Embedded newform 69.5.h.a.2.10

• Label: 69.5.h.a.2.10
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85051 + 2.87944i) q^{2} +(0.149878 - 8.99875i) q^{3} +(1.77982 + 3.89726i) q^{4} +(13.0909 - 44.5836i) q^{5} +(25.6340 + 17.0838i) q^{6} +(-46.3292 + 53.4668i) q^{7} +(-68.7229 - 9.88086i) q^{8} +(-80.9551 - 2.69742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85051	+	2.87944i	−0.462627	+	0.719861i	−0.991680	−	0.128728i	\(-0.958911\pi\)
							0.529053	+	0.848588i	\(0.322547\pi\)
\(3\)	0.149878	−	8.99875i	0.0166531	−	0.999861i
							\(5\)	13.0909	−	44.5836i	0.523638	−	1.78335i	−0.0925065	−	0.995712i	\(-0.529488\pi\)
0.616144	−	0.787634i	\(-0.288694\pi\)
\(7\)	−46.3292	+	53.4668i	−0.945494	+	1.09116i	0.0502260	+	0.998738i	\(0.484006\pi\)
							−0.995720	+	0.0924204i	\(0.970540\pi\)
\(11\)	17.7376	+	27.6002i	0.146591	+	0.228101i	0.906782	−	0.421599i	\(-0.138531\pi\)
							−0.760191	+	0.649700i	\(0.774895\pi\)
\(13\)	−149.338	−	172.345i	−0.883654	−	1.01979i	−0.999648	−	0.0265411i	\(-0.991551\pi\)
							0.115994	−	0.993250i	\(-0.462995\pi\)
\(17\)	−135.211	−	61.7488i	−0.467858	−	0.213664i	0.167514	−	0.985870i	\(-0.446426\pi\)
							−0.635372	+	0.772206i	\(0.719153\pi\)
\(19\)	10.2987	+	22.5510i	0.0285283	+	0.0624682i	0.923360	−	0.383936i	\(-0.125432\pi\)
							−0.894831	+	0.446404i	\(0.852704\pi\)
\(23\)	155.292	−	505.693i	0.293558	−	0.955941i
							\(29\)	−712.225	−	325.262i	−0.846878	−	0.386756i	−0.0557782	−	0.998443i	\(-0.517764\pi\)
−0.791100	+	0.611687i	\(0.790491\pi\)
\(31\)	−137.489	+	956.257i	−0.143069	+	0.995065i	0.784157	+	0.620562i	\(0.213096\pi\)
							−0.927226	+	0.374502i	\(0.877814\pi\)
\(37\)	2489.61	−	731.016i	1.81856	−	0.533978i	0.819336	−	0.573313i	\(-0.194342\pi\)
							0.999226	+	0.0393350i	\(0.0125240\pi\)
\(41\)	−56.8853	+	193.734i	−0.0338402	+	0.115249i	−0.974681	−	0.223600i	\(-0.928219\pi\)
							0.940841	+	0.338848i	\(0.110037\pi\)
\(43\)	−39.1456	−	272.264i	−0.0211712	−	0.147249i	0.976494	−	0.215545i	\(-0.0691529\pi\)
							−0.997665	+	0.0682960i	\(0.978244\pi\)
\(47\)		−	163.297i		−	0.0739234i	−0.999317	−	0.0369617i	\(-0.988232\pi\)
							0.999317	−	0.0369617i	\(-0.0117680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.10	✓	300
3.2	odd	2	inner	69.5.h.a.2.21	yes	300
23.12	even	11	inner	69.5.h.a.35.21	yes	300
69.35	odd	22	inner	69.5.h.a.35.10	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.10	✓	300	1.1	even	1	trivial
69.5.h.a.2.21	yes	300	3.2	odd	2	inner
69.5.h.a.35.10	yes	300	69.35	odd	22	inner
69.5.h.a.35.21	yes	300	23.12	even	11	inner