Embedded newform 23.4.c.a.8.3

• Label: 23.4.c.a.8.3
• Level: $23$
• Weight: $4$
• Character: 23.8
• Analytic conductor: $1.357$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.35704393013\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			8.3
Character	\(\chi\)	\(=\)	23.8
Dual form			23.4.c.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0282746 - 0.196654i) q^{2} +(-3.66027 - 8.01487i) q^{3} +(7.63807 + 2.24274i) q^{4} +(-6.59261 - 7.60828i) q^{5} +(-1.67965 + 0.493191i) q^{6} +(21.3237 + 13.7039i) q^{7} +(1.31727 - 2.88443i) q^{8} +(-33.1594 + 38.2680i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0282746	−	0.196654i	0.00999659	−	0.0695278i	−0.984212	−	0.176991i	\(-0.943364\pi\)
							0.994209	+	0.107463i	\(0.0342728\pi\)
\(3\)	−3.66027	−	8.01487i	−0.704419	−	1.54246i	−0.834529	−	0.550964i	\(-0.814260\pi\)
							0.130110	−	0.991500i	\(-0.458467\pi\)
\(5\)	−6.59261	−	7.60828i	−0.589661	−	0.680505i	0.379992	−	0.924990i	\(-0.375927\pi\)
							−0.969653	+	0.244484i	\(0.921381\pi\)
\(7\)	21.3237	+	13.7039i	1.15137	+	0.739940i	0.969911	−	0.243458i	\(-0.0782819\pi\)
							0.181458	+	0.983399i	\(0.441918\pi\)
\(11\)	2.01428	+	14.0097i	0.0552118	+	0.384006i	0.998627	+	0.0523914i	\(0.0166843\pi\)
							−0.943415	+	0.331615i	\(0.892407\pi\)
\(13\)	−3.85568	+	2.47789i	−0.0822594	+	0.0528649i	−0.581124	−	0.813815i	\(-0.697387\pi\)
							0.498865	+	0.866680i	\(0.333751\pi\)
\(17\)	98.0272	−	28.7834i	1.39853	−	0.410647i	0.506354	−	0.862326i	\(-0.330993\pi\)
							0.892180	+	0.451679i	\(0.149175\pi\)
\(19\)	−66.4831	−	19.5212i	−0.802751	−	0.235709i	−0.145478	−	0.989361i	\(-0.546472\pi\)
							−0.657273	+	0.753653i	\(0.728290\pi\)
\(23\)	−83.9685	+	71.5282i	−0.761246	+	0.648464i
							\(29\)	−202.341	+	59.4126i	−1.29565	+	0.380436i	−0.855646	−	0.517562i	\(-0.826840\pi\)
−0.440000	+	0.897998i	\(0.645021\pi\)
\(31\)	−117.310	+	256.874i	−0.679664	+	1.48826i	0.183335	+	0.983051i	\(0.441311\pi\)
							−0.862999	+	0.505206i	\(0.831417\pi\)
\(37\)	114.021	−	131.587i	0.506619	−	0.584669i	−0.443611	−	0.896219i	\(-0.646303\pi\)
							0.950230	+	0.311550i	\(0.100848\pi\)
\(41\)	−66.9116	−	77.2201i	−0.254874	−	0.294140i	0.613864	−	0.789412i	\(-0.289614\pi\)
							−0.868738	+	0.495271i	\(0.835069\pi\)
\(43\)	53.4045	+	116.940i	0.189398	+	0.414724i	0.980380	−	0.197115i	\(-0.0631573\pi\)
							−0.790982	+	0.611839i	\(0.790430\pi\)
\(47\)	130.950			0.406406			0.203203	−	0.979137i	\(-0.434865\pi\)
							0.203203	+	0.979137i	\(0.434865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.8.3	yes	50
3.2	odd	2		207.4.i.a.100.3		50
23.3	even	11	inner	23.4.c.a.3.3	✓	50
23.7	odd	22		529.4.a.m.1.13		25
23.16	even	11		529.4.a.n.1.13		25
69.26	odd	22		207.4.i.a.118.3		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.3.3	✓	50	23.3	even	11	inner
23.4.c.a.8.3	yes	50	1.1	even	1	trivial
207.4.i.a.100.3		50	3.2	odd	2
207.4.i.a.118.3		50	69.26	odd	22
529.4.a.m.1.13		25	23.7	odd	22
529.4.a.n.1.13		25	23.16	even	11