Embedded newform 23.4.c.a.6.1

• Label: 23.4.c.a.6.1
• 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			6.1
Character	\(\chi\)	\(=\)	23.6
Dual form			23.4.c.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10729 + 4.61431i) q^{2} +(8.43369 + 2.47635i) q^{3} +(-11.6123 - 13.4013i) q^{4} +(-5.02341 + 3.22835i) q^{5} +(-29.1989 + 33.6973i) q^{6} +(3.12135 - 21.7095i) q^{7} +(47.3705 - 13.9092i) q^{8} +(42.2809 + 27.1723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10729	+	4.61431i	−0.745038	+	1.63141i	0.0300472	+	0.999548i	\(0.490434\pi\)
							−0.775085	+	0.631857i	\(0.782293\pi\)
\(3\)	8.43369	+	2.47635i	1.62306	+	0.476575i	0.961840	−	0.273613i	\(-0.0882188\pi\)
							0.661225	+	0.750188i	\(0.270037\pi\)
\(5\)	−5.02341	+	3.22835i	−0.449307	+	0.288752i	−0.745660	−	0.666326i	\(-0.767866\pi\)
							0.296353	+	0.955078i	\(0.404230\pi\)
\(7\)	3.12135	−	21.7095i	0.168537	−	1.17220i	−0.713373	−	0.700785i	\(-0.752833\pi\)
							0.881910	−	0.471418i	\(-0.156258\pi\)
\(11\)	4.09803	+	8.97343i	0.112327	+	0.245963i	0.957444	−	0.288619i	\(-0.0931963\pi\)
							−0.845116	+	0.534582i	\(0.820469\pi\)
\(13\)	2.55600	+	17.7774i	0.0545312	+	0.379273i	0.998751	+	0.0499580i	\(0.0159087\pi\)
							−0.944220	+	0.329315i	\(0.893182\pi\)
\(17\)	17.5335	−	20.2347i	0.250147	−	0.288684i	−0.616764	−	0.787148i	\(-0.711557\pi\)
							0.866911	+	0.498464i	\(0.166102\pi\)
\(19\)	−45.4702	−	52.4754i	−0.549030	−	0.633614i	0.411627	−	0.911352i	\(-0.364961\pi\)
							−0.960657	+	0.277738i	\(0.910415\pi\)
\(23\)	−93.1498	−	59.0772i	−0.844482	−	0.535585i
							\(29\)	26.8171	−	30.9486i	0.171717	−	0.198173i	−0.663367	−	0.748294i	\(-0.730873\pi\)
0.835085	+	0.550122i	\(0.185419\pi\)
\(31\)	85.9826	−	25.2468i	0.498159	−	0.146273i	−0.0229945	−	0.999736i	\(-0.507320\pi\)
							0.521154	+	0.853463i	\(0.325502\pi\)
\(37\)	−372.889	−	239.641i	−1.65683	−	1.06478i	−0.922513	−	0.385967i	\(-0.873868\pi\)
							−0.734313	−	0.678811i	\(-0.762496\pi\)
\(41\)	−238.779	+	153.454i	−0.909535	+	0.584523i	−0.909604	−	0.415477i	\(-0.863615\pi\)
							6.83865e−5		1.00000i	\(0.499978\pi\)
\(43\)	510.652	+	149.941i	1.81101	+	0.531762i	0.998677	−	0.0514252i	\(-0.0163764\pi\)
							0.812338	+	0.583187i	\(0.198195\pi\)
\(47\)	−228.723			−0.709845			−0.354922	−	0.934896i	\(-0.615493\pi\)
							−0.354922	+	0.934896i	\(0.615493\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.6.1	yes	50
3.2	odd	2		207.4.i.a.190.5		50
23.2	even	11		529.4.a.n.1.2		25
23.4	even	11	inner	23.4.c.a.4.1	✓	50
23.21	odd	22		529.4.a.m.1.2		25
69.50	odd	22		207.4.i.a.73.5		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.4.1	✓	50	23.4	even	11	inner
23.4.c.a.6.1	yes	50	1.1	even	1	trivial
207.4.i.a.73.5		50	69.50	odd	22
207.4.i.a.190.5		50	3.2	odd	2
529.4.a.m.1.2		25	23.21	odd	22
529.4.a.n.1.2		25	23.2	even	11