Embedded newform 115.4.g.a.6.7

• Label: 115.4.g.a.6.7
• Level: $115$
• Weight: $4$
• Character: 115.6
• 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			6.7
Character	\(\chi\)	\(=\)	115.6
Dual form			115.4.g.a.96.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.274193 - 0.600398i) q^{2} +(0.833837 + 0.244837i) q^{3} +(4.95359 + 5.71675i) q^{4} +(4.20627 - 2.70320i) q^{5} +(0.375631 - 0.433502i) q^{6} +(-4.72226 + 32.8441i) q^{7} +(9.85703 - 2.89429i) q^{8} +(-22.0785 - 14.1890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.274193	−	0.600398i	0.0969417	−	0.212273i	−0.854948	−	0.518714i	\(-0.826411\pi\)
							0.951890	+	0.306441i	\(0.0991383\pi\)
\(3\)	0.833837	+	0.244837i	0.160472	+	0.0471188i	0.360982	−	0.932573i	\(-0.382442\pi\)
							−0.200510	+	0.979692i	\(0.564260\pi\)
\(5\)	4.20627	−	2.70320i	0.376220	−	0.241782i
							\(7\)	−4.72226	+	32.8441i	−0.254978	+	1.77341i	0.312396	+	0.949952i	\(0.398868\pi\)
−0.567374	+	0.823460i	\(0.692041\pi\)
\(11\)	17.0273	+	37.2845i	0.466719	+	1.02197i	0.985904	+	0.167311i	\(0.0535083\pi\)
							−0.519185	+	0.854662i	\(0.673764\pi\)
\(13\)	−4.80412	−	33.4134i	−0.102494	−	0.712862i	−0.974666	−	0.223664i	\(-0.928198\pi\)
							0.872172	−	0.489199i	\(-0.162711\pi\)
\(17\)	21.3976	−	24.6941i	0.305275	−	0.352306i	−0.582296	−	0.812977i	\(-0.697846\pi\)
							0.887571	+	0.460671i	\(0.152391\pi\)
\(19\)	44.5912	+	51.4610i	0.538417	+	0.621367i	0.958145	−	0.286283i	\(-0.0924198\pi\)
							−0.419728	+	0.907650i	\(0.637874\pi\)
\(23\)	52.9307	+	96.7747i	0.479862	+	0.877344i
							\(29\)	164.016	−	189.285i	1.05024	−	1.21205i	0.0735756	−	0.997290i	\(-0.476559\pi\)
0.976668	−	0.214756i	\(-0.0688956\pi\)
\(31\)	134.639	−	39.5336i	0.780060	−	0.229046i	0.132625	−	0.991166i	\(-0.457660\pi\)
							0.647436	+	0.762120i	\(0.275841\pi\)
\(37\)	−369.717	−	237.603i	−1.64273	−	1.05572i	−0.938225	−	0.346025i	\(-0.887531\pi\)
							−0.704508	−	0.709696i	\(-0.748832\pi\)
\(41\)	179.988	−	115.671i	0.685596	−	0.440606i	−0.150922	−	0.988546i	\(-0.548224\pi\)
							0.836518	+	0.547940i	\(0.184588\pi\)
\(43\)	−73.7935	−	21.6677i	−0.261707	−	0.0768441i	0.148247	−	0.988950i	\(-0.452637\pi\)
							−0.409954	+	0.912106i	\(0.634455\pi\)
\(47\)	−419.749			−1.30270			−0.651348	−	0.758779i	\(-0.725796\pi\)
							−0.651348	+	0.758779i	\(0.725796\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.6.7	✓	110
23.4	even	11	inner	115.4.g.a.96.7	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.6.7	✓	110	1.1	even	1	trivial
115.4.g.a.96.7	yes	110	23.4	even	11	inner