Embedded newform 23.14.c.a.2.2

• Label: 23.14.c.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	23.2
Dual form			23.14.c.a.12.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-124.099 - 79.7538i) q^{2} +(186.867 + 1299.69i) q^{3} +(5636.91 + 12343.1i) q^{4} +(49603.5 + 14564.9i) q^{5} +(80464.9 - 176193. i) q^{6} +(166766. - 192458. i) q^{7} +(112891. - 785176. i) q^{8} +(-124521. + 36562.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 250 q - 11 q^{2} - 1207 q^{3} - 114699 q^{4} - 41875 q^{5} - 125668 q^{6} + 240331 q^{7} + 562996 q^{8} - 13202058 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−124.099	−	79.7538i	−1.37112	−	0.881164i	−0.372222	−	0.928144i	\(-0.621404\pi\)
							−0.998896	+	0.0469800i	\(0.985040\pi\)
\(3\)	186.867	+	1299.69i	0.147994	+	1.02932i	0.919498	+	0.393095i	\(0.128596\pi\)
							−0.771504	+	0.636224i	\(0.780495\pi\)
\(5\)	49603.5	+	14564.9i	1.41973	+	0.416872i	0.899413	−	0.437099i	\(-0.143994\pi\)
							0.520321	+	0.853971i	\(0.325812\pi\)
\(7\)	166766.	−	192458.i	0.535758	−	0.618298i	−0.421747	−	0.906714i	\(-0.638583\pi\)
							0.957505	+	0.288415i	\(0.0931284\pi\)
\(11\)	5.36215e6	−	3.44605e6i	0.912614	−	0.586501i	0.00210790	−	0.999998i	\(-0.499329\pi\)
							0.910506	+	0.413497i	\(0.135693\pi\)
\(13\)	1.35169e7	+	1.55993e7i	0.776685	+	0.896342i	0.996865	−	0.0791169i	\(-0.0252101\pi\)
							−0.220180	+	0.975459i	\(0.570665\pi\)
\(17\)	−2.86045e7	+	6.26351e7i	−0.287420	+	0.629361i	−0.997177	−	0.0750850i	\(-0.976077\pi\)
							0.709758	+	0.704446i	\(0.248804\pi\)
\(19\)	−1.50339e8	−	3.29196e8i	−0.733117	−	1.60530i	−0.794556	−	0.607191i	\(-0.792296\pi\)
							0.0614394	−	0.998111i	\(-0.480431\pi\)
\(23\)	8.29562e7	−	7.05092e8i	0.116847	−	0.993150i
							\(29\)	7.05372e8	−	1.54455e9i	0.220207	−	0.482186i	−0.766997	−	0.641651i	\(-0.778250\pi\)
0.987204	+	0.159465i	\(0.0509769\pi\)
\(31\)	8.24597e8	−	5.73519e9i	0.166875	−	1.16064i	−0.718420	−	0.695609i	\(-0.755135\pi\)
							0.885295	−	0.465030i	\(-0.153956\pi\)
\(37\)	1.38410e10	−	4.06408e9i	0.886861	−	0.260406i	0.193589	−	0.981083i	\(-0.437987\pi\)
							0.693271	+	0.720677i	\(0.256169\pi\)
\(41\)	−1.50468e10	−	4.41813e9i	−0.494707	−	0.145259i	0.0248561	−	0.999691i	\(-0.492087\pi\)
							−0.519564	+	0.854432i	\(0.673905\pi\)
\(43\)	3.09464e9	+	2.15237e10i	0.0746562	+	0.519245i	0.992494	+	0.122290i	\(0.0390239\pi\)
							−0.917838	+	0.396955i	\(0.870067\pi\)
\(47\)	1.21584e10			0.164528			0.0822639	−	0.996611i	\(-0.473785\pi\)
							0.0822639	+	0.996611i	\(0.473785\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.14.c.a.2.2	✓	250
23.12	even	11	inner	23.14.c.a.12.2	yes	250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.14.c.a.2.2	✓	250	1.1	even	1	trivial
23.14.c.a.12.2	yes	250	23.12	even	11	inner