Embedded newform 23.13.d.a.5.8

• Label: 23.13.d.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-54.9700 - 16.1406i) q^{2} +(-407.479 + 470.255i) q^{3} +(-684.594 - 439.962i) q^{4} +(1588.57 + 228.402i) q^{5} +(29989.3 - 19273.0i) q^{6} +(-79757.6 + 36424.1i) q^{7} +(184202. + 212581. i) q^{8} +(20530.6 + 142794. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−54.9700	−	16.1406i	−0.858906	−	0.252198i	−0.177516	−	0.984118i	\(-0.556806\pi\)
							−0.681390	+	0.731920i	\(0.738624\pi\)
\(3\)	−407.479	+	470.255i	−0.558956	+	0.645069i	−0.962946	−	0.269694i	\(-0.913077\pi\)
							0.403990	+	0.914763i	\(0.367623\pi\)
\(5\)	1588.57	+	228.402i	0.101668	+	0.0146177i	0.192961	−	0.981206i	\(-0.438191\pi\)
							−0.0912928	+	0.995824i	\(0.529100\pi\)
\(7\)	−79757.6	+	36424.1i	−0.677928	+	0.309599i	−0.724459	−	0.689317i	\(-0.757911\pi\)
							0.0465313	+	0.998917i	\(0.485183\pi\)
\(11\)	719281.	+	2.44965e6i	0.406016	+	1.38276i	0.868304	+	0.496032i	\(0.165210\pi\)
							−0.462288	+	0.886730i	\(0.652972\pi\)
\(13\)	688560.	−	1.50774e6i	0.142653	−	0.312367i	−0.824797	−	0.565429i	\(-0.808710\pi\)
							0.967450	+	0.253062i	\(0.0814378\pi\)
\(17\)	1.01506e7	+	1.57946e7i	0.420529	+	0.654356i	0.985288	−	0.170899i	\(-0.0546673\pi\)
							−0.564759	+	0.825256i	\(0.691031\pi\)
\(19\)	209029.	−	325255.i	0.00444309	−	0.00691358i	−0.839025	−	0.544093i	\(-0.816874\pi\)
							0.843468	+	0.537179i	\(0.180510\pi\)
\(23\)	−1.23888e8	+	8.10331e7i	−0.836879	+	0.547388i
							\(29\)	1.11954e8	−	7.19484e7i	0.188214	−	0.120958i	−0.443141	−	0.896452i	\(-0.646136\pi\)
0.631354	+	0.775494i	\(0.282499\pi\)
\(31\)	−6.69890e8	−	7.73095e8i	−0.754803	−	0.871089i	0.240222	−	0.970718i	\(-0.422780\pi\)
							−0.995025	+	0.0996290i	\(0.968234\pi\)
\(37\)	3.30351e9	−	4.74974e8i	1.28756	−	0.185122i	0.535676	−	0.844424i	\(-0.320057\pi\)
							0.751879	+	0.659301i	\(0.229148\pi\)
\(41\)	−5.59203e8	+	3.88934e9i	−0.117724	+	0.818790i	0.842327	+	0.538967i	\(0.181185\pi\)
							−0.960051	+	0.279824i	\(0.909724\pi\)
\(43\)	6.28287e8	+	5.44414e8i	0.0993911	+	0.0861229i	0.703139	−	0.711052i	\(-0.251781\pi\)
							−0.603748	+	0.797175i	\(0.706327\pi\)
\(47\)	−1.06843e10			−0.991193			−0.495597	−	0.868553i	\(-0.665051\pi\)
							−0.495597	+	0.868553i	\(0.665051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.8	✓	230
23.14	odd	22	inner	23.13.d.a.14.8	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.8	✓	230	1.1	even	1	trivial
23.13.d.a.14.8	yes	230	23.14	odd	22	inner