Embedded newform 115.4.g.a.16.7

• Label: 115.4.g.a.16.7
• Level: $115$
• Weight: $4$
• Character: 115.16
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $110$
• CM: no
• 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			16.7
Character	\(\chi\)	\(=\)	115.16
Dual form			115.4.g.a.36.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694897 - 0.801954i) q^{2} +(-7.82619 + 5.02959i) q^{3} +(0.978270 + 6.80402i) q^{4} +(2.07708 + 4.54816i) q^{5} +(-1.40490 + 9.77129i) q^{6} +(-10.2408 + 3.00696i) q^{7} +(13.2778 + 8.53312i) q^{8} +(24.7363 - 54.1649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.694897	−	0.801954i	0.245683	−	0.283534i	−0.619492	−	0.785003i	\(-0.712661\pi\)
							0.865176	+	0.501469i	\(0.167207\pi\)
\(3\)	−7.82619	+	5.02959i	−1.50615	+	0.967944i	−0.512112	+	0.858919i	\(0.671137\pi\)
							−0.994039	+	0.109026i	\(0.965227\pi\)
\(5\)	2.07708	+	4.54816i	0.185779	+	0.406800i
							\(7\)	−10.2408	+	3.00696i	−0.552950	+	0.162361i	−0.546258	−	0.837617i	\(-0.683948\pi\)
−0.00669189	+	0.999978i	\(0.502130\pi\)
\(11\)	−34.3010	−	39.5855i	−0.940195	−	1.08504i	−0.996242	−	0.0866151i	\(-0.972395\pi\)
							0.0560465	−	0.998428i	\(-0.482150\pi\)
\(13\)	0.320353	+	0.0940641i	0.00683461	+	0.00200682i	0.285148	−	0.958483i	\(-0.407957\pi\)
							−0.278313	+	0.960490i	\(0.589775\pi\)
\(17\)	−14.2892	+	99.3835i	−0.203861	+	1.41788i	0.588828	+	0.808258i	\(0.299590\pi\)
							−0.792689	+	0.609626i	\(0.791320\pi\)
\(19\)	−8.47821	−	58.9673i	−0.102370	−	0.712001i	−0.974771	−	0.223209i	\(-0.928347\pi\)
							0.872400	−	0.488792i	\(-0.162562\pi\)
\(23\)	62.2038	−	91.0916i	0.563930	−	0.825823i
							\(29\)	10.5088	−	73.0904i	0.0672910	−	0.468019i	−0.928116	−	0.372290i	\(-0.878573\pi\)
0.995407	−	0.0957290i	\(-0.0305182\pi\)
\(31\)	−193.744	−	124.512i	−1.12250	−	0.721385i	−0.158516	−	0.987356i	\(-0.550671\pi\)
							−0.963981	+	0.265971i	\(0.914307\pi\)
\(37\)	−74.9816	+	164.187i	−0.333160	+	0.729518i	−0.999875	−	0.0157944i	\(-0.994972\pi\)
							0.666716	+	0.745312i	\(0.267700\pi\)
\(41\)	132.082	+	289.219i	0.503116	+	1.10167i	0.975444	+	0.220247i	\(0.0706864\pi\)
							−0.472328	+	0.881423i	\(0.656586\pi\)
\(43\)	25.0013	−	16.0673i	0.0886664	−	0.0569824i	−0.495556	−	0.868576i	\(-0.665036\pi\)
							0.584223	+	0.811593i	\(0.301399\pi\)
\(47\)	−456.903			−1.41800			−0.709001	−	0.705207i	\(-0.750854\pi\)
							−0.709001	+	0.705207i	\(0.750854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.16.7	✓	110
23.13	even	11	inner	115.4.g.a.36.7	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.16.7	✓	110	1.1	even	1	trivial
115.4.g.a.36.7	yes	110	23.13	even	11	inner