Embedded newform 69.5.h.a.2.2

• Label: 69.5.h.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.08077 + 6.34980i) q^{2} +(-1.82217 + 8.81361i) q^{3} +(-17.0207 - 37.2701i) q^{4} +(-1.39862 + 4.76328i) q^{5} +(-48.5289 - 47.5367i) q^{6} +(54.0931 - 62.4268i) q^{7} +(186.576 + 26.8256i) q^{8} +(-74.3594 - 32.1197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.08077	+	6.34980i	−1.02019	+	1.58745i	−0.231336	+	0.972874i	\(0.574310\pi\)
							−0.788857	+	0.614577i	\(0.789327\pi\)
\(3\)	−1.82217	+	8.81361i	−0.202463	+	0.979290i
							\(5\)	−1.39862	+	4.76328i	−0.0559449	+	0.190531i	−0.982724	−	0.185075i	\(-0.940747\pi\)
0.926779	+	0.375606i	\(0.122565\pi\)
\(7\)	54.0931	−	62.4268i	1.10394	−	1.27402i	0.145303	−	0.989387i	\(-0.453584\pi\)
							0.958638	−	0.284629i	\(-0.0918703\pi\)
\(11\)	−27.9081	−	43.4258i	−0.230645	−	0.358891i	0.706570	−	0.707644i	\(-0.250242\pi\)
							−0.937215	+	0.348753i	\(0.886605\pi\)
\(13\)	−118.780	−	137.080i	−0.702841	−	0.811122i	0.286292	−	0.958142i	\(-0.407577\pi\)
							−0.989133	+	0.147020i	\(0.953032\pi\)
\(17\)	−272.746	−	124.559i	−0.943756	−	0.430999i	−0.116731	−	0.993164i	\(-0.537241\pi\)
							−0.827025	+	0.562165i	\(0.809969\pi\)
\(19\)	96.3375	+	210.950i	0.266863	+	0.584348i	0.994863	−	0.101229i	\(-0.0322774\pi\)
							−0.728000	+	0.685577i	\(0.759550\pi\)
\(23\)	329.628	−	413.747i	0.623115	−	0.782130i
							\(29\)	−1244.06	−	568.145i	−1.47927	−	0.675559i	−0.497819	−	0.867281i	\(-0.665865\pi\)
−0.981449	+	0.191722i	\(0.938593\pi\)
\(31\)	85.0949	−	591.848i	0.0885483	−	0.615867i	−0.896430	−	0.443186i	\(-0.853848\pi\)
							0.984978	−	0.172681i	\(-0.0552428\pi\)
\(37\)	−126.272	+	37.0767i	−0.0922365	+	0.0270831i	−0.327525	−	0.944842i	\(-0.606215\pi\)
							0.235289	+	0.971926i	\(0.424396\pi\)
\(41\)	572.741	−	1950.58i	0.340714	−	1.16037i	−0.593851	−	0.804575i	\(-0.702393\pi\)
							0.934565	−	0.355791i	\(-0.115789\pi\)
\(43\)	87.9991	+	612.047i	0.0475928	+	0.331015i	0.999683	+	0.0251963i	\(0.00802108\pi\)
							−0.952090	+	0.305819i	\(0.901070\pi\)
\(47\)			341.898i			0.154775i	0.997001	+	0.0773875i	\(0.0246579\pi\)
							−0.997001	+	0.0773875i	\(0.975342\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.2	✓	300
3.2	odd	2	inner	69.5.h.a.2.29	yes	300
23.12	even	11	inner	69.5.h.a.35.29	yes	300
69.35	odd	22	inner	69.5.h.a.35.2	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.2	✓	300	1.1	even	1	trivial
69.5.h.a.2.29	yes	300	3.2	odd	2	inner
69.5.h.a.35.2	yes	300	69.35	odd	22	inner
69.5.h.a.35.29	yes	300	23.12	even	11	inner