Embedded newform 69.6.g.a.5.14

• Label: 69.6.g.a.5.14
• Level: $69$
• Weight: $6$
• Character: 69.5
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $380$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(380\)
Relative dimension:	\(38\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.14
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13703 + 3.87238i) q^{2} +(-14.7589 - 5.01748i) q^{3} +(13.2177 + 8.49447i) q^{4} +(6.88235 - 47.8678i) q^{5} +(36.2109 - 51.4470i) q^{6} +(-29.0361 + 13.2603i) q^{7} +(-145.526 + 126.099i) q^{8} +(192.650 + 148.105i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13703	+	3.87238i	−0.201001	+	0.684546i	0.795868	+	0.605470i	\(0.207015\pi\)
							−0.996869	+	0.0790755i	\(0.974803\pi\)
\(3\)	−14.7589	−	5.01748i	−0.946783	−	0.321871i
							\(5\)	6.88235	−	47.8678i	0.123115	−	0.856285i	−0.830878	−	0.556455i	\(-0.812161\pi\)
0.953993	−	0.299830i	\(-0.0969298\pi\)
\(7\)	−29.0361	+	13.2603i	−0.223972	+	0.102284i	−0.524241	−	0.851570i	\(-0.675651\pi\)
							0.300269	+	0.953855i	\(0.402924\pi\)
\(11\)	606.675	−	178.136i	1.51173	−	0.443884i	0.582328	−	0.812954i	\(-0.302142\pi\)
							0.929402	+	0.369070i	\(0.120324\pi\)
\(13\)	−194.435	+	425.752i	−0.319091	+	0.698713i	−0.999415	−	0.0341990i	\(-0.989112\pi\)
							0.680324	+	0.732912i	\(0.261839\pi\)
\(17\)	−1751.05	+	1125.33i	−1.46952	+	0.944405i	−0.471479	+	0.881877i	\(0.656280\pi\)
							−0.998043	+	0.0625271i	\(0.980084\pi\)
\(19\)	−1018.97	+	1585.55i	−0.647559	+	1.00762i	0.349929	+	0.936776i	\(0.386206\pi\)
							−0.997487	+	0.0708447i	\(0.977431\pi\)
\(23\)	−1616.26	+	1955.51i	−0.637078	+	0.770799i
							\(29\)	−609.607	−	948.566i	−0.134603	−	0.209446i	0.767408	−	0.641159i	\(-0.221546\pi\)
−0.902011	+	0.431712i	\(0.857910\pi\)
\(31\)	7004.37	+	8083.47i	1.30908	+	1.51075i	0.678267	+	0.734816i	\(0.262731\pi\)
							0.630809	+	0.775938i	\(0.282723\pi\)
\(37\)	12032.4	−	1729.99i	1.44493	−	0.207749i	0.625184	−	0.780478i	\(-0.285024\pi\)
							0.819745	+	0.572728i	\(0.194115\pi\)
\(41\)	9662.26	+	1389.22i	0.897675	+	0.129066i	0.575680	−	0.817675i	\(-0.304737\pi\)
							0.321995	+	0.946741i	\(0.395647\pi\)
\(43\)	−10163.7	−	8806.89i	−0.838263	−	0.726359i	0.125794	−	0.992056i	\(-0.459852\pi\)
							−0.964056	+	0.265697i	\(0.914398\pi\)
\(47\)			18133.1i			1.19737i	0.800986	+	0.598683i	\(0.204309\pi\)
							−0.800986	+	0.598683i	\(0.795691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.14	✓	380
3.2	odd	2	inner	69.6.g.a.5.25	yes	380
23.14	odd	22	inner	69.6.g.a.14.25	yes	380
69.14	even	22	inner	69.6.g.a.14.14	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.14	✓	380	1.1	even	1	trivial
69.6.g.a.5.25	yes	380	3.2	odd	2	inner
69.6.g.a.14.14	yes	380	69.14	even	22	inner
69.6.g.a.14.25	yes	380	23.14	odd	22	inner