Embedded newform 115.4.g.a.16.4

• Label: 115.4.g.a.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	115.16
Dual form			115.4.g.a.36.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69252 + 1.95327i) q^{2} +(-4.05620 + 2.60676i) q^{3} +(0.187875 + 1.30670i) q^{4} +(2.07708 + 4.54816i) q^{5} +(1.77348 - 12.3348i) q^{6} +(-18.0212 + 5.29151i) q^{7} +(-20.2644 - 13.0231i) q^{8} +(-1.55865 + 3.41296i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69252	+	1.95327i	−0.598395	+	0.690584i	−0.971455	−	0.237223i	\(-0.923763\pi\)
							0.373061	+	0.927807i	\(0.378308\pi\)
\(3\)	−4.05620	+	2.60676i	−0.780616	+	0.501672i	−0.869238	−	0.494394i	\(-0.835390\pi\)
							0.0886218	+	0.996065i	\(0.471754\pi\)
\(5\)	2.07708	+	4.54816i	0.185779	+	0.406800i
							\(7\)	−18.0212	+	5.29151i	−0.973055	+	0.285715i	−0.729355	−	0.684136i	\(-0.760180\pi\)
−0.243701	+	0.969851i	\(0.578361\pi\)
\(11\)	23.4071	+	27.0133i	0.641592	+	0.740437i	0.979655	−	0.200687i	\(-0.0643173\pi\)
							−0.338064	+	0.941123i	\(0.609772\pi\)
\(13\)	12.6925	+	3.72685i	0.270790	+	0.0795110i	0.414308	−	0.910137i	\(-0.364024\pi\)
							−0.143519	+	0.989648i	\(0.545842\pi\)
\(17\)	5.64595	−	39.2685i	0.0805497	−	0.560235i	−0.909083	−	0.416615i	\(-0.863216\pi\)
							0.989633	−	0.143621i	\(-0.0458745\pi\)
\(19\)	−19.5882	−	136.239i	−0.236519	−	1.64502i	−0.668915	−	0.743339i	\(-0.733241\pi\)
							0.432396	−	0.901684i	\(-0.357668\pi\)
\(23\)	−38.5347	−	103.354i	−0.349350	−	0.936992i
							\(29\)	−37.7953	+	262.872i	−0.242014	+	1.68325i	0.399969	+	0.916529i	\(0.369021\pi\)
−0.641983	+	0.766719i	\(0.721888\pi\)
\(31\)	219.068	+	140.787i	1.26922	+	0.815678i	0.989518	−	0.144409i	\(-0.0461282\pi\)
							0.279701	+	0.960087i	\(0.409765\pi\)
\(37\)	−165.735	+	362.908i	−0.736395	+	1.61248i	0.0529995	+	0.998595i	\(0.483122\pi\)
							−0.789395	+	0.613886i	\(0.789605\pi\)
\(41\)	−190.422	−	416.965i	−0.725338	−	1.58827i	−0.806270	−	0.591548i	\(-0.798517\pi\)
							0.0809319	−	0.996720i	\(-0.474210\pi\)
\(43\)	−92.4431	+	59.4096i	−0.327848	+	0.210695i	−0.694201	−	0.719781i	\(-0.744242\pi\)
							0.366354	+	0.930476i	\(0.380606\pi\)
\(47\)	−169.504			−0.526058			−0.263029	−	0.964788i	\(-0.584721\pi\)
							−0.263029	+	0.964788i	\(0.584721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.16.4	✓	110
23.13	even	11	inner	115.4.g.a.36.4	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.16.4	✓	110	1.1	even	1	trivial
115.4.g.a.36.4	yes	110	23.13	even	11	inner