Embedded newform 23.13.d.a.5.3

• Label: 23.13.d.a.5.3
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-102.129 - 29.9877i) q^{2} +(821.786 - 948.392i) q^{3} +(6085.27 + 3910.76i) q^{4} +(12183.7 + 1751.75i) q^{5} +(-112368. + 72214.7i) q^{6} +(-67347.8 + 30756.7i) q^{7} +(-218700. - 252393. i) q^{8} +(-148483. - 1.03272e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−102.129	−	29.9877i	−1.59576	−	0.468558i	−0.641400	−	0.767206i	\(-0.721646\pi\)
							−0.954363	+	0.298648i	\(0.903464\pi\)
\(3\)	821.786	−	948.392i	1.12728	−	1.30095i	0.178880	−	0.983871i	\(-0.442753\pi\)
							0.948399	−	0.317078i	\(-0.102702\pi\)
\(5\)	12183.7	+	1751.75i	0.779758	+	0.112112i	0.520689	−	0.853747i	\(-0.325675\pi\)
							0.259069	+	0.965859i	\(0.416584\pi\)
\(7\)	−67347.8	+	30756.7i	−0.572447	+	0.261428i	−0.680537	−	0.732713i	\(-0.738254\pi\)
							0.108091	+	0.994141i	\(0.465526\pi\)
\(11\)	450147.	+	1.53306e6i	0.254096	+	0.865372i	0.983441	+	0.181226i	\(0.0580066\pi\)
							−0.729345	+	0.684146i	\(0.760175\pi\)
\(13\)	199150.	−	436077.i	0.0412591	−	0.0903448i	−0.887880	−	0.460075i	\(-0.847823\pi\)
							0.929139	+	0.369730i	\(0.120550\pi\)
\(17\)	−2.47972e7	−	3.85852e7i	−1.02733	−	1.59855i	−0.776082	−	0.630631i	\(-0.782796\pi\)
							−0.251246	−	0.967923i	\(-0.580840\pi\)
\(19\)	3.02361e7	−	4.70483e7i	0.642693	−	1.00005i	−0.355176	−	0.934800i	\(-0.615579\pi\)
							0.997869	−	0.0652508i	\(-0.0207847\pi\)
\(23\)	−1.45387e8	−	2.78812e7i	−0.982104	−	0.188341i
							\(29\)	1.00573e8	−	6.46345e7i	0.169081	−	0.108662i	−0.453364	−	0.891326i	\(-0.649776\pi\)
0.622444	+	0.782664i	\(0.286140\pi\)
\(31\)	−7.54486e8	−	8.70723e8i	−0.850122	−	0.981093i	0.149849	−	0.988709i	\(-0.452121\pi\)
							−0.999971	+	0.00761611i	\(0.997576\pi\)
\(37\)	4.05261e8	−	5.82678e7i	0.157952	−	0.0227101i	−0.0628851	−	0.998021i	\(-0.520030\pi\)
							0.220837	+	0.975311i	\(0.429121\pi\)
\(41\)	9.01162e8	−	6.26772e9i	0.189714	−	1.31949i	−0.643034	−	0.765838i	\(-0.722324\pi\)
							0.832748	−	0.553653i	\(-0.186766\pi\)
\(43\)	6.47282e9	+	5.60873e9i	1.02396	+	0.887266i	0.993677	−	0.112275i	\(-0.0358139\pi\)
							0.0302825	+	0.999541i	\(0.490359\pi\)
\(47\)	−1.44907e10			−1.34432			−0.672161	−	0.740405i	\(-0.734634\pi\)
							−0.672161	+	0.740405i	\(0.734634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.3	✓	230
23.14	odd	22	inner	23.13.d.a.14.3	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.3	✓	230	1.1	even	1	trivial
23.13.d.a.14.3	yes	230	23.14	odd	22	inner