Embedded newform 23.15.d.a.5.11

• Label: 23.15.d.a.5.11
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-72.6742 - 21.3391i) q^{2} +(-2557.19 + 2951.16i) q^{3} +(-8956.91 - 5756.26i) q^{4} +(-5706.48 - 820.468i) q^{5} +(248817. - 159905. i) q^{6} +(-701947. + 320568. i) q^{7} +(1.34076e6 + 1.54732e6i) q^{8} +(-1.48941e6 - 1.03591e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−72.6742	−	21.3391i	−0.567768	−	0.166712i	−0.0147647	−	0.999891i	\(-0.504700\pi\)
							−0.553003	+	0.833179i	\(0.686518\pi\)
\(3\)	−2557.19	+	2951.16i	−1.16927	+	1.34941i	−0.244149	+	0.969738i	\(0.578508\pi\)
							−0.925121	+	0.379672i	\(0.876037\pi\)
\(5\)	−5706.48	−	820.468i	−0.0730430	−	0.0105020i	0.105696	−	0.994398i	\(-0.466293\pi\)
							−0.178739	+	0.983896i	\(0.557202\pi\)
\(7\)	−701947.	+	320568.i	−0.852350	+	0.389255i	−0.793176	−	0.608993i	\(-0.791574\pi\)
							−0.0591740	+	0.998248i	\(0.518847\pi\)
\(11\)	−2.98706e6	−	1.01730e7i	−0.153284	−	0.522036i	0.846666	−	0.532125i	\(-0.178607\pi\)
							−0.999949	+	0.0100896i	\(0.996788\pi\)
\(13\)	4.41232e7	−	9.66162e7i	0.703174	−	1.53974i	−0.132905	−	0.991129i	\(-0.542431\pi\)
							0.836079	−	0.548608i	\(-0.184842\pi\)
\(17\)	−2.60890e8	−	4.05952e8i	−0.635791	−	0.989310i	−0.998354	−	0.0573510i	\(-0.981735\pi\)
							0.362563	−	0.931959i	\(-0.381902\pi\)
\(19\)	−2.28273e8	+	3.55200e8i	−0.255376	+	0.397373i	−0.945142	−	0.326659i	\(-0.894077\pi\)
							0.689766	+	0.724032i	\(0.257713\pi\)
\(23\)	−8.31874e8	+	3.30164e9i	−0.244322	+	0.969694i
							\(29\)	1.19952e10	−	7.70886e9i	0.695380	−	0.446894i	−0.144615	−	0.989488i	\(-0.546194\pi\)
0.839995	+	0.542594i	\(0.182558\pi\)
\(31\)	1.84809e10	+	2.13281e10i	0.671726	+	0.775213i	0.984645	−	0.174567i	\(-0.0558525\pi\)
							−0.312920	+	0.949780i	\(0.601307\pi\)
\(37\)	−1.55862e11	+	2.24096e10i	−1.64183	+	0.236060i	−0.900431	−	0.435000i	\(-0.856748\pi\)
							−0.741403	+	0.671060i	\(0.765839\pi\)
\(41\)	−4.42497e9	+	3.07763e10i	−0.0227208	+	0.158027i	−0.998023	−	0.0628474i	\(-0.979982\pi\)
							0.975302	+	0.220874i	\(0.0708909\pi\)
\(43\)	−2.49653e11	−	2.16326e11i	−0.918455	−	0.795846i	0.0608687	−	0.998146i	\(-0.480613\pi\)
							−0.979324	+	0.202300i	\(0.935158\pi\)
\(47\)	3.05421e10			0.0602857			0.0301428	−	0.999546i	\(-0.490404\pi\)
							0.0301428	+	0.999546i	\(0.490404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.11	✓	270
23.14	odd	22	inner	23.15.d.a.14.11	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.11	✓	270	1.1	even	1	trivial
23.15.d.a.14.11	yes	270	23.14	odd	22	inner