Embedded newform 23.4.c.a.4.5

• Label: 23.4.c.a.4.5
• Level: $23$
• Weight: $4$
• Character: 23.4
• 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			4.5
Character	\(\chi\)	\(=\)	23.4
Dual form			23.4.c.a.6.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61009 + 3.52560i) q^{2} +(3.27467 - 0.961529i) q^{3} +(-4.59861 + 5.30707i) q^{4} +(-10.7624 - 6.91659i) q^{5} +(8.66248 + 9.99703i) q^{6} +(-0.249078 - 1.73238i) q^{7} +(3.63606 + 1.06764i) q^{8} +(-12.9149 + 8.29992i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61009	+	3.52560i	0.569252	+	1.24649i	0.947194	+	0.320660i	\(0.103905\pi\)
							−0.377942	+	0.925829i	\(0.623368\pi\)
\(3\)	3.27467	−	0.961529i	0.630210	−	0.185046i	0.0489986	−	0.998799i	\(-0.484397\pi\)
							0.581212	+	0.813752i	\(0.302579\pi\)
\(5\)	−10.7624	−	6.91659i	−0.962620	−	0.618638i	−0.0378983	−	0.999282i	\(-0.512066\pi\)
							−0.924722	+	0.380643i	\(0.875703\pi\)
\(7\)	−0.249078	−	1.73238i	−0.0134490	−	0.0935395i	0.981991	−	0.188926i	\(-0.0605005\pi\)
							−0.995440	+	0.0953861i	\(0.969591\pi\)
\(11\)	21.0855	−	46.1708i	0.577956	−	1.26555i	−0.364496	−	0.931205i	\(-0.618759\pi\)
							0.942452	−	0.334342i	\(-0.108514\pi\)
\(13\)	−2.51752	+	17.5097i	−0.0537103	+	0.373563i	0.945183	+	0.326540i	\(0.105883\pi\)
							−0.998894	+	0.0470237i	\(0.985026\pi\)
\(17\)	8.23961	+	9.50902i	0.117553	+	0.135663i	0.811476	−	0.584386i	\(-0.198665\pi\)
							−0.693923	+	0.720049i	\(0.744119\pi\)
\(19\)	−100.660	+	116.167i	−1.21542	+	1.40267i	−0.326124	+	0.945327i	\(0.605743\pi\)
							−0.889293	+	0.457338i	\(0.848803\pi\)
\(23\)	92.7393	−	59.7195i	0.840760	−	0.541408i
							\(29\)	−119.037	−	137.376i	−0.762231	−	0.879661i	0.233463	−	0.972366i	\(-0.424994\pi\)
−0.995694	+	0.0927046i	\(0.970449\pi\)
\(31\)	84.3584	+	24.7699i	0.488749	+	0.143510i	0.516816	−	0.856096i	\(-0.327117\pi\)
							−0.0280673	+	0.999606i	\(0.508935\pi\)
\(37\)	253.260	−	162.760i	1.12529	−	0.723179i	0.160717	−	0.987001i	\(-0.448619\pi\)
							0.964572	+	0.263821i	\(0.0849830\pi\)
\(41\)	392.115	+	251.997i	1.49361	+	0.959886i	0.995698	+	0.0926587i	\(0.0295365\pi\)
							0.497913	+	0.867227i	\(0.334100\pi\)
\(43\)	−48.0075	+	14.0963i	−0.170257	+	0.0499921i	−0.365750	−	0.930713i	\(-0.619188\pi\)
							0.195493	+	0.980705i	\(0.437369\pi\)
\(47\)	−189.313			−0.587534			−0.293767	−	0.955877i	\(-0.594909\pi\)
							−0.293767	+	0.955877i	\(0.594909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.4.5	✓	50
3.2	odd	2		207.4.i.a.73.1		50
23.6	even	11	inner	23.4.c.a.6.5	yes	50
23.11	odd	22		529.4.a.m.1.21		25
23.12	even	11		529.4.a.n.1.21		25
69.29	odd	22		207.4.i.a.190.1		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.4.5	✓	50	1.1	even	1	trivial
23.4.c.a.6.5	yes	50	23.6	even	11	inner
207.4.i.a.73.1		50	3.2	odd	2
207.4.i.a.190.1		50	69.29	odd	22
529.4.a.m.1.21		25	23.11	odd	22
529.4.a.n.1.21		25	23.12	even	11