018\pi\)
							−0.997549	+	0.0699702i	\(0.977710\pi\)
\(41\)	−25.1319	−	55.0311i	−0.0957302	−	0.209620i	0.855709	−	0.517458i	\(-0.173122\pi\)
							−0.951439	+	0.307838i	\(0.900394\pi\)
\(43\)	367.457	−	236.150i	1.30318	−	0.837502i	0.309624	−	0.950859i	\(-0.399797\pi\)
							0.993554	+	0.113357i	\(0.0361604\pi\)
\(47\)	576.204			1.78825			0.894127	−	0.447813i	\(-0.147797\pi\)
							0.894127	+	0.447813i	\(0.147797\pi\)