Embedded newform 69.5.h.a.2.1

• Label: 69.5.h.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.11610 + 6.40478i) q^{2} +(8.97488 - 0.671914i) q^{3} +(-17.4323 - 38.1713i) q^{4} +(6.99421 - 23.8201i) q^{5} +(-32.6381 + 60.2478i) q^{6} +(-31.6181 + 36.4893i) q^{7} +(195.658 + 28.1313i) q^{8} +(80.0971 - 12.0607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.11610	+	6.40478i	−1.02903	+	1.60119i	−0.256149	+	0.966637i	\(0.582454\pi\)
							−0.772876	+	0.634557i	\(0.781183\pi\)
\(3\)	8.97488	−	0.671914i	0.997209	−	0.0746571i
							\(5\)	6.99421	−	23.8201i	0.279769	−	0.952804i	−0.692983	−	0.720954i	\(-0.743704\pi\)
0.972752	−	0.231850i	\(-0.0744779\pi\)
\(7\)	−31.6181	+	36.4893i	−0.645268	+	0.744679i	−0.980297	−	0.197529i	\(-0.936708\pi\)
							0.335029	+	0.942208i	\(0.391254\pi\)
\(11\)	106.255	+	165.336i	0.878140	+	1.36641i	0.929923	+	0.367755i	\(0.119873\pi\)
							−0.0517824	+	0.998658i	\(0.516490\pi\)
\(13\)	71.4206	+	82.4238i	0.422607	+	0.487715i	0.926629	−	0.375976i	\(-0.122692\pi\)
							−0.504022	+	0.863691i	\(0.668147\pi\)
\(17\)	300.650	+	137.302i	1.04031	+	0.475094i	0.860950	−	0.508689i	\(-0.169870\pi\)
							0.179361	+	0.983783i	\(0.442597\pi\)
\(19\)	−67.1655	−	147.072i	−0.186054	−	0.407401i	0.793504	−	0.608566i	\(-0.208255\pi\)
							−0.979558	+	0.201164i	\(0.935528\pi\)
\(23\)	475.350	−	232.128i	0.898582	−	0.438805i
							\(29\)	−85.8436	−	39.2035i	−0.102073	−	0.0466153i	0.363724	−	0.931507i	\(-0.381505\pi\)
−0.465797	+	0.884891i	\(0.654232\pi\)
\(31\)	189.963	−	1321.22i	0.197672	−	1.37484i	−0.613343	−	0.789817i	\(-0.710176\pi\)
							0.811015	−	0.585025i	\(-0.198915\pi\)
\(37\)	−712.259	+	209.138i	−0.520276	+	0.152767i	−0.531319	−	0.847172i	\(-0.678303\pi\)
							0.0110421	+	0.999939i	\(0.496485\pi\)
\(41\)	−599.664	+	2042.27i	−0.356730	+	1.21491i	0.564349	+	0.825537i	\(0.309127\pi\)
							−0.921079	+	0.389376i	\(0.872691\pi\)
\(43\)	−116.359	−	809.296i	−0.0629309	−	0.437694i	−0.996791	−	0.0800541i	\(-0.974491\pi\)
							0.933860	−	0.357640i	\(-0.116418\pi\)
\(47\)		−	1200.46i		−	0.543442i	−0.962376	−	0.271721i	\(-0.912407\pi\)
							0.962376	−	0.271721i	\(-0.0875928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.1	✓	300
3.2	odd	2	inner	69.5.h.a.2.30	yes	300
23.12	even	11	inner	69.5.h.a.35.30	yes	300
69.35	odd	22	inner	69.5.h.a.35.1	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.1	✓	300	1.1	even	1	trivial
69.5.h.a.2.30	yes	300	3.2	odd	2	inner
69.5.h.a.35.1	yes	300	69.35	odd	22	inner
69.5.h.a.35.30	yes	300	23.12	even	11	inner