Embedded newform 23.10.c.a.2.6

• Label: 23.10.c.a.2.6
• Level: $23$
• Weight: $10$
• Character: 23.2
• Analytic conductor: $11.846$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8458242318\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.6
Character	\(\chi\)	\(=\)	23.2
Dual form			23.10.c.a.12.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.3223 - 9.20438i) q^{2} +(-1.55598 - 10.8221i) q^{3} +(-92.2849 - 202.076i) q^{4} +(1543.83 + 453.310i) q^{5} +(-77.3256 + 169.319i) q^{6} +(-1288.91 + 1487.48i) q^{7} +(-1778.78 + 12371.7i) q^{8} +(18771.0 - 5511.66i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q - 43 q^{2} - 157 q^{3} - 4619 q^{4} + 2265 q^{5} + 1532 q^{6} + 8605 q^{7} - 28412 q^{8} - 84756 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.3223	−	9.20438i	−0.632962	−	0.406780i	0.184443	−	0.982843i	\(-0.440952\pi\)
							−0.817405	+	0.576063i	\(0.804588\pi\)
\(3\)	−1.55598	−	10.8221i	−0.0110907	−	0.0771376i	0.983524	−	0.180776i	\(-0.0578608\pi\)
							−0.994615	+	0.103638i	\(0.966952\pi\)
\(5\)	1543.83	+	453.310i	1.10468	+	0.324362i	0.782708	−	0.622388i	\(-0.213838\pi\)
							0.321968	+	0.946751i	\(0.395656\pi\)
\(7\)	−1288.91	+	1487.48i	−0.202899	+	0.234158i	−0.848075	−	0.529876i	\(-0.822239\pi\)
							0.645176	+	0.764034i	\(0.276784\pi\)
\(11\)	21846.0	−	14039.6i	0.449889	−	0.289126i	−0.296011	−	0.955185i	\(-0.595656\pi\)
							0.745899	+	0.666059i	\(0.232020\pi\)
\(13\)	−49497.3	−	57122.9i	−0.480658	−	0.554709i	0.462687	−	0.886522i	\(-0.346885\pi\)
							−0.943345	+	0.331812i	\(0.892340\pi\)
\(17\)	157630.	−	345162.i	0.457740	−	1.00231i	−0.530257	−	0.847837i	\(-0.677904\pi\)
							0.987997	−	0.154474i	\(-0.0493683\pi\)
\(19\)	−447368.	−	979600.i	−0.787543	−	1.72448i	−0.683555	−	0.729899i	\(-0.739567\pi\)
							−0.103988	−	0.994579i	\(-0.533160\pi\)
\(23\)	1.33744e6	−	111337.i	0.996553	−	0.0829595i
							\(29\)	−1.53747e6	+	3.36659e6i	−0.403660	+	0.883892i	0.593226	+	0.805036i	\(0.297854\pi\)
−0.996886	+	0.0788559i	\(0.974873\pi\)
\(31\)	−208368.	+	1.44923e6i	−0.0405232	+	0.281845i	0.959477	+	0.281787i	\(0.0909273\pi\)
							−1.00000		5.72403e-5i	\(0.999982\pi\)
\(37\)	1.14120e7	−	3.35087e6i	1.00105	−	0.293934i	0.260161	−	0.965565i	\(-0.416224\pi\)
							0.740885	+	0.671631i	\(0.234406\pi\)
\(41\)	4.90407e6	+	1.43996e6i	0.271037	+	0.0795837i	0.414427	−	0.910083i	\(-0.363982\pi\)
							−0.143390	+	0.989666i	\(0.545800\pi\)
\(43\)	−5.25308e6	−	3.65360e7i	−0.234318	−	1.62972i	−0.679075	−	0.734069i	\(-0.737619\pi\)
							0.444757	−	0.895651i	\(-0.353290\pi\)
\(47\)	−6.30219e6			−0.188387			−0.0941936	−	0.995554i	\(-0.530027\pi\)
							−0.0941936	+	0.995554i	\(0.530027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.10.c.a.2.6	✓	170
23.12	even	11	inner	23.10.c.a.12.6	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.10.c.a.2.6	✓	170	1.1	even	1	trivial
23.10.c.a.12.6	yes	170	23.12	even	11	inner