Embedded newform 23.14.c.a.2.8

• Label: 23.14.c.a.2.8
• Level: $23$
• Weight: $14$
• Character: 23.2
• Analytic conductor: $24.663$
• Analytic rank: $0$
• Dimension: $250$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.6631136589\)
Analytic rank:	\(0\)
Dimension:	\(250\)
Relative dimension:	\(25\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.8
Character	\(\chi\)	\(=\)	23.2
Dual form			23.14.c.a.12.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-52.7445 - 33.8968i) q^{2} +(-17.4363 - 121.272i) q^{3} +(-1770.09 - 3875.97i) q^{4} +(8956.10 + 2629.75i) q^{5} +(-3191.06 + 6987.45i) q^{6} +(66066.1 - 76244.3i) q^{7} +(-111116. + 772826. i) q^{8} +(1.51534e6 - 444944. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 250 q - 11 q^{2} - 1207 q^{3} - 114699 q^{4} - 41875 q^{5} - 125668 q^{6} + 240331 q^{7} + 562996 q^{8} - 13202058 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−52.7445	−	33.8968i	−0.582750	−	0.374510i	0.215802	−	0.976437i	\(-0.430764\pi\)
							−0.798551	+	0.601927i	\(0.794400\pi\)
\(3\)	−17.4363	−	121.272i	−0.0138091	−	0.0960443i	0.981751	−	0.190169i	\(-0.0609037\pi\)
							−0.995560	+	0.0941251i	\(0.969995\pi\)
\(5\)	8956.10	+	2629.75i	0.256338	+	0.0752678i	0.407377	−	0.913260i	\(-0.366444\pi\)
							−0.151038	+	0.988528i	\(0.548262\pi\)
\(7\)	66066.1	−	76244.3i	0.212247	−	0.244946i	−0.639636	−	0.768678i	\(-0.720915\pi\)
							0.851883	+	0.523732i	\(0.175461\pi\)
\(11\)	−2.96578e6	+	1.90599e6i	−0.504762	+	0.324391i	−0.768118	−	0.640308i	\(-0.778807\pi\)
							0.263357	+	0.964699i	\(0.415170\pi\)
\(13\)	1.18570e7	+	1.36837e7i	0.681308	+	0.786272i	0.986101	−	0.166147i	\(-0.0531327\pi\)
							−0.304793	+	0.952419i	\(0.598587\pi\)
\(17\)	7.93148e6	−	1.73675e7i	0.0796959	−	0.174510i	−0.865589	−	0.500754i	\(-0.833056\pi\)
							0.945285	+	0.326245i	\(0.105783\pi\)
\(19\)	1.16215e8	+	2.54475e8i	0.566714	+	1.24093i	0.948529	+	0.316691i	\(0.102572\pi\)
							−0.381815	+	0.924239i	\(0.624701\pi\)
\(23\)	6.30806e8	−	3.25761e8i	0.888515	−	0.458847i
							\(29\)	1.51414e9	−	3.31551e9i	0.472694	−	1.03505i	−0.511714	−	0.859156i	\(-0.670989\pi\)
0.984408	−	0.175899i	\(-0.0562833\pi\)
\(31\)	1.34094e9	−	9.32646e9i	0.271368	−	1.88741i	−0.162929	−	0.986638i	\(-0.552094\pi\)
							0.434297	−	0.900770i	\(-0.356997\pi\)
\(37\)	4.42491e8	−	1.29927e8i	0.0283526	−	0.00832508i	−0.267526	−	0.963551i	\(-0.586206\pi\)
							0.295878	+	0.955226i	\(0.404388\pi\)
\(41\)	3.80587e10	+	1.11751e10i	1.25129	+	0.367413i	0.839247	−	0.543750i	\(-0.182996\pi\)
							0.412046	+	0.911163i	\(0.364814\pi\)
\(43\)	−5.64855e9	−	3.92865e10i	−0.136267	−	0.947760i	−0.937147	−	0.348935i	\(-0.886543\pi\)
							0.800880	−	0.598825i	\(-0.204366\pi\)
\(47\)	−5.51323e10			−0.746053			−0.373027	−	0.927821i	\(-0.621680\pi\)
							−0.373027	+	0.927821i	\(0.621680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.14.c.a.2.8	✓	250
23.12	even	11	inner	23.14.c.a.12.8	yes	250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.14.c.a.2.8	✓	250	1.1	even	1	trivial
23.14.c.a.12.8	yes	250	23.12	even	11	inner