Newform orbit 123.2.o.a

• Label: 123.2.o.a
• Level: $123$
• Weight: $2$
• Character orbit: 123.o
• Analytic conductor: $0.982$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	123.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.982159944862\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(12\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−0.417164	−	2.63387i	−0.443101	+	1.67441i	−4.86113	+	1.57948i	−3.09789	−	1.57845i	4.59503	+	0.468565i	−1.40062	+	0.336260i	3.76671	+	7.39258i	−2.60732	−	1.48387i	−2.86511	+	8.81791i
11.2	−0.359723	−	2.27120i	0.374897	−	1.69099i	−3.12684	+	1.01597i	0.136866	+	0.0697367i	−3.97544	−	0.243178i	1.38834	−	0.333310i	1.34435	+	2.63844i	−2.71890	−	1.26790i	0.109152	−	0.335936i
11.3	−0.301761	−	1.90525i	1.62073	+	0.610933i	−1.63679	+	0.531826i	1.77427	+	0.904038i	0.674904	−	3.27224i	−1.07322	+	0.257656i	−0.244312	−	0.479489i	2.25352	+	1.98031i	1.18701	−	3.65323i
11.4	−0.204191	−	1.28921i	−1.35293	+	1.08147i	0.281736	−	0.0915416i	1.07673	+	0.548623i	1.67050	+	1.52339i	3.44592	−	0.827292i	−1.36072	−	2.67056i	0.660841	−	2.92631i	0.487432	−	1.50016i
11.5	−0.161468	−	1.01947i	−1.38568	−	1.03918i	0.888868	−	0.288811i	−2.26856	−	1.15589i	−0.835670	+	1.58045i	−1.08656	+	0.260860i	−1.37515	−	2.69889i	0.840207	+	2.87994i	−0.812093	+	2.49937i
11.6	−0.0177628	−	0.112150i	0.265930	+	1.71151i	1.88985	−	0.614050i	0.831663	+	0.423754i	0.187222	−	0.0602252i	−3.16486	+	0.759817i	−0.205534	−	0.403382i	−2.85856	+	0.910285i	0.0327512	−	0.100798i
11.7	0.0177628	+	0.112150i	1.39826	−	1.02218i	1.88985	−	0.614050i	−0.831663	−	0.423754i	0.139475	+	0.138658i	−3.16486	+	0.759817i	0.205534	+	0.403382i	0.910285	−	2.85856i	0.0327512	−	0.100798i
11.8	0.161468	+	1.01947i	−1.71463	−	0.245010i	0.888868	−	0.288811i	2.26856	+	1.15589i	−0.0270780	−	1.78758i	−1.08656	+	0.260860i	1.37515	+	2.69889i	2.87994	+	0.840207i	−0.812093	+	2.49937i
11.9	0.204191	+	1.28921i	−0.191951	−	1.72138i	0.281736	−	0.0915416i	−1.07673	−	0.548623i	2.18003	−	0.598957i	3.44592	−	0.827292i	1.36072	+	2.67056i	−2.92631	+	0.660841i	0.487432	−	1.50016i
11.10	0.301761	+	1.90525i	1.57802	+	0.714033i	−1.63679	+	0.531826i	−1.77427	−	0.904038i	−0.884223	+	3.22199i	−1.07322	+	0.257656i	0.244312	+	0.479489i	1.98031	+	2.25352i	1.18701	−	3.65323i
11.11	0.359723	+	2.27120i	−0.930619	+	1.46080i	−3.12684	+	1.01597i	−0.136866	−	0.0697367i	−3.65254	−	1.58814i	1.38834	−	0.333310i	−1.34435	−	2.63844i	−1.26790	−	2.71890i	0.109152	−	0.335936i
11.12	0.417164	+	2.63387i	0.870670	−	1.49731i	−4.86113	+	1.57948i	3.09789	+	1.57845i	4.30693	+	1.66861i	−1.40062	+	0.336260i	−3.76671	−	7.39258i	−1.48387	−	2.60732i	−2.86511	+	8.81791i
17.1	−2.58379	+	0.409231i	1.72931	−	0.0974314i	4.60636	−	1.49670i	0.849438	−	1.66712i	−4.42829	+	0.959429i	−3.09560	−	1.89699i	−6.62762	+	3.37694i	2.98101	−	0.336978i	−1.51253	+	4.65509i
17.2	−1.78320	+	0.282431i	0.939299	−	1.45524i	1.19792	−	0.389226i	−1.65937	+	3.25669i	−1.26395	+	2.86026i	2.93212	+	1.79680i	1.19110	−	0.606897i	−1.23543	−	2.73381i	2.03919	−	6.27598i
17.3	−1.51342	+	0.239702i	0.304533	+	1.70507i	0.330876	−	0.107508i	−0.908306	+	1.78265i	−0.869597	−	2.50749i	−2.71397	−	1.66312i	2.25557	−	1.14927i	−2.81452	+	1.03850i	0.947344	−	2.91562i
17.4	−1.43806	+	0.227766i	−1.68236	+	0.411905i	0.114026	−	0.0370494i	0.734786	−	1.44210i	2.32552	−	0.975528i	−0.197333	−	0.120926i	2.43905	−	1.24276i	2.66067	−	1.38594i	−0.728205	+	2.24118i
17.5	−0.708816	+	0.112265i	1.68556	+	0.398604i	−1.41230	+	0.458883i	0.808261	−	1.58630i	−1.23950	−	0.0933069i	2.77131	+	1.69826i	2.22841	−	1.13543i	2.68223	+	1.34374i	−0.394821	+	1.21514i
17.6	−0.108588	+	0.0171986i	−1.29435	−	1.15093i	−1.89062	+	0.614299i	−1.34206	+	2.63394i	0.160346	+	0.102716i	−1.15051	−	0.705034i	0.390650	−	0.199046i	0.350708	+	2.97943i	0.100431	−	0.309095i
17.7	0.108588	−	0.0171986i	0.101415	−	1.72908i	−1.89062	+	0.614299i	1.34206	−	2.63394i	−0.0187254	−	0.189501i	−1.15051	−	0.705034i	−0.390650	+	0.199046i	−2.97943	−	0.350708i	0.100431	−	0.309095i
17.8	0.708816	−	0.112265i	−0.910016	+	1.47373i	−1.41230	+	0.458883i	−0.808261	+	1.58630i	−0.479585	+	1.14676i	2.77131	+	1.69826i	−2.22841	+	1.13543i	−1.34374	−	2.68223i	−0.394821	+	1.21514i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
41.h	odd	40	1	inner
123.o	even	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	123.2.o.a	✓	192
3.b	odd	2	1	inner	123.2.o.a	✓	192
41.h	odd	40	1	inner	123.2.o.a	✓	192
123.o	even	40	1	inner	123.2.o.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
123.2.o.a	✓	192	1.a	even	1	1	trivial
123.2.o.a	✓	192	3.b	odd	2	1	inner
123.2.o.a	✓	192	41.h	odd	40	1	inner
123.2.o.a	✓	192	123.o	even	40	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(123, [\chi])\).