Embedded newform 23.13.d.a.5.12

• Label: 23.13.d.a.5.12
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.12
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.705443 - 0.207137i) q^{2} +(890.238 - 1027.39i) q^{3} +(-3445.32 - 2214.17i) q^{4} +(-536.567 - 77.1467i) q^{5} +(-840.822 + 540.363i) q^{6} +(114910. - 52477.8i) q^{7} +(3943.94 + 4551.55i) q^{8} +(-187373. - 1.30321e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.705443	−	0.207137i	−0.0110225	−	0.00323651i	0.276217	−	0.961095i	\(-0.410919\pi\)
							−0.287239	+	0.957859i	\(0.592737\pi\)
\(3\)	890.238	−	1027.39i	1.22118	−	1.40931i	0.337428	−	0.941352i	\(-0.390443\pi\)
							0.883749	−	0.467961i	\(-0.155011\pi\)
\(5\)	−536.567	−	77.1467i	−0.0343403	−	0.00493739i	0.125123	−	0.992141i	\(-0.460067\pi\)
							−0.159464	+	0.987204i	\(0.550976\pi\)
\(7\)	114910.	−	52477.8i	0.976721	−	0.446054i	0.137891	−	0.990447i	\(-0.455968\pi\)
							0.838830	+	0.544394i	\(0.183240\pi\)
\(11\)	−144045.	−	490573.i	−0.0813098	−	0.276916i	0.908796	−	0.417242i	\(-0.137003\pi\)
							−0.990105	+	0.140326i	\(0.955185\pi\)
\(13\)	2.08316e6	−	4.56149e6i	0.431582	−	0.945032i	−0.561486	−	0.827486i	\(-0.689770\pi\)
							0.993068	−	0.117545i	\(-0.0375026\pi\)
\(17\)	6.13517e6	+	9.54651e6i	0.254175	+	0.395504i	0.944768	−	0.327739i	\(-0.106287\pi\)
							−0.690593	+	0.723244i	\(0.742650\pi\)
\(19\)	−3.70586e7	+	5.76643e7i	−0.787712	+	1.22570i	0.182442	+	0.983217i	\(0.441600\pi\)
							−0.970155	+	0.242488i	\(0.922037\pi\)
\(23\)	−1.45528e8	+	2.71318e7i	−0.983061	+	0.183278i
							\(29\)	4.43883e8	−	2.85266e8i	0.746243	−	0.479582i	−0.111433	−	0.993772i	\(-0.535544\pi\)
0.857676	+	0.514190i	\(0.171908\pi\)
\(31\)	5.91527e8	+	6.82658e8i	0.666506	+	0.769189i	0.983826	−	0.179130i	\(-0.0573281\pi\)
							−0.317319	+	0.948319i	\(0.602783\pi\)
\(37\)	1.81249e9	−	2.60597e8i	0.706425	−	0.101569i	0.220267	−	0.975440i	\(-0.429307\pi\)
							0.486158	+	0.873871i	\(0.338398\pi\)
\(41\)	3.29661e8	−	2.29284e9i	0.0694008	−	0.482693i	−0.925247	−	0.379366i	\(-0.876142\pi\)
							0.994647	−	0.103327i	\(-0.0329488\pi\)
\(43\)	−4.64147e9	−	4.02186e9i	−0.734252	−	0.636233i	0.205277	−	0.978704i	\(-0.434191\pi\)
							−0.939528	+	0.342471i	\(0.888736\pi\)
\(47\)	3.22630e9			0.299308			0.149654	−	0.988738i	\(-0.452184\pi\)
							0.149654	+	0.988738i	\(0.452184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.12	✓	230
23.14	odd	22	inner	23.13.d.a.14.12	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.12	✓	230	1.1	even	1	trivial
23.13.d.a.14.12	yes	230	23.14	odd	22	inner