Embedded newform 23.15.d.a.5.1

• Label: 23.15.d.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-239.810 - 70.4144i) q^{2} +(1474.41 - 1701.56i) q^{3} +(38767.3 + 24914.3i) q^{4} +(40294.5 + 5793.47i) q^{5} +(-473392. + 304231. i) q^{6} +(341082. - 155767. i) q^{7} +(-4.86086e6 - 5.60973e6i) q^{8} +(-40735.0 - 283318. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−239.810	−	70.4144i	−1.87351	−	0.550113i	−0.997725	−	0.0674159i	\(-0.978525\pi\)
							−0.875787	−	0.482697i	\(-0.839657\pi\)
\(3\)	1474.41	−	1701.56i	0.674171	−	0.778034i	−0.310852	−	0.950458i	\(-0.600614\pi\)
							0.985023	+	0.172424i	\(0.0551599\pi\)
\(5\)	40294.5	+	5793.47i	0.515769	+	0.0741564i	0.395284	−	0.918559i	\(-0.370646\pi\)
							0.120485	+	0.992715i	\(0.461555\pi\)
\(7\)	341082.	−	155767.i	0.414164	−	0.189142i	−0.197425	−	0.980318i	\(-0.563258\pi\)
							0.611589	+	0.791176i	\(0.290531\pi\)
\(11\)	−5.58644e6	−	1.90257e7i	−0.286673	−	0.976317i	−0.969368	−	0.245614i	\(-0.921010\pi\)
							0.682695	−	0.730703i	\(-0.260808\pi\)
\(13\)	−1.14100e7	+	2.49844e7i	−0.181837	+	0.398166i	−0.978497	−	0.206260i	\(-0.933871\pi\)
							0.796661	+	0.604427i	\(0.206598\pi\)
\(17\)	2.74749e8	+	4.27518e8i	0.669567	+	1.04187i	0.995341	+	0.0964188i	\(0.0307388\pi\)
							−0.325774	+	0.945448i	\(0.605625\pi\)
\(19\)	4.09916e8	−	6.37842e8i	0.458585	−	0.713572i	−0.532555	−	0.846396i	\(-0.678768\pi\)
							0.991140	+	0.132823i	\(0.0424043\pi\)
\(23\)	1.31456e9	−	3.14082e9i	0.386088	−	0.922462i
							\(29\)	2.02035e10	−	1.29840e10i	1.17123	−	0.752702i	0.197474	−	0.980308i	\(-0.436726\pi\)
0.973753	+	0.227606i	\(0.0730900\pi\)
\(31\)	1.72324e10	+	1.98873e10i	0.626346	+	0.722841i	0.976899	−	0.213701i	\(-0.0685520\pi\)
							−0.350553	+	0.936543i	\(0.614007\pi\)
\(37\)	−3.53874e10	+	5.08793e9i	−0.372766	+	0.0535956i	−0.326152	−	0.945317i	\(-0.605752\pi\)
							−0.0466139	+	0.998913i	\(0.514843\pi\)
\(41\)	2.90029e10	−	2.01719e11i	0.148920	−	1.03576i	−0.769071	−	0.639163i	\(-0.779281\pi\)
							0.917991	−	0.396601i	\(-0.129810\pi\)
\(43\)	2.48997e11	+	2.15757e11i	0.916043	+	0.793755i	0.978915	−	0.204266i	\(-0.0654808\pi\)
							−0.0628728	+	0.998022i	\(0.520026\pi\)
\(47\)	−6.01540e11			−1.18735			−0.593676	−	0.804704i	\(-0.702324\pi\)
							−0.593676	+	0.804704i	\(0.702324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.1	✓	270
23.14	odd	22	inner	23.15.d.a.14.1	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.1	✓	270	1.1	even	1	trivial
23.15.d.a.14.1	yes	270	23.14	odd	22	inner