Embedded newform 23.6.c.a.2.7

• Label: 23.6.c.a.2.7
• Level: $23$
• Weight: $6$
• Character: 23.2
• Analytic conductor: $3.689$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.68882785570\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(9\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	23.2
Dual form			23.6.c.a.12.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.39264 + 3.46564i) q^{2} +(2.36907 + 16.4773i) q^{3} +(3.77666 + 8.26972i) q^{4} +(27.3965 + 8.04434i) q^{5} +(-44.3288 + 97.0664i) q^{6} +(-81.9154 + 94.5354i) q^{7} +(20.8990 - 145.356i) q^{8} +(-32.7312 + 9.61074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.39264	+	3.46564i	0.953294	+	0.612645i	0.922134	−	0.386870i	\(-0.126444\pi\)
							0.0311594	+	0.999514i	\(0.490080\pi\)
\(3\)	2.36907	+	16.4773i	0.151976	+	1.05702i	0.912902	+	0.408179i	\(0.133836\pi\)
							−0.760926	+	0.648839i	\(0.775255\pi\)
\(5\)	27.3965	+	8.04434i	0.490084	+	0.143902i	0.517432	−	0.855724i	\(-0.326888\pi\)
							−0.0273482	+	0.999626i	\(0.508706\pi\)
\(7\)	−81.9154	+	94.5354i	−0.631860	+	0.729205i	−0.977914	−	0.209010i	\(-0.932976\pi\)
							0.346054	+	0.938215i	\(0.387521\pi\)
\(11\)	300.766	−	193.291i	0.749458	−	0.481647i	−0.109313	−	0.994007i	\(-0.534865\pi\)
							0.858771	+	0.512360i	\(0.171229\pi\)
\(13\)	16.3698	+	18.8917i	0.0268648	+	0.0310036i	0.769023	−	0.639221i	\(-0.220743\pi\)
							−0.742158	+	0.670225i	\(0.766198\pi\)
\(17\)	503.714	−	1102.98i	0.422729	−	0.925647i	−0.571722	−	0.820447i	\(-0.693724\pi\)
							0.994451	−	0.105200i	\(-0.0335483\pi\)
\(19\)	−183.859	−	402.595i	−0.116842	−	0.255849i	0.842171	−	0.539211i	\(-0.181278\pi\)
							−0.959013	+	0.283362i	\(0.908550\pi\)
\(23\)	−2536.95	+	14.4590i	−0.999984	+	0.00569926i
							\(29\)	−470.657	+	1030.59i	−0.103922	+	0.227558i	−0.954449	−	0.298373i	\(-0.903556\pi\)
0.850527	+	0.525932i	\(0.176283\pi\)
\(31\)	−1283.74	+	8928.61i	−0.239924	+	1.66870i	0.412582	+	0.910921i	\(0.364627\pi\)
							−0.652505	+	0.757784i	\(0.726282\pi\)
\(37\)	−323.366	+	94.9489i	−0.0388320	+	0.0114021i	−0.301091	−	0.953595i	\(-0.597351\pi\)
							0.262259	+	0.964998i	\(0.415533\pi\)
\(41\)	−2981.06	−	875.319i	−0.276956	−	0.0813217i	0.140305	−	0.990108i	\(-0.455192\pi\)
							−0.417261	+	0.908787i	\(0.637010\pi\)
\(43\)	−2170.08	−	15093.2i	−0.178980	−	1.24483i	−0.859129	−	0.511759i	\(-0.828994\pi\)
							0.680149	−	0.733074i	\(-0.261915\pi\)
\(47\)	−21454.3			−1.41667			−0.708337	−	0.705875i	\(-0.750554\pi\)
							−0.708337	+	0.705875i	\(0.750554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.6.c.a.2.7	✓	90
23.9	even	11		529.6.a.j.1.35		45
23.12	even	11	inner	23.6.c.a.12.7	yes	90
23.14	odd	22		529.6.a.k.1.35		45

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.c.a.2.7	✓	90	1.1	even	1	trivial
23.6.c.a.12.7	yes	90	23.12	even	11	inner
529.6.a.j.1.35		45	23.9	even	11
529.6.a.k.1.35		45	23.14	odd	22