Newform orbit 69.2.g.a

• Label: 69.2.g.a
• Level: $69$
• Weight: $2$
• Character orbit: 69.g
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−0.694600	+	2.36559i	0.236423	+	1.71584i	−3.43104	−	2.20500i	0.0722440	−	0.502468i	−4.22319	−	0.632543i	3.47479	−	1.58688i	3.87278	−	3.35579i	−2.88821	+	0.811326i	1.13845	+	0.519914i
5.2	−0.417491	+	1.42184i	1.61954	−	0.614080i	−0.164830	−	0.105930i	0.394240	−	2.74200i	0.196983	+	2.55910i	−3.84519	+	1.75604i	−2.02041	+	1.75070i	2.24581	−	1.98905i	3.73410	+	1.70530i
5.3	−0.298617	+	1.01700i	−1.66399	+	0.480761i	0.737395	+	0.473895i	−0.228939	+	1.59231i	0.00796426	−	1.83584i	−1.28446	+	0.586593i	−2.30424	+	1.99663i	2.53774	−	1.59996i	−1.55101	−	0.708322i
5.4	0.298617	−	1.01700i	−0.239057	+	1.71547i	0.737395	+	0.473895i	0.228939	−	1.59231i	1.67325	+	0.755391i	−1.28446	+	0.586593i	2.30424	−	1.99663i	−2.88570	−	0.820191i	−1.55101	−	0.708322i
5.5	0.417491	−	1.42184i	0.377345	−	1.69045i	−0.164830	−	0.105930i	−0.394240	+	2.74200i	−2.24601	−	1.24227i	−3.84519	+	1.75604i	2.02041	−	1.75070i	−2.71522	−	1.27576i	3.73410	+	1.70530i
5.6	0.694600	−	2.36559i	−1.73202	+	0.0101732i	−3.43104	−	2.20500i	−0.0722440	+	0.502468i	−1.17900	+	4.10432i	3.47479	−	1.58688i	−3.87278	+	3.35579i	2.99979	−	0.0352405i	1.13845	+	0.519914i
11.1	−1.13923	−	1.77267i	0.311887	−	1.70374i	−1.01370	+	2.21969i	1.10513	−	0.324494i	−3.37548	+	1.38807i	−1.21996	+	1.05710i	0.918158	−	0.132011i	−2.80545	−	1.06275i	−1.83421	−	1.58935i
11.2	−0.848533	−	1.32034i	0.567934	+	1.63629i	−0.192468	+	0.421446i	1.50619	−	0.442257i	1.67856	−	2.13832i	1.83527	−	1.59027i	−2.38727	+	0.343238i	−2.35490	+	1.85861i	−1.86198	−	1.61342i
11.3	−0.0493131	−	0.0767326i	1.73042	−	0.0750729i	0.827374	−	1.81170i	−2.86859	+	0.842294i	−0.0910930	−	0.129078i	−1.75762	+	1.52299i	−0.360384	+	0.0518154i	2.98873	−	0.259816i	0.206090	+	0.178578i
11.4	0.0493131	+	0.0767326i	−1.63918	−	0.559548i	0.827374	−	1.81170i	2.86859	−	0.842294i	−0.0378973	−	0.153371i	−1.75762	+	1.52299i	0.360384	−	0.0518154i	2.37381	+	1.83440i	0.206090	+	0.178578i
11.5	0.848533	+	1.32034i	−1.00593	+	1.41000i	−0.192468	+	0.421446i	−1.50619	+	0.442257i	−2.71525	−	0.131731i	1.83527	−	1.59027i	2.38727	−	0.343238i	−0.976228	−	2.83672i	−1.86198	−	1.61342i
11.6	1.13923	+	1.77267i	0.180745	−	1.72259i	−1.01370	+	2.21969i	−1.10513	+	0.324494i	3.25951	−	1.64203i	−1.21996	+	1.05710i	−0.918158	+	0.132011i	−2.93466	−	0.622700i	−1.83421	−	1.58935i
14.1	−0.694600	−	2.36559i	0.236423	−	1.71584i	−3.43104	+	2.20500i	0.0722440	+	0.502468i	−4.22319	+	0.632543i	3.47479	+	1.58688i	3.87278	+	3.35579i	−2.88821	−	0.811326i	1.13845	−	0.519914i
14.2	−0.417491	−	1.42184i	1.61954	+	0.614080i	−0.164830	+	0.105930i	0.394240	+	2.74200i	0.196983	−	2.55910i	−3.84519	−	1.75604i	−2.02041	−	1.75070i	2.24581	+	1.98905i	3.73410	−	1.70530i
14.3	−0.298617	−	1.01700i	−1.66399	−	0.480761i	0.737395	−	0.473895i	−0.228939	−	1.59231i	0.00796426	+	1.83584i	−1.28446	−	0.586593i	−2.30424	−	1.99663i	2.53774	+	1.59996i	−1.55101	+	0.708322i
14.4	0.298617	+	1.01700i	−0.239057	−	1.71547i	0.737395	−	0.473895i	0.228939	+	1.59231i	1.67325	−	0.755391i	−1.28446	−	0.586593i	2.30424	+	1.99663i	−2.88570	+	0.820191i	−1.55101	+	0.708322i
14.5	0.417491	+	1.42184i	0.377345	+	1.69045i	−0.164830	+	0.105930i	−0.394240	−	2.74200i	−2.24601	+	1.24227i	−3.84519	−	1.75604i	2.02041	+	1.75070i	−2.71522	+	1.27576i	3.73410	−	1.70530i
14.6	0.694600	+	2.36559i	−1.73202	−	0.0101732i	−3.43104	+	2.20500i	−0.0722440	−	0.502468i	−1.17900	−	4.10432i	3.47479	+	1.58688i	−3.87278	−	3.35579i	2.99979	+	0.0352405i	1.13845	−	0.519914i
17.1	−1.87897	−	0.858095i	−1.18942	−	1.25908i	1.48446	+	1.71316i	−1.27941	+	0.822226i	1.15448	+	3.38640i	−3.90579	−	0.561567i	−0.155288	−	0.528861i	−0.170547	+	2.99515i	3.10951	−	0.447081i
17.2	−1.69911	−	0.775958i	−0.953890	+	1.44572i	0.975144	+	1.12538i	3.49515	−	2.24620i	2.74258	−	1.71625i	2.15133	+	0.309315i	0.268869	+	0.915684i	−1.18019	−	2.75811i	−7.68160	+	1.10445i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
23.d	odd	22	1	inner
69.g	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	69.2.g.a	✓	60
3.b	odd	2	1	inner	69.2.g.a	✓	60
23.d	odd	22	1	inner	69.2.g.a	✓	60
69.g	even	22	1	inner	69.2.g.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
69.2.g.a	✓	60	1.a	even	1	1	trivial
69.2.g.a	✓	60	3.b	odd	2	1	inner
69.2.g.a	✓	60	23.d	odd	22	1	inner
69.2.g.a	✓	60	69.g	even	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(69, [\chi])\).