Embedded newform 23.4.c.a.6.4

• Label: 23.4.c.a.6.4
• Level: $23$
• Weight: $4$
• Character: 23.6
• Analytic conductor: $1.357$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.35704393013\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			6.4
Character	\(\chi\)	\(=\)	23.6
Dual form			23.4.c.a.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.49850 - 3.28126i) q^{2} +(-7.64281 - 2.24413i) q^{3} +(-3.28227 - 3.78794i) q^{4} +(15.0482 - 9.67090i) q^{5} +(-18.8163 + 21.7152i) q^{6} +(-2.56161 + 17.8164i) q^{7} +(10.3412 - 3.03646i) q^{8} +(30.6625 + 19.7056i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.49850	−	3.28126i	0.529800	−	1.16010i	−0.435794	−	0.900046i	\(-0.643532\pi\)
							0.965594	−	0.260054i	\(-0.0837403\pi\)
\(3\)	−7.64281	−	2.24413i	−1.47086	−	0.431883i	−0.554481	−	0.832197i	\(-0.687083\pi\)
							−0.916378	+	0.400314i	\(0.868901\pi\)
\(5\)	15.0482	−	9.67090i	1.34595	−	0.864992i	0.348570	−	0.937283i	\(-0.386667\pi\)
							0.997384	+	0.0722911i	\(0.0230311\pi\)
\(7\)	−2.56161	+	17.8164i	−0.138314	+	0.961993i	0.795938	+	0.605379i	\(0.206978\pi\)
							−0.934251	+	0.356615i	\(0.883931\pi\)
\(11\)	−10.3803	−	22.7298i	−0.284526	−	0.623026i	0.712365	−	0.701809i	\(-0.247624\pi\)
							−0.996892	+	0.0787829i	\(0.974897\pi\)
\(13\)	7.78384	+	54.1378i	0.166065	+	1.15501i	0.886920	+	0.461923i	\(0.152840\pi\)
							−0.720855	+	0.693086i	\(0.756250\pi\)
\(17\)	−35.0472	+	40.4466i	−0.500011	+	0.577044i	−0.948513	−	0.316738i	\(-0.897412\pi\)
							0.448502	+	0.893782i	\(0.351958\pi\)
\(19\)	−11.3172	−	13.0608i	−0.136650	−	0.157702i	0.683300	−	0.730138i	\(-0.260544\pi\)
							−0.819950	+	0.572435i	\(0.805999\pi\)
\(23\)	26.7730	+	107.006i	0.242720	+	0.970096i
							\(29\)	55.4848	−	64.0329i	0.355285	−	0.410021i	−0.549769	−	0.835316i	\(-0.685284\pi\)
0.905055	+	0.425296i	\(0.139830\pi\)
\(31\)	−100.127	+	29.3999i	−0.580107	+	0.170335i	−0.558605	−	0.829434i	\(-0.688663\pi\)
							−0.0215024	+	0.999769i	\(0.506845\pi\)
\(37\)	−160.158	−	102.927i	−0.711617	−	0.457329i	0.134095	−	0.990969i	\(-0.457187\pi\)
							−0.845712	+	0.533640i	\(0.820824\pi\)
\(41\)	−93.7947	+	60.2782i	−0.357275	+	0.229607i	−0.706949	−	0.707265i	\(-0.749929\pi\)
							0.349674	+	0.936872i	\(0.386292\pi\)
\(43\)	279.375	+	82.0318i	0.990796	+	0.290924i	0.736674	−	0.676248i	\(-0.236395\pi\)
							0.254122	+	0.967172i	\(0.418213\pi\)
\(47\)	−504.689			−1.56631			−0.783154	−	0.621828i	\(-0.786390\pi\)
							−0.783154	+	0.621828i	\(0.786390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.6.4	yes	50
3.2	odd	2		207.4.i.a.190.2		50
23.2	even	11		529.4.a.n.1.20		25
23.4	even	11	inner	23.4.c.a.4.4	✓	50
23.21	odd	22		529.4.a.m.1.20		25
69.50	odd	22		207.4.i.a.73.2		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.4.4	✓	50	23.4	even	11	inner
23.4.c.a.6.4	yes	50	1.1	even	1	trivial
207.4.i.a.73.2		50	69.50	odd	22
207.4.i.a.190.2		50	3.2	odd	2
529.4.a.m.1.20		25	23.21	odd	22
529.4.a.n.1.20		25	23.2	even	11