Embedded newform 23.15.d.a.5.15

• Label: 23.15.d.a.5.15
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(41.2203 + 12.1034i) q^{2} +(2006.50 - 2315.62i) q^{3} +(-12230.5 - 7860.05i) q^{4} +(-102785. - 14778.2i) q^{5} +(110735. - 71165.2i) q^{6} +(373207. - 170438. i) q^{7} +(-869944. - 1.00397e6i) q^{8} +(-655385. - 4.55830e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	41.2203	+	12.1034i	0.322033	+	0.0945575i	0.438754	−	0.898607i	\(-0.355420\pi\)
							−0.116721	+	0.993165i	\(0.537238\pi\)
\(3\)	2006.50	−	2315.62i	0.917466	−	1.05881i	−0.0806058	−	0.996746i	\(-0.525685\pi\)
							0.998072	−	0.0620664i	\(-0.0197691\pi\)
\(5\)	−102785.	−	14778.2i	−1.31565	−	0.189161i	−0.551484	−	0.834186i	\(-0.685938\pi\)
							−0.764162	+	0.645024i	\(0.776847\pi\)
\(7\)	373207.	−	170438.i	0.453172	−	0.206957i	−0.175735	−	0.984438i	\(-0.556230\pi\)
							0.628907	+	0.777481i	\(0.283503\pi\)
\(11\)	−1.62863e6	−	5.54659e6i	−0.0835743	−	0.284628i	0.907093	−	0.420930i	\(-0.138296\pi\)
							−0.990667	+	0.136302i	\(0.956478\pi\)
\(13\)	−2.56358e7	+	5.61345e7i	−0.408548	+	0.894595i	0.587784	+	0.809018i	\(0.300001\pi\)
							−0.996332	+	0.0855771i	\(0.972727\pi\)
\(17\)	−7.17991e7	−	1.11722e8i	−0.174975	−	0.272267i	0.742678	−	0.669649i	\(-0.233555\pi\)
							−0.917653	+	0.397382i	\(0.869919\pi\)
\(19\)	2.81067e8	−	4.37349e8i	0.314438	−	0.489275i	−0.647679	−	0.761914i	\(-0.724260\pi\)
							0.962116	+	0.272639i	\(0.0878964\pi\)
\(23\)	1.02531e9	+	3.24678e9i	0.301135	+	0.953582i
							\(29\)	−1.19181e10	+	7.65931e9i	−0.690910	+	0.444021i	−0.838409	−	0.545041i	\(-0.816514\pi\)
0.147499	+	0.989062i	\(0.452878\pi\)
\(31\)	5.00236e9	+	5.77303e9i	0.181821	+	0.209832i	0.839342	−	0.543604i	\(-0.182941\pi\)
							−0.657521	+	0.753436i	\(0.728395\pi\)
\(37\)	−7.32672e10	+	1.05342e10i	−0.771787	+	0.110966i	−0.516948	−	0.856017i	\(-0.672932\pi\)
							−0.254840	+	0.966983i	\(0.582023\pi\)
\(41\)	2.73075e10	−	1.89928e11i	0.140215	−	0.975219i	−0.791277	−	0.611458i	\(-0.790583\pi\)
							0.931492	−	0.363761i	\(-0.118508\pi\)
\(43\)	−1.32414e10	−	1.14738e10i	−0.0487142	−	0.0422111i	0.630163	−	0.776463i	\(-0.282988\pi\)
							−0.678878	+	0.734251i	\(0.737533\pi\)
\(47\)	−8.05656e11			−1.59025			−0.795124	−	0.606447i	\(-0.792594\pi\)
							−0.795124	+	0.606447i	\(0.792594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.15	✓	270
23.14	odd	22	inner	23.15.d.a.14.15	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.15	✓	270	1.1	even	1	trivial
23.15.d.a.14.15	yes	270	23.14	odd	22	inner